E DE SANTIAGO DE COMPOSTELA FACULTADE DE F´ISICA DEPARTAMENTO DE F´ISICA DA MATERIA CONDENSADA GRUPO DE F´ISICA NON LINEAL
Coupled oscillators: chaotic synchronization, high-dimensional chaos and wavefronts in bistable media
Memoria presentada por Diego Paz´ o Bueno para optar ´ o grao de Doutor en Ciencias F´ısicas pola Universidade de Santiago de Compostela. Xaneiro, 2003
Dissertation of the Faculty of Physics, University of Santiago de Compostela, Spain Diego Paz´ o Bueno (
[email protected]) Coupled oscillators: chaotic synchronization, high-dimensional chaos and wavefronts in bistable media Santiago de Compostela, 2003. This document has been created with PDFTEX, Version 3.14.
Vicente P´erez Mu˜ nuzuri, profesor titular da Universidade de Santiago de Compostela,
CERTIFICA
que a presente memoria, titulada “Coupled oscillators: chaotic synchronization, high-dimensional chaos and wavefronts in bistable media”, foi realizada por Diego Paz´ o Bueno baixo a s´ ua direcci´on, e que concl´ ue a Tese que presenta para optar ´o grao de Doutor en Ciencias F´ısicas.
E, para que as´ı conste, asina a presente en Santiago de Compostela a 9 de Xaneiro de 2003.
V◦ . e prace Vicente P´erez Mu˜ nuzuri
Asdo.: Diego Paz´o Bueno
A mis padres.
A
gradecimientos cknowledgments
Haber acabado esta tesis me hace sentir que estos u ´ltimos a˜ nos han servido para algo. El camino no ha estado exento de sinsabores ni de momentos en los que me he tenido que armar de paciencia. Sin embargo, es justo acordarse ahora de todo aqu´ello que me han llenado de satisfacci´on tanto a nivel humano como cient´ıfico. Primeramente, he conocido a muy diversas personas de lugares aqu´ı y all´a, y de todas he aprendido algo. Haber tenido la oportunidad de visitar otros sitios y otras culturas me ha enriquecido enormemente. Por otro lado, en no pocas veces, aquello nuevo que he apendido de f´ısica o matem´aticas me ha maravillado. ¡Una ecuaci´on puede ser bella! Los siguientes p´arrafos me dan la oportunidad de expresar mi gratitud a todos aquellos a los que se la debo. Al profesor Vicente P´erez Villar, el ‘jefe’ del Grupo de F´ısica no Lineal (GFNL), le doy las gracias por haberme dado la oportunidad de trabajar en su Grupo y por prestarme su ayuda cuando la he precisado. A mi director Vicente P´erez Mu˜ nuzuri le estoy agradecido por guiarme y aconsejarme en mis primeros pasos en el mundo de la ciencia, y por emplear su tiempo en supervisar esta tesis. Tambi´en agradezco a Alberto P´erez Mu˜ nuzuri y a Moncho G´omez Gesteira, la ambilidad y la simpat´ıa con la que siempre me han tratado. No puedo pasar por alto a la ‘masa obrera’ del Grupo con quienes he compartido, en muchos casos, nuestra condici´on becaria (o precaria) adem´as de muchos buenos momentos. Mi primera menci´on debe ir hacia aqu´ellas que han compartido conmigo durante m´as o menos tiempo su pertenencia al ‘grup´ usculo ca´otico’: Nieves, In´es y Noelia. Les doy las gracias por ayudarme tantas y tantas veces. Tambi´en quiero destacar a los compa˜ neros con quienes he convivido en el despacho: Adolfo, Irene y David. Fue una gran suerte tener un espacio de trabajo tan agradable y tan lleno de ganas de echar un mano cuando la precis´e. Por supuesto, tambi´en estoy en deuda con el resto de gente con la que he coincidido en el Grupo: Bea, Carlos, Chus, Edu, Iv´an, Jose Manuel, Juan, Pablo, Pedro, Manuel, Maite, Nico y Roi. Todos ellos me ayudaron en m´as de una ocasi´on y no recuerdo ni una mala cara.
Tambi´en merecen una rese˜ na aqu´ellos que visitaron el GFNL durante estos a˜ nos pasados. A todos ellos les agradezco su simpat´ıa. En especial a Roberto Deza le agradezco sus sugerencias en la elaboraci´on de mis art´ıculos y de esta tesis. Por lo que respecta a la gente ajena al Grupo merece una menci´on especial Manuel Mat´ıas con quien llevo colaborando desde hace ya varios an˜os. Le estoy agradecido por haber aprendido tantas cosas y por su hospitalidad en mis estancias en Salamanca primero, y en Palma de Mallorca despu´es. En este punto, tambi´en quiero hacer part´ıcipes de mi gratitud al resto de miembros del imedea de Palma. Ich m¨ochte auch J¨ urgen Kurths and Misha Zaks f¨ ur ihre Hilfe und Mitarbeitung danken. Ich fand meine deutsche Erfahrung in der Potsdam Universit¨at sehr ergiebig und interessant. Mi estancia en Potsdam me obliga a acordarme de m´as gente a la que estoy agradecida, principalmente de Ernest Montbri´o gracias al cual habituarme a un nuevo pa´ıs me result´o mucho m´as f´acil. Tambi´en me gustar´ıa aprovechar estas l´ıneas para agradecer la amistad que me une a tanta gente fenomenal en muchos sitios (Le´on, Madrid, Pontevedra, ...). Mi mayor gratitud es para mis padres, por preopuparse por m´ı y quererme tanto. Para rematar, desexo agradecer ´ as instituci´ ons oficiais o apoio ´ econ´ omico sen o cal ser´ıa imposible para min ter feita esta tese. Estas son a Secretar´ıa Xeral de Investigaci´ on e Desenvolvemento da Xunta de Galicia e a Universidade de Santiago de Compostela a trav´es do seu Vicerrectorado de Investigaci´ on.
Santiago de Compostela, Enero 2003
Diego
Contents Resumen
xix
Summary
xxvii
1 El Caos: Fundamentos 1.1 Introducci´on . . . . . . . . . . . . . . . . . 1.2 Bifurcaciones locales . . . . . . . . . . . . 1.3 Rutas al caos . . . . . . . . . . . . . . . . 1.3.1 Cascada de duplicaci´on de periodo 1.3.2 Intermitencia . . . . . . . . . . . . 1.3.3 Cuasiperiodicidad . . . . . . . . . 1.3.4 Crisis . . . . . . . . . . . . . . . . 1.3.5 Bifurcaciones globales . . . . . . . 1.3.5.a Sistema de Lorenz . . . . 1.3.5.b Caos de Shil’nikov . . . . ´ 1.4 Orbitas peri´odicas inestables (UPOs) . . . 1.4.1 M´etodo de Newton-Raphson . . . 1.5 Sincronizaci´on ca´otica . . . . . . . . . . .
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2 Role of Unstable Periodic Orbits in Phase and Lag Synchronization between Coupled Chaotic Oscillators 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Periodic orbits in the absence of coupling . . . . . . . . . . 2.3 Attractors of a coupled system: role of unstable tori in synchronization transitions . . . . . . . . . . . . . . . . . . 2.4 Phase synchronization . . . . . . . . . . . . . . . . . . . . . 2.5 Lag synchronization . . . . . . . . . . . . . . . . . . . . . . 2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 1 5 8 9 11 12 15 16 17 20 20 21 23
27 27 29 32 36 42 47
xii
CONTENTS
3 Transition to High-Dimensional Chaos through a Global Bifurcation 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 System and overall picture . . . . . . . . . . . . . . . . . . . 3.2.1 Lyapunov exponents and attractors . . . . . . . . . 3.2.2 Behaviors along the line σ = 20 . . . . . . . . . . . . 3.3 The centered periodic rotating wave: analytical solution . . 3.4 Transition to quasiperiodic behavior . . . . . . . . . . . . . 3.5 Numerical evidences of the route to chaos exhibited by the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Coexistence between 3D-torus and CRW . . . . . . . 3.5.2 Heteroclinic explosion . . . . . . . . . . . . . . . . . 3.5.3 Four-dimensional branched manifold . . . . . . . . . 3.5.4 Boundary crisis and power law of chaotic transients 3.6 Description in terms of a return map . . . . . . . . . . . . . 3.7 Route to chaos: theoretical analysis . . . . . . . . . . . . . 3.8 Further remarks . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Onset of Traveling Fronts in an Array of Coupled Symmetric Bistable Units 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The system . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Standing, oscillating and traveling fronts . . . . . . . . . . . 4.4 Gluing of cycles . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Cylindrical coordinates . . . . . . . . . . . . . . . . 4.4.2 Velocity of the front as a function of D − Dth . . . . 4.4.3 Quantitative analysis . . . . . . . . . . . . . . . . . . 4.5 Exotic front dynamics . . . . . . . . . . . . . . . . . . . . . 4.5.1 r = 20 (δ > 1) . . . . . . . . . . . . . . . . . . . . . 4.5.2 r = 23 (δ < 1) . . . . . . . . . . . . . . . . . . . . . 4.6 The effect of parameter mismatch and asymmetry . . . . . 4.7 Universality? . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Other couplings and systems . . . . . . . . . . . . . 4.7.2 Large D . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.3 The discrete FitzHugh-Nagumo model . . . . . . . . 4.7.3.a Transition to traveling front . . . . . . . . 4.7.3.b The continuum limit . . . . . . . . . . . . . 4.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51 51 54 55 58 61 65 68 69 72 72 76 77 79 80 83
85 85 87 88 90 91 93 94 97 100 101 104 105 105 106 107 107 109 112
xiii
CONTENTS
5 Spatio-Temporal Patterns in an Array of Coupled Lorenz Oscillators 5.1 Model . . . . . . . . . . . . . . . . . . . . 5.2 Bistable region . . . . . . . . . . . . . . . 5.2.1 Traveling fronts . . . . . . . . . . . 5.2.2 Short wavelength bifurcation . . . 5.3 Chaotic region . . . . . . . . . . . . . . . 5.4 Discussion . . . . . . . . . . . . . . . . . . 6 Conclusions and Outlook
Non-Diagonally . . . . . .
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115 115 116 116 119 121 126 127
Appendix A: The FitzHugh-Nagumo cell: a case of a TakensBogdanov codimension-two point . . . . . . . . . . . . . . . . . 131 REFERENCES
151
LIST OF PUBLICATIONS
154
INDEX
155
xiv
CONTENTS
List of Figures 1.1
Saddle-node and pitchfork bifurcation diagrams . . . . . . . . .
6
1.2
Hopf bifurcation diagram . . . . . . . . . . . . . . . . . . . . .
7
1.3
Period-doubling cascade in the R¨ossler oscillator . . . . . . . .
11
1.4
Curry-Yorke route to chaos . . . . . . . . . . . . . . . . . . . .
14
1.5
Sketch of a saddle-loop bifurcation . . . . . . . . . . . . . . . .
16
1.6
Route to chaos in the Lorenz model . . . . . . . . . . . . . . .
19
1.7
Homoclinic orbit to a saddle-focus . . . . . . . . . . . . . . . .
21
2.1
Poincar´e section and return map of a R¨ossler oscillator . . . . .
30
2.2
Frequencies of UPOs embedded into the R¨ossler attractor . . .
32
2.3
Configurations favorable for the locking on the torus . . . . . .
34
2.4
Statistics of phase slips
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37
2.5
Bifurcation diagram showing UPOs of length 1 and 2 . . . . . .
39
2.6
Bifurcation diagram showing UPOs of length 3 . . . . . . . . .
40
2.7
Snapshot of two coupled R¨ossler oscillators at the beginning of the phase jump . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
2.8
Phase difference between subsystems . . . . . . . . . . . . . . .
43
2.9
Return map for different values of ε . . . . . . . . . . . . . . .
44
2.10 Periodic orbits at ε = 0.15 . . . . . . . . . . . . . . . . . . . . .
45
2.11 Poincar´e section showing the “ghost” of the length-5 window .
46
2.12 Role of the out-of-phase UPO of length 2 . . . . . . . . . . . .
48
3.1
Regions of the (R, σ) plane with different states . . . . . . . . .
57
3.2
Bifurcation diagram. Points represent intersections with the Poinacar´e section Im(X1 ) = 0 . . . . . . . . . . . . . . . . . . .
59
Time series corresponding to different behaviors
60
3.3
. . . . . . . .
xvi 3.4
LIST OF FIGURES
The four largest Lyapunov exponents of a ring of three unidirectionally coupled Lorenz systems . . . . . . . . . . . . .
61
Diagram representing schematically the transitions from synchronous chaos to a PRW . . . . . . . . . . . . . . . . . . . . .
62
Numerical and theoretical results for the frequency ω and the max. and the min. values of the amplitude of the coordinate X1 as function of R. . . . . . . . . . . . . . . . . . . . . . . . .
65
3.7
Poincar´e section of two- and three-torus attractors . . . . . . .
67
3.8
Blowout of the four largest Lyapunov exponents for the T3 . .
70
3.9
Schematic of a ‘slave locking’ . . . . . . . . . . . . . . . . . . .
71
3.10 Numerical experiment showing trajectories starting in an initial condition in the symmetric unstable PRW . . . . . . . . . . . .
73
3.11 Determination of the correlation dimension . . . . . . . . . . .
75
3.12 Average chaotic transient as a function of the distance to the critical point Rbc . . . . . . . . . . . . . . . . . . . . . . . . . .
76
3.13 Numerical experiment for the boundary crisis . . . . . . . . . .
77
3.14 Return map of the maxima of the variable Z0 satisfying to be larger than their adjacent maxima . . . . . . . . . . . . . . . .
78
3.15 Three-dimensional representation of the proposed heteroclinic route to create the high-dimensional chaotic attractor . . . . .
79
3.16 Two-dimensional representation of the proposed heteroclinic route to create the high-dimensional chaotic attractor . . . . .
81
4.1
3D visualization of a front . . . . . . . . . . . . . . . . . . . . .
89
4.2
Regions with different front dynamics in the plane r − D
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89
4.3
Motions of different units for standing, oscillating and traveling regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
4.4
Solutions continuation of the uncoupled limit . . . . . . . . . .
91
4.5
Front dynamics in the reduced cylindrical phase space ξ − η . .
93
4.6
Sketch of a gluing bifurcation . . . . . . . . . . . . . . . . . . .
94
4.7
Velocity and oscillation period near the critical point . . . . . .
95
4.8
Eigenvalues of the B-state for different values of D and r. . . .
96
4.9
Spiraling approach of a oscillating state to the B-state . . . . .
97
4.10 Sketch of a gluing bifurcation mediated by a saddle-focus . . .
98
4.11 Saddle index and imaginary part for different values of r . . . .
98
3.5 3.6
xvii
LIST OF FIGURES
4.12 Velocity and oscillation period for r = 20 . . . . . . . . . . . . 101 4.13 Chaotic motion of a front for r = 20 . . . . . . . . . . . . . . . 102 4.14 Period-doubling cascade leading to chaos
. . . . . . . . . . . . 102
4.15 Velocity and oscillation periods for r = 23 . . . . . . . . . . . . 103 4.16 Subsidiary oscillations of the front . . . . . . . . . . . . . . . . 104 4.17 Velocity vs. D in the FHN model . . . . . . . . . . . . . . . . . 108 4.18 Diagram showing regions with different front dynamics in the FHN model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.1
Phase space, for the off-diagonal coupling, of the patterns in the plane r − D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.2
Projection onto the x − y plane of the trajectory followed by the oscillators of the array . . . . . . . . . . . . . . . . . . . . . . . 118
5.3
Logarithmic law of the front velocity for the off-diagonal coupling 119
5.4
Velocity of the front for r = 8 . . . . . . . . . . . . . . . . . . . 120
5.5
Largest eigenvalues at the uniform states in C± as a function of the wave number k . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.6
Different behaviors for several couplings (D) for r = 28
5.7
Reference framework where phase can be readily computed . . 123
5.8
x variable, amplitude and two-color-discretized phase for D = 8 and r = 25, 26, 27, 28 . . . . . . . . . . . . . . . . . . . . . . . . 125
. . . . 122
A.1 Regions of the parameter space of the FHN cell . . . . . . . . . 132
Resumen Contexto del trabajo La cooperaci´on entre matem´aticos, f´ısicos, ingenieros, bi´ologos y cient´ıficos de otras ´areas del conocimiento ha dado lugar a lo que actualmente se conoce como inter-disciplinearidad. En este campo, la ciencia no lineal –o simplemente la no-linealidad– se ha convertido en el punto de encuentro que proporciona un lenguaje com´ un, facilitando el entendimiento entre las diferentes disciplinas. A la luz de las mismas herramientas matem´aticas, se explican las relaciones inesperadas entre fen´omenos con una f´ısica (subyacente) distinta. Junto con los antiguos m´etodos estad´ısticos, la teor´ıa de bifurcaciones ha resultado ser la piedra angular que unifica en el mismo marco matem´atico los resultados de diversidad de experimentos. Probablemente el fen´omeno de la din´amica no lineal m´as fascinante, y el m´as popular, es el caos. La imposibilidad de predicciones a largo plazo es la consecuencia del caos. Esto fue primeramente reconocido por Poincar´e, que qued´o asombrado con su complejidad geom´etrica. En la primera mitad del siglo XX, el caos en los sistemas conservativos fue el principal tema de inter´es te´orico (Birkhoff, Kolmogorov, Arnol’d, Moser, ...). Al mismo tiempo, hubo un creciente inter´es en los experimentos con osciladores no lineales disipativos (Van der Pol, Appleton...). El art´ıculo de Lorenz de 1963 revel´o la naturaleza omnipresente del caos y la utilidad de los ordenadores, ya que los experimentos num´ericos pasaron a ser f´acilmente asequibles. De esta forma, desde los a˜ nos setenta la din´amica ca´otica ha sido objeto de significativo inter´es para una amplia comunidad de cient´ıficos. Las ideas de Cantor sobre conjuntos fractales fueron recuperadas por Mandelbrot, la conexi´on entre caos y turbulencia fue apuntada por Ruelle y Takens, y Feigenbaum introdujo los conceptos de universalidad y renormalizaci´ on en el contexto de los sistemas din´amicos.
xx
Resumen
Hoy en d´ıa, las transiciones que originan el caos de baja dimensi´on a partir del movimiento regular (orden) est´an bien caracterizadas; y la corriente principal de investigaci´on se ha desplazado hacia otros temas como el caos de alta dimensi´on, la sincronizaci´on ca´otica, la formaci´on de estructuras, y las aplicaciones del caos a la biolog´ıa y a otras ciencias.
Resumen de la memoria El cap´ıtulo 1 pretende dar una visi´on personal del mundo del caos. Es el u ´nico que no est´a escrito en ingl´es (el idioma hegem´onico de la ciencia y de la f´ısica en particular) en tanto en cuanto est´a pensado como un manual para aquellos hispanohablantes que llegan al mundo del caos por primera vez. El cap´ıtulo consiste en una introducci´on a los conceptos fundamentales del caos as´ı como de las ´areas relacionadas con ´el del an´alisis matem´atico. Se presta una especial atenci´on a las ideas m´as simples de la teor´ıa de bifurcaciones. Esto proporciona las herramientas b´asicas para entender los diferentes escenarios que llevan del movimiento regular (est´atico, peri´odico o cuasiperi´odico) al caos. Tambi´en se incluye una breve descripci´on de los principales tipos de sincronizaci´on ca´otica. En el cap´ıtulo 2 se trata el problema de las transiciones a las sincronizaciones de fase y de retardo [PZK03] entre dos osciladores ca´oticos no id´enticos acoplados. Como modelo de oscilador ca´otico con fase coherente se toma el oscilador de R¨ossler. Para explicar las transiciones observadas se recurre a la ayuda de las ‘´orbitas peri´odicas inestables’ (UPOs) que se conoce se encuentran inmersas en todo oscilador ca´otico de baja dimensi´on. Se muestra que el comienzo de la sincronizaci´on de fase corresponde a la aparici´on de infinitas UPOs (con enganches 1:1) en la superficie de los toros invariantes que existen en el l´ımite de ausencia de acoplamiento entre ambos osciladores. Debido al surgimiento no simult´aneo de estas UPOs el sistema exhibe –para un rango del acoplamiento– sincronizaci´on de fase intermitente. En este estado, largos intervalos de tiempo donde los osciladores tienen sus fases sincronizadas son interrumpidos por saltos de 2π en sus fases relativas. Estos saltos ocurren cada vez con menos frecuencia, a la vez que el par´ametro de acoplamiento se acerca al par´ametro cr´ıtico por encima del cual se establece una sincronizaci´on de fase perfecta. Los saltos obedecen una ley de escala que se ha denominado ‘intermitencia de rendija’. De acuerdo con esto, parece que los saltos en la fase ocurren cuando la trayectoria pasa cerca de aquellos toros que permanecen desenganchados (o est´an enganchados con un n´ umero de rotaci´on diferente de uno). Por
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otro lado, una descripci´on de la transici´on a la sincronizaci´on de retardo por medio de las UPOs, deber´ıa clarificar la aparente reducci´on en la complejidad del atractor. Se observa que la mayor´ıa de las UPOs que hicieron posible la aparici´on de la sincronizaci´on de fase no est´an presentes cuando se alcanza la sincronizaci´on de retardo. M´as exactamente, las UPOs con una estructura no adecuada para la sincronizaci´on de retardo desaparecen en diversas bifurcaciones. Aquellas UPOs fuera-de-fase que sobreviven para valores altos del par´ametro de acoplamiento dan lugar a un fen´omeno conocido como ‘sincronizaci´on de fase intermitente’. As´ı, el sistema exhibe, por momentos, configuraciones con un retardo ex´otico mientras que permanece sincronizado con un peque˜ no retardo la mayor parte del tiempo. Finalmente, para un acoplamiento suficientemente grande todas las UPOs est´an asociadas a (aproximadamente) el mismo retardo, y la complejidad del atractor es la misma que la de un u ´nico oscilador. El cap´ıtulo 3 se dedica a una novedosa transici´on al caos de alta dimensi´on [PSM01, PM, SPM] en un sistema compuesto de tres osciladores de Lorenz. Los osciladores est´an acoplados unidireccionalmente, y de esta manera forman una geometr´ıa circular con simetr´ıa c´ıclica. En un plano definido por dos par´ametros se identifican las regiones con diferentes comportamientos. Nos concentramos en una l´ınea que pasa a trav´es de las diversas regiones con la meta de describir las transiciones entre ellas. Para los valores m´as peque˜ nos del par´ametro de control R tenemos caos sincronizado. En este estado los tres osciladores siguen la misma trayectoria dentro del atractor de Lorenz. Por otro lado, para valores altos de R existe un movimiento peri´odico conocido como ‘onda peri´odica rotante’ (PRW) donde cada oscilador sigue la misma ´orbita peri´odica pero con la particularidad de que existe una diferencia de fase de 2π/3 entre ellos. Se entiende mejor el sistema cuando las ecuaciones diferenciales ordinarias que describen la din´amica del anillo se expresan en t´erminos de dos modos de Fourier discretos k = 0, 1. As´ı, existe una soluci´on anal´ıtica muy aproximada de la PRW, que consiste en un estado est´atico para el modo uniforme k = 0, m´as una soluci´on sinusoidal para el modo espacial k = 1. Por lo que concierne al caos sincronizado, este estado corresponde a un movimiento ca´otico del modo k = 0, y un modo k = 1 nulo. La p´erdida del caos sincronizado, lleva a un atractor ca´otico de alta dimensi´on conocido como onda ca´otica rotante (CRW) donde el modo k = 1 pasa a ser distinto de cero y oscilante. Entonces la din´amica es ca´otica al tiempo que se observa, superpuesta al modo k = 0, una din´amica oscilatoria con desplazamientos en la fase de 2π/3 entre las unidades adyacentes.
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Cuando R disminuye, la PRW sufre una bifurcaci´on horquilla que da como resultado dos PRWs estables relacionadas por simetr´ıa. Con una mayor disminuci´on de R las PRWs se inestabilizan, dando cuasiperiodicidad a dos frecuencias, que tambi´en acaba por inestabilizarse a consecuencia de una bifurcaci´on de Hopf secundaria. Como resultado, aparecen dos atractores cuasiperi´odicos a tres frecuencias. Finalmente esto dos 3-toros parecen fundirse, y como resultado surge la CRWs. Sin embargo, un examen detallado nos permite obtener una descripci´on mucho m´as detallada de la formaci´on del atractor tipo CRW. Con este prop´osito, usamos diferentes t´ecnicas para arrojar alguna luz sobre este problema: exponentes de Lyapunov, secciones de Poincar´e, estad´ısticas de los transitorios ca´oticos, una medida de la dimensi´on de correlaci´on, un mapa de retorno, ... La primera conclusi´on es que el conjunto ca´otico de alta dimensi´on –que se manifiesta como un transitorio ca´otico– se crea para un valor de R donde los atractores del sistema son un par de toros. Esto ocurre a trav´es de una doble conexi´on heterocl´ınica de las PRWs asim´etricas con la sim´etrica. As´ı, de forma an´aloga al sistema de Lorenz, llamamos a este mecanismo ‘explosi´on heterocl´ınica’. En esta explosi´on se crea un n´ umero infinito de toros inestables tridimensionales. El conjunto ca´otico se vuelve atractor en una crisis de borde (de cuenca de atracci´on) que involucra los dos 3toros inestables m´as simples. Para un valor ligeramente inferior de R, los dos 3-toros estables mencionados m´as arriba desaparecen al fusionarse con dos inestables en sendas bifurcaciones silla-nodo gemelas. Por tanto, existe un peque˜ no intervalo de R donde el atractor ca´otico de alta dimensi´on coexiste con la cuasiperiodicidad a tres frecuencias. La importancia de esta ruta al caos de alta dimensi´on reside en la ausencia de un atractor ca´otico de baja dimensi´on intermedio. Una secci´on del cap´ıtulo se dedica a analizar la robustez de la descripci´on ofrecida aqu´ı, y hasta qu´e punto puede considerarse como una aproximaci´on de la ruta real (que es probablemente imposible de describir debido a su complejidad inherente). En el cap´ıtulo 4 [PP01, PP] se trata el problema de la aparici´on de un frente viajero en un sistema reacci´on-difusi´on discreto con biestabilidad sim´etrica. Nos restringimos a sistemas cuya din´amica local consiste de dos puntos fijos estables relacionados por simetr´ıa, y un punto de equilibrio inestable situado entre medias. Para sistemas continuos una bifurcaci´on por rotura de paridad del frente puede inestabilizar el frente est´atico creando dos frentes viajeros que se propagan en sentidos opuestos. La situaci´on para un sistema discreto es m´as complicada. Concretamente, para un conjunto de unidades de Lorenz biestables acoplados en una dimensi´on se encuentra que
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variando un par´ametro (por ejemplo el acoplamiento) la transici´on ocurre siguiendo la siguiente ruta: frente est´ atico → oscilante → viajero. En el primer paso el frente est´atico se inestabiliza por una bifurcaci´on de Hopf supercr´ıtica, dando lugar a un frente oscilante. El periodo de oscilaci´on crece hasta que finalmente diverge a infinito al llegar a un punto cr´ıtico. M´as all´a de este punto aparecen dos soluciones contra-propagantes, con unas velocidades que crecen desde cero de una forma muy abrupta. La dependencia funcional, del periodo de oscilaci´on y de la velocidad del frente, en el acoplamiento se explica en el contexto de una bifurcaci´on collage entre ciclos. Para ver esto se construye un nuevo espacio de fases cil´ındrico definiendo dos variables reducidas. La transici´on de frente oscilante a viajero presenta en este nuevo espacio de fases una gran analog´ıa con con la transici´on de libraci´on a rotaci´on en el cl´asico problema del p´endulo. As´ı en el punto de transici´on existe una doble conexi´on homocl´ınica de la dislocaci´on (est´atica) inestable consigo misma. El acercamiento de una ´orbita peri´odica a un punto de silla viene caracterizado por una divergencia logar´ıtmica de su periodo. Mostramos que el periodo de oscilaci´on, y la inversa de la velocidad del frente exhiben esta dependencia funcional. No obstante, la situaci´on puede llegar a ser mucho m´as complicada si la soluci´on inestable que media el proceso de “pegado” es un punto silla-foco en vez de una silla. En este caso, dependiendo del valor de una cantidad conocida como ´ındice de silla, uno puede observar din´amicas del frente ex´oticas. Hace a˜ nos, Shil’nikov prob´o que una conexi´on homocl´ınica a un punto silla-foco puede generar una herradura de Smale y, por tanto, caos. Esto explica por qu´e se observan diversos reg´ımenes oscilantes y viajeros nuevos, incluyendo un movimiento err´atico (que refleja una din´amica ca´otica subyacente). Los resultados obtenidos para el conjunto de osciladores de Lorenz biestables se reproducen para diferentes tipos de acoplamiento y tambi´en para una l´ınea de modelos de dinamo biestables. Hemos estudiado tambi´en la versi´on discreta y sim´etrica del bien conocido modelo de FitzHugh-Nagumo (FHN). Para este modelo la transici´on es algo diferente. En vez de un r´egimen oscilante, hay una zona del espacio de par´ametros donde los frentes est´atico y viajero coexisten. La soluci´on viajera aparece en una bifurcaci´on silla-nodo de ciclos. Para un acoplamiento mayor, las dos soluciones viajeras inestables relacionadas por simetr´ıa (creadas en bifurcaciones silla-nodo gemelas y que se mueven en direcciones contrarias) se fusionan y –como resultado– se crea una soluci´on oscilante inestable. Finalmente esta soluci´on inestable se encoge hasta que colapsa a un punto. Este punto es la soluci´on frente est´atico estable que
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pasa a ser inestable a partir de esta colisi´on. Este u ´ltimo paso no es m´as que una bifurcaci´ on de Hopf subcr´ıtica, a partir de la cual la soluci´on viajera es la u ´nica estable. La dependencia de la velocidad para esta ruta presenta rasgos especiales. Primeramente, debido a su aparici´on en una bifurcaci´on silla-nodo, la soluci´on viajera nace con una velocidad distinta de cero. La dependencia funcional de la velocidad es –de acuerdo con esto– una ra´ız cuadrada. No obstante, es subrayable que m´as all´a de un cierto punto puede reconocerse una dependencia logar´ıtmica (que es consecuencia de la proximidad al punto de silla). Tanto para el modelo consistente en sistemas de Lorenz acoplados como para el modelo FHN, hemos investigado el l´ımite de muy alta difusi´on, teniendo en cuenta el hecho de que en el l´ımite de difusi´on infinita se recupera, para estos sistemas, el continuo. Las dislocaciones estable e inestable se funden para formar la soluci´on continua. Por tanto, desde un punto de vista formal existe una bifurcaci´on en ese l´ımite. A causa de las condiciones de simetr´ıa, ´esta puede considerarse un bifurcaci´on horquilla. Las l´ıneas de bifurcaci´on observadas a difusi´on finita para ambos modelos (ristra de osciladores de Lorenz y FHN) convergen (para acoplamiento infinito) a un punto, que en el caso del modelo de FitzHugh-Nagumo es el punto que fue previamente encontrado al estudiar la versi´on continua. Desde el punto de vista de los modelos discretos esta singularidad situada en el infinito es un punto de codimensi´on dos donde dos autovalores de las dos soluciones est´aticas se anulan. Esta bifurcaci´on es conocida como TakensBogdanov (TB) y ha sido estudiada por diversos autores. Sin embargo, el caso con el que nos hemos topado aqu´ı es tan particular que no ha sido tratado previamente. As´ı, hemos visto la forma normal contenida en el libro de Guckenheimer y Holmes que describe una TB con la simetr´ıa que nuestro problema, pero con el inconveniente de que no considera un espacio de fases cil´ındrico. Dependiendo de un par´ametro interno aparecen dos posibles conjuntos de bifurcaciones cerca del punto de codimensi´on dos. Cada uno de ellos muestra fuertes analog´ıas con las bifurcaciones observadas en nuestros experimentos num´ericos. Por tanto, conjeturamos que cualquier rotura de paridad de un frente que d´e lugar a frentes viajeros en un sistema continuo biestable sim´etrico se transforma –al discretizarlo– en una de las dos rutas que se describen en esta tesis. A este respecto, es importante resaltar que la implementaci´on num´erica de un sistema continuo siempre lleva consigo alg´ un tipo de discretizaci´on, y por tanto esperamos que los fen´omenos estudiados aqu´ı se manifiesten en un escala fina. El cap´ıtulo 5 [PMP01] estudia un sistema 1D de osciladores de Lorenz
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acoplados que muestra frentes viajeros y que est´a caracterizado por una matriz de acoplamiento no diagonal. En la regi´on en la que el oscilador de Lorenz es biestable, la transici´on a la propagaci´on, demuestra ser equivalente a la mostrada en el cap´ıtulo 4 para osciladores acoplados diagonalmente. De acuerdo con esto, se obtienen perfiles logar´ıtmicos para la velocidad del frente y para el periodo de oscilaci´on. Hemos ampliado nuestro ´ambito de estudio a la regi´on de par´ametros donde los puntos fijos del oscilador de Lorenz se vuelven inestables, de tal forma que ´este pasa a ser ca´otico. En esta regi´on (ca´otica), el sistema exhibe dos tipos de caos espaciotemporal dependiendo de que exista o no propagaci´on de frentes. En la regi´on con propagaci´on encontramos dos procesos caracter´ısticos: creaci´on espont´anea de frentes contra-propagantes, e inversi´on del movimiento de los frentes. La l´ınea de bifurcaci´on que separaba regiones con y sin propagaci´on en el caso biestable marca ahora la frontera entre dos tipos de caos espaciotemporal. Adem´as, hemos encontrado tambi´en que por encima de un cierto acoplamiento, tanto en la regi´on biestable como en la ca´otica, el sistema sufre una bifurcaci´ on de onda corta. Este tipo de bifurcaci´on, y la aparici´on de la propagaci´on de frentes por la ruta explicada arriba, se observan solamente en sistemas discretos. Tambi´en hemos comprobado que el patr´on de longitud de onda corta (que emerge de la bifurcaci´on hom´onima) inhibe el caos espacio-temporal, dando lugar a un patr´on ordenado. Finalmente, las conclusiones principales de esta tesis, as´ı como algunas perspectivas, se re´ unen en el cap´ıtulo 6.
Summary Context of the work The cooperation among mathematicians, physicists, engineers, biologists and scientists from other areas of knowledge has given rise to what is nowadays known as cross-disciplinarity. In its realm, nonlinear science –or simply nonlinearity– has become a meeting point that provides a common language thus facilitating the understanding among disciplines. Under the light of common mathematical tools, unexpected relations between phenomena with different underlying physics are explained. Together with the old statistical methods, bifurcation theory has become the keystone that unifies into a common mathematical framework the results of a variety of experiments. Probably the most exciting, and the most popular, phenomenon of nonlinear dynamics is chaos. The impossibility of long-term predictions is the consequence of chaos. It was first noticed by Poincar´e, who became astonished with its geometric complexity. In the first half of the 20th century, chaos in conservative systems was the main object of theoretical interest (Birkhoff, Kolmogorov, Arnol’d, Moser, ...). At the same time, there was a growing interest in experiments with dissipative nonlinear oscillators (Van der Pol, Appleton...). The paper by Lorenz in 1963 revealed the pervasive nature of chaos and the utility of computers, as numerical experiments were becoming pretty accessible. In this way, since the 1970s chaotic dynamics has been the subject of significant interest by a wide community of scientists. The ideas of Cantor on fractal sets were recovered by Mandelbrot, the connection between chaos and turbulence was pointed out by Ruelle and Takens, and Feigenbaum introduced the concepts of universality and renormalization in the context of dynamical systems. Nowadays, the transitions that originate low-dimensional chaos from
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regular motion (order) are well characterized; and the mainstream has shifted to other subjects as high-dimensional chaos, chaotic synchronization, pattern formation, and applications of chaos to biology and other sciences.
Summary of the thesis Chapter 1 intends to give a personal view of the world of chaos. I have chosen to write it in Spanish to best serve the purpose of being a primer for (Spanish-speaker) newcomers to the world of chaos. It consists in an introduction to the most fundamental concepts of chaos and related areas of mathematical analysis. Special emphasis is given to the simplest ideas of bifurcation theory. This provides the basic tools to understand the different scenarios leading from regular motion (steady state, periodicity or quasiperiodicity) to chaos. Also, a short description of the main types of chaotic synchronization is included. In Chapter 2, the problem of the transitions to phase and lag synchronization in coupled non-identical chaotic oscillators is addressed [PZK03]. The R¨ossler oscillator is taken as a model of a phase-coherent chaotic oscillator. The explanation of the observed transitions is attempted, with the help of the ‘unstable periodic orbits’ (UPOs) that are known to be embedded in every low-dimensional chaotic attractor. It is shown that the onset of phase synchronization corresponds to the appearance of an infinity of UPOs (with 1:1 locking ratios) on the surface of the invariant tori existing in the limit of no coupling between both oscillators. Due to the non-simultaneous emergence of these UPOs the system exhibits –in some range of the coupling parameter– intermittent phase synchronization. In this state, long time intervals where the oscillators are phase synchronized are interrupted by jumps of 2π in their relative phases. These jumps occur less and less frequently as the coupling parameter approaches the critical value above which perfect phase synchronization settles down. The jumps obey a scaling law what has been denominated ‘eyelet intermittency’. Accordingly, it seems that phase jumps occur when the trajectory passes near those tori that remain unlocked (or are locked with a rotation number different of one). On the other hand, a description of the transition to lag synchronization by means of the UPOs, should clarify the apparent reduction in the complexity of the attractor. It is shown that most of the UPOs that made possible the onset of phase synchronization are not present when lag synchronization is reached. More precisely, the UPOs
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with a badly suited structure for lag synchronization disappear in several bifurcations. Those out-of-phase UPOs that survive for larger values of the coupling strength give rise to a phenomenon known as ‘intermittent lag synchronization’, where the system exhibits exotic lag configurations at some moments, whereas remains lag synchronized (with small lag) most of the time. Finally, for a large enough value of the coupling all the UPOs are associated to (very approximately) the same lag, and the complexity of the attractor is the same as that of a single oscillator. Chapter 3 is devoted to a novel transition to high-dimensional chaos [PSM01, PM, SPM] in a system composed of three coupled Lorenz oscillators. The oscillators are unidirectionally coupled, and hence they form a ring geometry with a cyclic symmetry. On a plane spanned by two parameters, regions with different behaviors are identified. We focus on a (monoparametric) line crossing along the different regions with the goal of describing the transitions among them. For the smallest values of the control parameter R we have synchronous chaos. In this state, the three oscillators follow the same trajectory into the Lorenz attractor. On the other hand, for large values of R there exists a periodic motion known as ‘periodic rotating wave’ (PRW) where each oscillator follows the same periodic orbit but with the particularity that a 2π/3 phase shift exists between them. The system is better understood when the ordinary differential equations describing the dynamics of the ring are cast in terms of two discrete Fourier modes k = 0, 1. Thus, there exists a very approximate analytical solution to the PRW, that consists in a steady state of the uniform k = 0 mode, plus a sinusoidal solution for the spatial k = 1 mode. Concerning synchronous chaos, this state corresponds to a chaotic motion of the k = 0 mode, and a vanishing k = 1 mode. Loss of synchronous chaos leads to a high-dimensional chaotic attractor known as ‘chaotic rotating wave’ (CRW) where the k = 1 mode becomes different from zero and oscillating. Then the dynamics is chaotic, at the same time that a superimposed oscillating dynamics with 2π/3 phase shifts between adjacent units is observed. As R is lowered, the PRW undergoes a pitchfork bifurcation that yields two stable symmetry-related PRWs. With a further decrease of R the PRWs become unstable, giving rise to two-frequency quasiperiodicity, which also becomes unstable through a secondary Hopf bifurcation. As a result, two symmetry-related three-frequency quasiperiodic attractors appear. Finally, these two three-tori seem to merge, and as a consequence the CRW emerges. However a detailed examination allows us a much more
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accurate description of the formation of the CRW attractor. With this aim, we used different techniques to shed some light into this problem: Lyapunov exponents, Poincar´e sections, statistics of the chaotic transients, a measure of the correlation dimension, a return map, ... The first conclusion is that the high-dimensional chaotic set –that manifests itself as a chaotic transient– is created at a value of R where the attractors of the system are a pair of two-tori. This occurs through a double heteroclinic connection of the asymmetric PRWs with the symmetric one. Thus, analogously to the Lorenz system, we call this mechanism ‘heteroclinic explosion’. In this explosion, an infinite number of unstable three-dimensional tori is created. The chaotic set becomes attracting at a boundary crisis that involves the two simplest unstable three-tori. At a slightly smaller value of R the stable three-tori mentioned above coalesce with the unstable ones in twin saddle-node bifurcations. Therefore, there exists a small interval of R where the high-dimensional chaotic attractor coexists with three-frequency quasiperiodicity. The importance of this route to high-dimensional chaos lies in the absence of an intermediate low-dimensional chaotic attractor. A section of the chapter is devoted to analyze the robustness of the description explained here, and to what extent can it be considered just an approximation of the real route (that is probably impossible to be described due to its inherent complexity). In Chapter 4 [PP01, PP] the problem of the onset of a traveling front in a discrete reaction-diffusion system with symmetric bistability is addressed. We restrict to systems whose local dynamics consists of two stable symmetry-related fixed points, and one unstable equilibrium located in between. For continuous systems, a parity-front bifurcation may render unstable the static front, creating two counterpropagating traveling fronts. The situation for a discrete system is more complicated. Concretely, for an array of bistable Lorenz units it is found that varying a parameter (the coupling strength, for example) the transition occurs following this route: static → oscillating → traveling front. In the first step the static front becomes unstable through a supercritical Hopf bifurcation, giving rise to an oscillating front. The oscillation period grows until it diverges to infinity at a critical point. Beyond this point two counter propagating front solutions appear, with their velocities growing from zero in a very abrupt way. The functional dependence of the oscillation period and the velocity of the front on the coupling is explained in the context of a gluing bifurcation of cycles. To see this, a new cylindrical phase space is constructed defining two reduced variables. The transition from oscillating to traveling front
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presents in this new phase space a strong analogy with the transition from libration to rotation in the classical pendulum problem. Thus, in the transition point there exists a double homoclinic connection of the unstable (static) dislocation of the front with itself. The approach of a periodic orbit to a saddle point is characterized by a logarithmic divergence of its period. We show that the oscillation period, and the inverse of the velocity of the front exhibit this functional dependence. Nonetheless, the situation may become much more convoluted if the unstable solution mediating the gluing process is not a saddle, but a saddle-focus instead. In this case, depending on a quantity known as saddle index, one may observe ‘exotic’ front dynamics of the front. It was proved years ago by Shil’nikov that a homoclinic connection to a saddle-focus point may generate a Smale horseshoe and, therefore, chaos. This explains why several new oscillating traveling regimes of the front are observed, including an erratic motion (reflecting an underlying chaotic dynamics). The results obtained for the array of bistable Lorenz oscillators are reproduced for different coupling types and also for an array of bistable dynamo models. We have also studied the discrete and symmetric version of the wellknown FitzHugh-Nagumo (FHN) model. For this model, the transition is somewhat different. Instead of an oscillating regime there is a zone in the parameter space where stable static and traveling fronts coexist. The traveling solution appears in a saddle-node bifurcation of cycles. For a larger coupling the two unstable symmetry-related traveling solutions (created in twin saddle-node bifurcations and moving in opposite directions) become glued and –as a result– an unstable oscillating solution is created. Finally, this unstable solution shrinks until it collapses to a point. This point is the stable static front solution which becomes unstable from this coalescence. This last step is nothing but a subcritical Hopf bifurcation, above which the traveling solution is the only stable one. The features of the velocity dependence for this route are special. First of all, because of its emergence in a saddle-node bifurcation, the traveling solution is born with a non-zero velocity. The functional dependence of the velocity is –accordingly– a square root. Nonetheless, it is remarkable that beyond a crossover, a logarithmic dependence (that is a consequence of the proximity to a saddle point) can be recognized. For both the model consisting in coupled Lorenz systems and the FHN model, we have investigated the limit of very large diffusion, taking into account the fact that in the limit of infinite diffusion, the continuum limit is recovered for these systems. The stable and unstable dislocations
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coalesce to form the continuous solution. Therefore, from a formal point of view there exists a bifurcation at that limit. Because of the symmetry conditions, this can be considered a pitchfork bifurcation. The bifurcation lines observed at finite diffusion for both models (Lorenz array and FHN) converge (for infinite coupling) at a point, that in the case of the FitzHughNagumo model is the point that was previously reported when studying the continuous version. From the point of view of the discrete models this singularity located at infinity is a codimension-two point where two eigenvalues of the static solutions become zero. This bifurcation is known as Takens-Bogdanov (TB) and has been studied by several authors. However, the case we have faced here is so particular that has not been addressed before. Thus, we have looked at the normal form provided in the book of Guckenheimer and Holmes that describes a TB with the same symmetry that our problem, but with the drawback that it does not considers a cylindrical phase space. Depending on an internal parameter two possible bifurcation sets close to the codimension-two point appear. Each of them shows strong analogies with the bifurcations observed in our numerical experiments. Therefore, we speculate that any parity front bifurcation leading to traveling fronts in a symmetric bistable continuous system is transformed –when discretized– in one of the two routes reported in this thesis. In this respect, it is important to emphasize that the numerical implementation of a continuous system always involves some kind of discretization, and therefore we expect the phenomena studied here to manifest at a fine scale. Chapter 5 [PMP01] studies an array of Lorenz oscillators, coupled through a non-diagonal matrix, where wavefront solutions arise. In the region where the Lorenz oscillator is bistable, the transition to propagation is demonstrated to be equivalent to that shown in Chapter 4 for on-diagonal coupled oscillators. Accordingly, logarithmic profiles for the velocity of the front and the oscillation period are obtained. We have broadened our scope to the parameter range where the fixed points of the Lorenz oscillator become unstable, such that it is chaotic. In this (chaotic) region, the system exhibits two types of spatio-temporal chaos depending on whether there exists or not front propagation. In the propagating region we find two characteristic processes: spontaneous creation of counter-propagating fronts, and front reversal. The bifurcation line that separated propagating from non-propagating regions in the bistable case marks now the boundary between two kinds of spatio-temporal chaos. In addition, we have also found that above a certain coupling, in the bistable as well as in the chaotic
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regions, the system undergoes a short wavelength bifurcation. This kind of bifurcation, and the onset of propagating fronts by the route explained above, are observed in discrete systems only. We have also noticed that the short wavelength pattern (emerging from the homonymous bifurcation) inhibits spatio-temporal chaos, giving rise to an ordered pattern. Finally, the main conclusions of this PhD Thesis, as well as an outlook, are gathered together in Chapter 6.
Cap´ıtulo 1
El Caos: Fundamentos 1.1
Introducci´ on
A finales del siglo XIX, el gran matem´atico Jules Henri Poincar´e par´ ticip´o en un concurso organizado por el rey Oscar II de Suecia. Dicho concurso consist´ıa en demostrar la estabilidad o inestabilidad del sistema solar. Ninguno de los participantes fue capaz de resolver el problema, lo cual no es sorprendente puesto que ni hoy mismo se conoce la respuesta con total seguridad, pero fue Poincar´e quien gan´o el concurso. La victoria fue merecida puesto que, aunque se limit´o a estudiar un problema que consist´ıa u ´nicamente en tres cuerpos que se atra´ıan mutuamente por la gravedad, Poincar´e demostr´o que un caso tan aparentemente sencillo no era integrable y que, adem´as, el comportamiento pod´ıa resultar tan complicado que resultase, en la pr´actica, impredecible [Poi93]. Esta impredecibilidad de la que hablaba Poincar´e, en un sistema determinista, ha recibido diversos nombres como, por ejemplo, ruido determinista, pero se conoce actualmente como ‘caos’, y la primera aparici´on en la literatura con este nombre se remonta a 1975 [LY75]. Aunque la palabra ‘caos’ evoca entre el com´ un de la gente conceptos como desorden o cataclismo, como sucede con muchos otros conceptos usados en f´ısica, existe una definici´on rigurosa del t´ermino que no coincide con la usada en el lenguaje corriente. El caos se define como un movimiento determinista que presenta sensibilidad exponencial a las perturbaciones. Es decir, dos condiciones iniciales ligeramente diferentes se alejar´an, en promedio, la una de la otra exponencialmente, de forma que tras no mucho tiempo ser´an claramente distinguibles (y adem´as estar´an descorrelacionadas) las trayectorias seguidas para cada condici´on inicial. Por tanto, el caos
2
El Caos: Fundamentos
introduce un l´ımite en la predictibilidad de un sistema f´ısico aunque se conozcan la leyes que gobiernan el sistema, ya que cualquier m´ınimo error en la determinaci´on de la condici´on inicial se amplifica sin freno. Esta influencia crucial de (casi) cualquier m´ınima perturbaci´on tiene importantes connotaciones en lo referente al poder de predicci´on que se le puede atribuir a una teor´ıa f´ısica, ya que aunque una teor´ıa o modelo describa perfectamente un determinado proceso, la existencia de caos limita la capacidad de predicci´on en funci´on del error en la determinaci´on de la condici´on inicial. Este concepto ha llegado al gran p´ ublico como ‘efecto mariposa’ ya que se supone que la atm´osfera tambi´en es un sistema que presenta caos y, por tanto, una ´ınfima perturbaci´on, como el simple aleteo de una mariposa puede provocar (o evitar) que haya, d´ıas o semanas m´as tarde, un tif´on en la otra esquina del mundo. Diversos libros de divulgaci´on se han ocupado del caos, entre los que cabe destacar los de Ruelle [Rue93] y Lorenz [Lor95]. Tambi´en resulta interesante, en una primera lectura, un art´ıculo publicado en Investigaci´ on y Ciencia en el a˜ no 1987 [CFPS87]. Cabe preguntarse si el caos es un fen´omeno frecuente o, si al contrario, se observa s´olo en condiciones muy especiales. Pues bien, la din´amica ca´otica ha demostrado ser un comportamiento muy frecuente en la naturaleza, y, prueba de ello es que se ha encontrado en multitud de sistemas f´ısicos (ver por ejemplo [Cvi84, Hao90, BGK+ 02]) y qu´ımicos [EKDO83]; e incluso, en la din´amica de poblaciones en ecolog´ıa [ERG98]. L´ogicamente, los modelos matem´aticos que describen estos procesos muestran caos al realizar una simulaci´on num´erica, independientemente de que se trate de sistemas din´amicos continuos (EDOs) o discretos (mapas). Por supuesto, tambi´en se encuentra caos cuando se tratan sistemas extendidos descritos por ecuaciones en derivadas parciales, pudi´endose encontrar, adem´as, caos espacio-temporal cuando existe dependencia sensible no s´olo en el tiempo sino tambi´en en el espacio. El caos se muestra al observador como un movimiento irregular, que podr´ıa inducir a sospechar la existencia de una componente estoc´astica (ruido). Pero, como hemos dicho, el caos es un fen´omeno puramente determinista, a pesar de que a primera vista haya una cierta aleatoriedad aparente. El que la transformada de Fourier de una se˜ nal ca´otica, compuesta por una variable del sistema frente al tiempo, sea de ancho espectro, a diferencia del movimiento peri´odico o cuasiperi´odico, no tiene por qu´e implicar la existencia de una componente estoc´astica. Es evidente que existe una diferencia fundamental entre el sistema planetario estudiado por Poincar´e y la atm´osfera, que no reside simplemente
3
1.1 Introducci´ on
en la complejidad de uno y otro problema. Mientras que Poincar´e estudi´o un sistema conservativo, la atm´osfera es un sistema disipativo que recibe energ´ıa constantemente del Sol y que la disipa irradi´andola hacia el exterior o por su fricci´on viscosa. En los sistemas conservativos, el volumen en el espacio de fases (definido, por ejemplo, por posiciones y velocidades) se conserva en el tiempo, como nos dice el teorema de Liouville. Por otro lado, en los sistemas disipativos un volumen en el espacio de fases, que evolucione con el flujo, se contrae a lo largo del tiempo y tiende a cero. Por esto, en los sistemas disipativos se puede hablar de atractores ya que las trayectorias tienden a una regi´on del espacio de fases de volumen cero, que puede ser un punto de equilibrio, un ciclo, la superficie de un toroide, o un objeto m´as complicado como un ‘fractal’, en el caso de que el atractor sea ca´otico. No merece la pena extenderse m´as, en la naturaleza (multi)fractal de los atractores ca´oticos (denominados por esto extra˜ nos, aunque ambos conceptos no son totalmente intercambiables), puesto que el tema rebasa el espacio propio de una introducci´on. Una referencia concreta sobre los fractales es el libro de Takayasu [Tak90], no obstante, la mayor´ıa de los libros sobre caos tratan este tema con cierta profundidad. La medida de cu´an ca´otico es un determinado atractor, viene dada por los exponentes de Lyapunov. Estos son un promedio adecuado de las tasas de divergencia y convergencia, en diferentes direcciones, de ´orbitas cercanas en el espacio de fases. En particular, si para un sistema k-dimensional de ecuaciones diferenciales de primer orden x˙ ≡
dx = F(x) dt
(1.1)
consideramos una ´orbita desplazada infinitesimalmente x(t) + δx(t) y el vector y(t) = δx(t)/|δx(0)|, para el cual tenemos la ecuaci´on que nos da la evoluci´on del desplazamiento infinitesimal y˙ = DF(x(t)) · y
(1.2)
donde DF(x(t)) denota las matriz Jacobiana k × k de derivadas parciales de F(x(t)). Los exponentes de Lyapunov estar´an dados entonces por 1 ln |y(t)| t→∞ t
λ(x(0), y(0)) = l´ım
(1.3)
La dependencia en y(0) es formal, puesto que el resultado es el mismo para (casi) cualquier perturbaci´on δx(t). Asimismo, el teorema erg´odico multiplicativo de Osedelec asegura que el resultado ser´a el mismo para
4
El Caos: Fundamentos
todo x(0) en la cuenca de atracci´on del atractor, excepto para un posible conjunto con medida de Lebesgue cero [ER85]. Un atractor ca´otico se caracteriza por tener un exponente de Lyapunov positivo, o m´as de uno, en cuyo caso se habla de atractor hiperca´otico. Los exponentes positivos dan cuenta de la dependencia sensible propia del caos. Adem´as si el sistema es aut´onomo tendr´a un exponente nulo correspondiente a las perturbaciones en la direcci´on del flujo. Por otro lado, habr´a uno o m´as exponentes negativos, necesarios para que el sistema sea disipativo (y el flujo contraiga el volumen). Por tanto, se necesitan k > 2 variables para poder encontrar caos. L´ogicamente, existe una definici´on an´aloga para los exponentes de Lyapunov en mapas. En estos sistemas, una sola variable es suficiente para generar caos, siempre y cuando el mapa sea no invertible. Otro aspecto importante es que el hecho de que la divergencia de las trayectorias en un atractor ca´otico sea justamente exponencial, y no, por ejemplo, simplemente polin´omica, tiene importantes consecuencias ya que, en el primer caso, el error no puede ser acotado durante un tiempo arbitrariamente grande, por mucha capacidad computacional que se utilice [For87]. ¿Quiere esto decir que cualquier simulaci´on num´erica de una trayectoria ca´otica carece entonces de sentido, debido a los errores de redondeo que introduce el ordenador al calcular cada paso de integraci´on? Afortunadamente, la mayor´ıa de los sistemas presentan lo que se conoce como ‘shadowing’. Esto implica que, aunque nuestra soluci´on num´erica no se corresponda con una determinada condici´on inicial, s´ı que, t´ıpicamente, se aproxima a (o sombrea), durante un largo periodo de tiempo, una soluci´on verdadera cuya condici´on inicial est´a cercana a la condici´on inicial considerada primeramente (ver por ejemplo [HGY88, GHSY90, SY91]). De esta forma, aunque nuestras soluciones num´ericas carezcan de utilidad predictiva, s´ı que representan soluciones reales del sistema, y por tanto, nos informan sobre la naturaleza del atractor ca´otico. En la secci´on 1.2 se introducen los conceptos b´asicos de la teor´ıa de bifurcaciones. A partir de ah´ı, se explican, en la secci´on 1.3, las diversas ‘rutas’ por las que un atractor no ca´otico puede transformarse en un atractor ca´otico. En la secci´on 1.4, se trata el tema de los invariantes inmersos en un atractor ca´otico, especialmente las ´orbitas peri´odicas inestables. Finalmente, la secci´on 1.5 describe los diversos tipos de sincronizaci´on ca´otica, que se pueden producir por interacci´on de dos o m´as sistemas ca´oticos.
5
1.2 Bifurcaciones locales
1.2
Bifurcaciones locales
En muchas ocasiones, no es necesario conocer la soluci´on exacta de una determinada ecuaci´on diferencial (o un mapa), sino que es suficiente con saber qu´e tipo de soluci´on presenta el sistema (punto fijo, ciclo l´ımite, etc). La rama de la matem´atica que se ocupa de las transiciones entre estados cualitativamente diferentes se conoce como teor´ıa de bifurcaciones. Una bifurcaci´on se puede producir en un sistema si se var´ıa alguno de sus par´ametros; pi´ensese, por ejemplo, en el numero de Reynolds de un fluido o en la resistencia de un resistor en un circuito el´ectrico. En esta secci´on se presentan las bifurcaciones llamadas ‘locales’ que se producen ‘gen´ericamente’ en sistemas diferenciables1 . Estas bifurcaciones pueden estudiarse considerando u ´nicamente el entorno de una soluci´on. Por otro lado, una bifurcaci´on es gen´erica cuando se encuentra t´ıpicamente, sin necesidad alguna de simetr´ıa o condici´on adicional. Tambi´en se puede decir que es estructuralmente estable porque no es destruida por una peque˜ na perturbaci´on. El apelativo de gen´ericas se puede sustituir por codimensi´onuno, que quiere decir que en un sistema n-param´etrico estas bifurciones se dan en variedades en el espacio de par´ametros de dimensi´on n − 1. Por ejemplo, si tenemos dos par´ametros, los estados cualitativamente diferentes se localizar´an en regiones del plano, u otra variedad bidimensional, que ambas definan. Por tanto, habr´a l´ıneas de bifurcaci´on que separen estados cualitativamente distintos. Normalmente, uno cruzar´a, al variar alguno de los par´ametros, una de esas l´ıneas para pasar de un estado a otro. Tambi´en existe la posibilidad, si uno tiene infinita precisi´on, de que al variar los par´ametros uno cruce dos l´ıneas a la vez, es decir, pase por un punto donde intersecten dos de ellas. En este caso, tan particular, tendremos una bifurcaci´on de codimensi´on dos (puede encontrarse un ejemplo en el Ap´endice). En sistemas continuos (EDOs, ecuaciones derivadas ordinarias) con una variable existe u ´nicamente un tipo de bifurcaci´on gen´erica que se conoce como bifurcaci´on tangente o silla-nodo (saddle-node). En esta bifurcaci´on aparecen una soluci´on estable y otra inestable. Est´a asociada a un autovalor nulo. Por ejemplo, la ecuaci´on x˙ = ax2 −
(1.4)
no tiene soluciones de equilibrio para < 0 (suponiendo a > 0), mientras 1
No repasamos, en cambio, el concepto de estabilidad asint´ otica y las condiciones de estabilidad lineal; si bien algunos apuntes aparecen en el Ap´endice.
6
El Caos: Fundamentos
x
bif. tangente
bif. horquilla supercritica
bif. horquilla subrcritica
0.4
0.4
0.4
0.2
0.2
0.2
0
0
0
−0.2
−0.2 a)
−0.4 −0.1
−0.2 b)
0 ε
0.1
−0.4 −0.1
c) 0 ε
0.1
−0.4 −0.1
0 ε
0.1
Figura 1.1: Diagramas de bifurcaci´ on de las Ecs. (1.4, 1.5). Las soluciones estables e inestables aparecen con l´ıneas continua y discontinua, respectivamente.
p p que para > 0 exiten dos soluciones x = − /a (estable) y x = /a (inestable). Aunque no es gen´erica, porque precisa la existencia de una simetr´ıa por reflexi´on, merece la pena mencionar aqu´ı la bifurcaci´on horquilla (pitchfork). En esta bifurcaci´on se pasa de tener una soluci´on a tener tres, dos de ellas relacionadas por simetr´ıa. Se tienen dos variantes, supercr´ıtica y subcr´ıtica. En la primera una soluci´on estable se inestabiliza para dar lugar a dos soluciones estables. En la variante subcr´ıtica una soluci´on estable se vuelve inestable cuando dos soluciones inestables colisionan con ella. Por ejemplo, la ecuaci´on siguiente presenta una bifurcaci´on horquilla en = 0: x˙ = x(ax2 + ).
(1.5)
que es supercr´ıtica para a < 0 y subcr´ıtica para a > 0. Los diagramas de bifurcaci´on de las Ecs. (1.4,1.5) se muestran en la Fig. 1.1. La segunda bifurcaci´on gen´erica es la bifurcaci´on de Andronov-Hopf, o en forma abreviada Hopf. Precisa de dos o m´as dimensiones y ocurre cuando un punto fijo cambia de estabilidad al cruzar dos autovalores complejos conjugados el eje imaginario. Entonces, en un entorno de la bifurcaci´on hay un ciclo l´ımite, con frecuencia angular aproximadamente igual al valor de la parte imaginaria de los autovalores. Al igual que la bifurcaci´on horquilla, tambi´en tiene variantes super y subcr´ıtica, dependiendo de que el ciclo l´ımite sea estable o inestable. Por ejemplo, presenta una bif. de Hopf en = 0 el siguiente sistema de ecuaciones: x˙ = −ωy + x + (x2 + y 2 )(ax − by) y˙ = ωx + y + (x2 + y 2 )(bx + ay).
(1.6)
Pasando a coordenadas polares queda m´as claro el significado de cada
7
1.2 Bifurcaciones locales
Figura 1.2: La bifurcaci´ on de Hopf (Ec. (1.6)). El punto de equilibrio aparece en negro o gris y la soluci´ on peri´ odica con trazo continuo o discontinuo seg´ un sean estables o inestables.
par´ametro: r˙ = r + ar3 θ˙ = ω + br2
(1.7)
En = 0 la soluci´on en el origen cambia su estabilidad. Para a < 0 (caso supercr´ıtico) un ciclo l´ımite estable aparece para > 0; en cambio, para a > 0 (caso subcr´ıtico) un ciclo l´ımite inestable colisiona con el punto fijo. Un esquema con los dos casos se puede ver en la Fig. 1.2. Las ecuaciones (1.4,1.5,1.6) se han presentado aqu´ı como casos particulares. Sin embargo, estas ecuaciones se conocen como formas normales de dichas bifurcaciones. Cualquier sistema que sufre una bifurcaci´on puede ser reducido a la forma normal correspondiente, tras sucesivos cambios de coordenadas y despreciando los t´erminos de orden superior [GH83]. En un entorno de la bifurcaci´on el sistema se comportar´a de forma muy similar a como indica su forma normal, ya que
8
El Caos: Fundamentos
los t´ermimos de orden superior son tan peque˜ nos como arbitrariamente se quiera. Las bifurcaciones que se dan en mapas son tambi´en importantes ya que, a partir de un flujo k-dimensional se puede construir un mapa con k − 1 variables. Para esto, se tiene que realizar una secci´on de Poincar´e que consiste en tomar los puntos donde el flujo intersecta, en un sentido, una (hiper)superficie (k − 1)-dimensional. De esta forma, una ´orbita peri´odica se convierte en el mapa (de Poincar´e) asociado (F) en un punto fijo (x∗ ): xn+1 = F(xn )
/
x∗ = F(x∗ )
(1.8)
La condici´on de estabilidad de un punto fijo de un mapa es que todos los autovalores de la matriz jacobiana en ese punto est´en dentro de la circunferencia de radio unidad en el plano complejo. Los mapas presentan bifurcaciones gen´ericas que son an´alogas a las de los flujos: la bifurcaci´on silla-nodo (asociada a un autovalor +1) y la bifurcaci´on de Hopf, tambi´en conocida como Neimark-Sacker (asociada a dos autovalores complejos conjugados que abandonan la mencionada circunferencia). Existe, adem´as, ´ una bifurcaci´on gen´erica que es genuina de los sistemas discretos. Esta es la bifurcaci´on de duplicaci´ on de periodo o flip. Se produce cuando un autovalor toma el valor -1. Existen las variantes supercr´ıtica y subcr´ıtica dependiendo de que la ´orbita de doble periodo (x1 , x2 , x1 , ...) = (x∗ + δx1 , x∗ + δx2 , x∗ + δx1 , ...) existente en un entorno de la bifurcaci´on sea estable o inestable. N´otese que la ´orbita de doble periodo da lugar a dos puntos fijos de la doble recurrencia del mapa: x1 = F(F(x1 ))
,
x2 = F(F(x2 ))
(1.9)
Aunque no hemos mostrado aqu´ı m´as que los m´ınimos fundamentos de la teor´ıa de bifurcaciones, esto nos servir´a como base para describir, en la pr´oxima secci´on, las formas en las que un sistema pasa de tener un comportamiento “ordenado” a tener un comportamiento ca´otico.
1.3
Rutas al caos
Como ya hemos mencionado, variando un par´ametro de control, un sistema transita entre estados cualitativamente diferentes a trav´es de bifurcaciones. Las bifurcaciones descritas en la secci´on anterior son universales, ya que se pueden dar gen´ericamente en cualquier sistema din´amico, no importa qu´e sistema f´ısico represente. Cabe preguntarse, por
1.3 Rutas al caos
9
tanto, qu´e tipo de bifurcaciones deben ocurrir en un sistema para que ’este pase a exhibir un comportamiento ca´otico. La transici´on al caos de baja dimensi´on se produce a trav´es de una serie de rutas (o escenarios) que han resultado ser universales. La descripci´on te´orica de estas rutas se ha realizado, normalmente a partir de mapas. Cabe resaltar aqu´ı que no deja de resultar sorprendente que mapas, en muchos casos con una sola variable, sean capaces de modelizar las transiciones al caos que se producen en sistemas tan sumamente complicados como un fluido. De hecho, gran parte de los primeros trabajos en este campo se concibieron con la intenci´on de describir la transici´on a la turbulencia. Era habitual usar los t´erminos caos y turbulencia de forma casi equivalente. Hoy en d´ıa, sin embargo, se suele reservar el t´ermino turbulencia para la din´amica observada a muy alto n´ umero de Reynolds que no se puede describir como un movimiento ca´otico de baja dimensi´on. La turbulencia, o tambi´en llamada turbulencia totalmente desarrollada, constituye a´ un uno de los retos m´as importantes de la f´ısica.
1.3.1
Cascada de duplicaci´ on de periodo
Muchos mapas en una variable (ver Ec. (1.8)) muestran una sucesi´on de bifurcaciones de duplicaci´on de periodo que desemboca en un estado ca´otico. Lo llamativo es que muchos sistemas f´ısicos complicados muestran una ruta coincidente con la que se observa en un mapa de gran simplicidad. Si miramos el espectro de Fourier de una determinada variable de un sistema tenemos que el movimiento peri´odico se manifiesta con un pico en la frecuencia caracter´ıstica, digamos f1 . Tras una duplicaci´on de periodo, aparece un subarm´onico, que se ve como un nuevo pico en f1 /2. La siguiente duplicaci´on nos da un nuevo pico en f1 /4 (as´ı como en 3f1 /4). Finalmente, la aparici´on de infinitos subarm´onicos nos lleva al caos caracterizado por tener un espectro continuo. Es esta visi´on en t´erminos de espectro de potencias lo que hace que esta ruta tambi´en se conozca como cascada subarm´onica. La ruta al caos por duplicaci´on de periodo est´a indiscutiblemente unida a Feigenbaum [Fei78]. El gran m´erito de Feigenbaum fue encontrar y explicar la universalidad en una clase de mapas unidimesionales en un intervalo. En concreto, todos los mapas en un intervalo que satisfagan ser unimodales. Unimodal quiere decir que tiene un solo m´aximo (que es aproximable por una cuadr´atica) y que la derivada Schwarziana es negativa
10
El Caos: Fundamentos
en el intervalo: SF (x) ≡
F 000 (x) 3 − F 0 (x) 2
F 00 (x) F 0 (x)
2 b + 1): rH =
σ(σ + b + 3) . σ−b−1
(1.16)
Para los par´ametros tomados en el art´ıculo de Lorenz σ = 10, b = 8/3, tenemos que rH ≈ 24.74. La bifurcaci´on de Hopf en rH es subcr´ıtica. Demostrarlo requiere un gran esfuerzo de c´alculo ya que tenemos tres dimensiones2 . 2
No es totalmente riguroso considerar la forma normal de la bifurcaci´ on (Ec. (1.6)) en el plano definido por los autovectores asociados a los autovalores imaginarios puros en rH . El c´ alculo de los t´erminos no lineales al orden m´ as bajo, para una bifurcaci´ on de Hopf en un espacio N -dimensional, puede encontrarse en [HW78]. Ref. [MM76] trata el caso concreto del Lorenz (ojo, tiene alg´ un error ya que la bifurcaci´ on de Hopf en el Lorenz es siempre subcr´ıtica).
19
1.3 Rutas al caos
a)
b)
c)
d)
e)
f)
g)
h)
Figura 1.6: Esquema (correspondiente a una proyecci´ on sobre el plano x − z) de la transici´ on al caos en el sistema de Lorenz, ver detalles en el texto.
Para entender como aparecen las ´orbitas peri´odicas inestables que participan en la inestabilizaci´on de C± en rH , y principalmente, para explicar la aparici´on del caos en el modelo de Lorenz hay que tener en cuenta las bifurcaciones globales. En la Fig. 1.6 se presenta un esquema correspondiente a una proyecci´on sobre el plano (x, z) de los puntos fijos, las variedades estable e inestables del origen y las ´orbitas inestables antes mencionadas. La Fig. 1.6(a) corresponde a r < 1, con un u ´nico punto fijo en el origen. Entre (a) y (b) se produce la bifurcaci´on horquilla que da lugar al nacimiento de C± . Cuando r se incrementa a´ un m´as, C± se vuelven focos (c). Cuando r alcanza un valor cr´ıtico se produce una doble conexi´on homocl´ınica (d). Este punto corresponde, para los par´ametros est´andar de σ y b, a r1 ≈ 13,926. Con la doble conexi´on homocl´ınica aparecen las dos ´orbitas inestables mostradas en (e) as´ı como un conjunto infinito3 de ´orbitas peri´odicas inestables (no mostradas) que puden clasificarse de acuerdo con la secuencia de giros en torno a C+ y a C− . Este conjunto infinito de ciclos inestables aparece gracias a que la conexi´on es doble, y su nacimiento coincide con la aparici´on del transitorio ca´otico. Por lo general, las trayectorias en el espacio de fases acaban por caer a C+ o C− tras un tiempo de comportamiento aparentemente ca´otico en el que la trayectoria visita ambos lados del sistema. A medida que r crece las ´orbitas mostradas en la Fig. 1.6(e) se hacen cada vez m´as peque˜ nas, lo que produce que 3
El nacimiento de este conjunto infinito de ciclos se conoce como ‘explosi´ on homocl´ınica’.
20
El Caos: Fundamentos
los transitorios ca´oticos sean m´as prolongados . Llegado un punto cr´ıtico r2 ≈ 24,06, se establece una conexi´on heterocl´ınica (doble) entre la variedad inestable del origen y las ´orbitas inestables, (f). Esto corresponde a una crisis [YY79, KY79b, GORY87], de tal forma que para r > r2 tenemos un atractor ca´otico (con forma de doble l´obulo, como se puede ver en la portada de esta tesis), mientras que para r < r2 s´olo existe un transitorio ca´otico. En el intervalo r2 < r < rH existe triestabilidad entre C± y el atractor extra˜ no, (g). Por encima de rH , C± son inestables y el atractor ca´otico resulta ser el u ´nico del sistema (h). Finalmente, cabe destacar que hasta muy recientemente no se ha podido demostrar, de forma matem´aticamente rigurosa, que el atractor de Lorenz existe como tal [Tuc99, Ste00]. La simple evidencia num´erica no satisface a los matem´aticos. 1.3.5.b
Caos de Shil’nikov
El caos de Shil’nikov tambi´en se conoce como caos homocl´ınico, aunque esta es una denominaci´on un tanto imprecisa ya que hay varios tipos de caos homocl´ınico. Se observ´o por primera vez en un sistema qu´ımico [AAR87] y en un l´aser [AMG87]. No obstante, el estudio te´orico hab´ıa sido iniciado a˜ nos atr´as por Shil’nikov. El problema de la din´amica en las cercan´ıas de una conexi´on homocl´ınica a un punto silla con autovalores complejos (ver Fig. 1.7) atrajo la atenci´on de Shil’nikov, quien primeramente lo estudi´o en 1965 [Shi65]. Shil’nikov demostr´o que, para el caso que se muestra en la figura, se forma una situaci´on propicia para la existencia de caos (concretamente, una herradura de Smale). Para esto debe cumplirse que el ´ındice de silla δ=−
ρ λu
(1.17)
sea menor que uno. Tambi´en es posible la situaci´on inversa en el tiempo: variedad estable unidimensional y variedad inestable bidimensional. En el art´ıculo de Glendinning y Sparrow [GS84], se estudia la jerarqu´ıa en la que infinitas ´orbitas inestables aparecen en un entorno del punto cr´ıtico del par´ametro de control donde se encuentra la conexi´on homocl´ınica.
1.4
´ Orbitas peri´ odicas inestables (UPOs)
Los atractores ca´oticos se caracterizan por tener un conjunto infinito de objetos inestables en su interior. Estos objetos suelen ser ´orbitas peri´odicas
´ 1.4 Orbitas peri´ odicas inestables (UPOs)
21
´ Figura 1.7: Orbita homocl´ınica (Γ) a un punto sillafoco. La ´ orbita sale del punto fijo en la direcci´ on tangente al autovector asociado al autovalor λu , y entra tangente al plano definido por los autovectores asociados al autovalor imaginario λs = ρ±iω (ρ < 0).
inestables (unstable periodic orbits, UPOs). Ya mencionamos este hecho al estudiar las rutas al caos en el sistema de Lorenz y en el escenario de Shil’nikov. Es f´acil percatarse de la existencia de infinitas UPOs en el caso de la ruta al caos por duplicaci´on de periodo, ya que cada duplicaci´on de periodo deja como residuo una ´orbita inestable. Si hablamos de sistemas que son disipativos en todo el espacio de fases, es claro que las UPOs ser´an objetos tipo silla; ya que si fueran totalmente inestables el volumen del espacio de fases se expandir´ıa en un entorno. As´ı pues, una trayectoria ca´otica puede entenderse como aqu´ella que va “saltando” de una UPO a otra. La variedad estable de una UPO le permite acercarse a ´esta hasta que es expulsada por su variedad inestable, para ser captada por otra UPO. Como las UPOs constituyen el esqueleto del atractor extra˜ no, el completo conocimiento de ´estas proporciona toda la informaci´on del atractor [ACE+ 87, Cvi88, GOY87]. Como veremos en la Sec. 1.5 y en el cap´ıtulo 2 las UPOs juegan un papel muy importante en la sincronizaci´on ca´otica. El inconveniente de las UPOs es que son infinitas y hallarlas para un determinado sistema no es una tarea f´acil. En la siguiente secci´on se describe el m´etodo m´as sencillo de estabilizaci´on de UPOs.
1.4.1
M´ etodo de Newton-Raphson
El m´etodo de Newton-Raphson se caracteriza por su r´apida convergencia, pero tiene el inconveniente de que solamente converge en un entorno peque˜ no de la soluci´on. Esto hace que sea necesario tener una buena estimaci´on inicial de la localizaci´on de la UPO. Para hallar las ´orbitas inestables de un sistema continuo k-dimensional, primeramente hay que realizar una secci´on de Poincar´e del sistema. Como la integraci´on num´erica de la trayectoria, por un m´etodo Runge-Kutta u otro, va “a saltos”, se recomienda utilizar alg´ un ardid como el de H´enon [Hen82] con objeto de obtener un resultado preciso. El mapa (en k − 1 variables) de Poincar´e asociado nos dice cual es la posici´on de cada intersecci´on en funci´on de la anterior. Por supuesto, no tenemos la f´ormula del mapa, as´ı que s´olo podemos integrar num´ericamente el sistema. Si
22
El Caos: Fundamentos
estamos interesados en una UPO que se cierra tras n cortes con la secci´on de Poincar´e, tendremos que encontrar alguno de los n puntos fijos de la n-´esima iteraci´on del mapa de Poincar´e, correspondientes a esa ´orbita. Si el atractor ca´otico tiene una dimensi´on fractal cercana a dos, puede obtenerse una estimaci´on bastante buena de la posici´on de las ´orbitas inestables, ya que la secci´on de Poincar´e del atractor es pr´acticamente unidimensional. De esta forma puede obtenerse un mapa de retorno en una variable a partir del cual interpolar la posici´on aproximada de las UPOs. Formalmente tenemos un (n-´esimo) mapa de Poincar´e F, tal que la UPO en la que estamos interesados intersecta en x∗ = (x∗1 , x∗2 , . . . , x∗k−1 )T = F(x∗ ). Supongamos que tenemos una estimaci´on razonable de x∗ , que llamaremos x0 , entonces se cumplir´a que: ¯ 0 = F(x0 ) ∼ x (1.18) = x∗ + A(x0 − x∗ ) donde A es la matriz jacobiana del mapa F. As´ı pues, el problema reside en encontrar la matriz A ya que: x∗ ∼ x0 − Ax0 ) = x1 (1.19) = (I − A)−1 (¯ lo que nos da una mejor estimaci´on x1 . Un proceso iterativo x0 , x1 , x2 , . . . nos permite aproximarnos a la soluci´on tanto como queramos: x∞ = x∗ . Para encontar los valores de los elementos de matriz de A = (aij ), hay que realizar una iteraci´on del mapa sobre una condici´on inicial ligeramente perturbada x0 + ~ : ˆ 0 = F(x0 + ~) ∼ x = x∗ + A(x0 + ~ − x∗ ) Sustrayendo la ec. (1.18) llegamos a que: ˆ0 − x ¯0 ∼ x = A~.
(1.20) (1.21)
Si la perturbaci´on, tiene s´olo el r-´esimo elemento distinto de cero: ~ = (0, . . . , , . . . , 0)T podremos estimar los elementos de matriz de la r-´esima columna de A, a partir de la ec. (1.21): x ˆ0 − x ¯0i a0ir ∼ . (1.22) = i Por tanto, para cada estimaci´on de la UPO tenemos que tomar k − 1 perturbaciones para estimar todos los elementos de matriz de A que nos permitan conseguir una mejor estimaci´on con la Ec. (1.19). Adem´as, dado que la matriz A es la jacobiana del mapa de Poincar´e asociado a esa ´orbita, el c´alculo de sus autovalores nos informa de sus propiedades de estabilidad. Al tratarse de una ´orbita inestable, al menos uno de sus autovalores estar´a fuera de la circunferencia unidad en el plano complejo.
1.5 Sincronizaci´ on ca´ otica
1.5
23
Sincronizaci´ on ca´ otica
Sincronizaci´on y caos parecen dos t´erminos contradictorios. Debido a la dependencia sensible del caos, parecer´ıa imposible que dos o m´as sistemas ca´oticos pudieran sincronizarse. Sin embargo, la interacci´on de dos sistemas ca´oticos puede, efectivamente, provocar una sincroniazci´on entre ellos. Los primeros art´ıculos sobre sincronizaci´on ca´otica [FY83, Pik84] no provocaron demasiado inter´es en la comunidad cient´ıfica. Es a partir del famoso art´ıculo de Pecora y Carroll [PC90], cuando comienza un estudio intensivo en la sincronizaci´on. Parte de este inter´es se debi´o a la posibilidad de usar el fen´omeno para comunicaciones encriptadas [CO93, Lor00] o multiplexaci´on [Mar99]. Por otro lado, el fen´omeno de la sincronizaci´on de osciladores acoplados ha sido de inter´es desde tiempos de Huygens [Huy86], tanto para la mec´anica [Ble88] como para la biolog´ıa [SS94, Gla01]. Por lo tanto, es natural extender estas teor´ıas al ´ambito de los osciladores ca´oticos. Fruto de este inter´es han aparecido multitud de publicaciones, entre las que podemos destacar el libro de Pikovsky, Rosenblum y Kurths [PRK01] y el art´ıculo de revisi´on de Boccaletti y otros [BKO+ 02]. Hasta aqu´ı no hemos hecho uso de la definici´on del t´ermino sincronizaci´on. As´ı pues, primeramente tendremos que introducir el significado de lo que entendemos por sincronizaci´on. Lo cierto es que, actualmente, se distinguen varios tipos de sincronizaci´on. Los fundamentales son los siguientes: • Sincronizaci´on completa: Dos sistemas ca´oticos id´enticos describen la misma trayectoria x1 (t) = x2 (t) debido a su interacci´on [PC90]. El sistema evoluciona ca´oticamente en el subespacio x1 = x2 , conocido como variedad de sincronizaci´on. El atractor en estado sincronizado es igual al de cada uno de los sistemas por separado y, por tanto, contiene el mismo espectro de ´orbitas inestables. Estas ´orbitas son inestables s´olo en una direcci´on que est´a contenida en la variedad de sincronizaci´on. Ahora bien, si al variar un par´ametro de control, alguna de estas UPOs se inestabiliza en una direcci´on transversa a la variedad de sincronizaci´on se producir´a el fen´omeno conocido como ‘attractor bubbling’ [ABS94]. Aunque en promedio el atractor sea transversalmente estable, si la trayectoria se acerca mucho a alguna de estas ´orbitas ser´a expelida lejos de la variedad de sincronizaci´on a lo largo de la direcci´on inestable tranversalmente. Esto produce que la sincronizaci´on sea interrumpida por episodios de p´erdida de moment´anea de ´esta. En resumen, entre la sincronizaci´on y la no
24
El Caos: Fundamentos
sincronizaci´on suele encontrarse una regi´on (m´as o menos peque˜ na) que muestra sincronizaci´on de “baja calidad”. • Sincronizaci´on generalizada: Dos sistemas ca´oticos no id´enticos ajustan sus din´amicas de tal forma que ambas quedan relacionadas por una funci´on χ, tal que: x1 (t) = χ(x2 (t)) [RSTA95]. • Sincronizaci´on de fase: S´olo se puede definir para aquellos sistemas ca´oticos que tienen una fase bien definida [PROK97]. Existir´a sincronizaci´on n : m si sus fases (definidas en la recta real, es decir cada giro suma 2π) satifacen |nφ1 − mφ2 | < cte. [RPK96], al mismo tiempo que sus amplitudes permanecen altamente descorrelacionadas. N´otese, que aunque los dos osciladores (ca´oticos) sean id´enticos esta condici´on no se satisfar´a al no ser que exista un acoplamiento superior a un umbral, debido a que la din´amica ca´otica hece que el periodo de cada giro fluct´ ue en cada oscilador. • Sincronizaci´on de retardo: se produce cuando un oscilador repite, aproximadamente, la trayectoria de otro con un retardo τ , es decir: x1 (t) ∼ = x2 (t + τ ) [RPK97]. En el cap´ıtulo 2 se estudiar´an en profundidad la sincronizaci´on de fase y de retardo. Por otro lado, en el cap´ıtulo 3 aparecer´a la sincronizaci´on completa entre osciladores de Lorenz.
1.5 Sincronizaci´ on ca´ otica
25
La mayor parte de los contenidos que se tratan en este cap´ıtulo se pueden encontrar en muchos libros. En la siguiente lista se incluyen aqu´ellos que son adecuados para una primera lectura: • Schuster [Sch88]: antiguo pero bastante completo. • Berg´e, Pomeau & Vidal [BPV88]: orientado hacia una visi´on f´ısica, especialmente experimental (ediciones en franc´es e ingl´es). • Alligood, Sauer & Yorke [ASY97]: muy pedag´ogico y con gran cantidad de problemas. Se estudian los mapas m´as que en el resto. • Glendinning [Gle94]: matem´aticamente riguroso sin resultar pesado. No trata todos los “tipos de caos”. • Strogatz [Str94]: pedag´ogico a la vez que amplio en el tratamiento de los diversos problemas de la f´ısica no lineal. • Sol´e & Manrubia [SM96]: trata muchos temas relativos a la f´ısica no lineal, incluido un cap´ıtulo bastante extenso al caos determinista. Tiene la ventaja de que es el u ´nico de la lista que est´a escrito en castellano. • Ott [Ott93]: probablemente el m´as completo en lo que al tema concreto del caos se refiere. La nueva edici´on (2002) trata la sincronizaci´on ca´otica.
26
El Caos: Fundamentos
Chapter 2
Role of Unstable Periodic Orbits in Phase and Lag Synchronization between Coupled Chaotic Oscillators Abstract. Interaction between chaotic oscillators leads to adjustment of their characteristics. Depending on the strength of the coupling, interacting subsystems can share different dynamical features. Under relatively weak coupling, only the timescales of chaotic motions get adjusted; this is known as “phase synchronization”. A stronger coupling can enforce a convergence between phase portraits: a subsystem imitates the sequence of states of the other one, either immediately (“complete synchronization”), or after a time shift (“lag synchronization”). With the help of unstable periodic orbits embedded into the chaotic attractor, we investigate the transition from nonsynchronized behavior to phase synchronizaton, and further to lag synchronization. We demonstrate that onset of phase synchronization requires locking on the surfaces of unstable tori, and relate intermittent phase jumps to local violations of this requirement. Further, we argue that onset of lag synchronization is preceded by the disappearance of many unstable periodic orbits whose geometry is incompatible with the lag configuration. We identify orbits which are responsible for intermittent deviations from the state of lag synchronization.
2.1
Introduction
Synchronization is a universal phenomenon that often occurs when two or more nonlinear oscillators are coupled. Its discovery dates back to Huygens[Huy86], who observed and explained the effect of mutual adjustment between two pendulum clocks hanging from a common support. For coupled periodic oscillators the effect of entrainment of frequencies is
28
Role of Unstable Periodic Orbits in Phase and Lag Synchronization ...
well understood and widely used in applications [Ble88]. Last years we have witnessed the successful extension of the basic ideas of synchronization to the realm of chaotic dynamics [PRK01]. Since chaotic oscillations are more complicated than periodic ones, such extension is neither obvious nor straightforward. The instantaneous state of a periodic process is adequately characterized by the current value of its phase; on the contrary, complete information about the state of a chaotic variable includes, in general, more characteristics. Different degrees of adjustment between these characteristics correspond to different kinds of synchrony: from complete synchronization where the difference between two chaotic signals virtually disappears [FY83, Pik84, PC90], through the “generalized” synchronization where the instantaneous states of subsystems are interrelated by a functional dependence [RSTA95, KP96], to phase synchronization. In the latter case coupled chaotic oscillators remain largely uncorrelated, but the mean timescales of their oscillations coincide or become commensurate. Phase synchronization appears to be the weakest form of synchrony between chaotic systems; it does not require the coupling to be strong. In certain situations, the increase of coupling leads through the further, more ordered stage of synchronized motion: the lag synchronization [RPK97]. In this state –which precedes the complete synchronization– phase portraits of subsystems are (nearly) the same, and the plot X1 (t) for a variable X1 from the first subsystem can be obtained from the plot of its counterpart X2 from the second subsystem by a mere time shift: X1 (t) = X2 (t + τ ). Different aspects of phase and lag synchronization have been investigated mostly from the point of view of global characteristics (Lyapunov exponents, distributions of phase jumps, statistics of intermittent violations of lag configuration, etc.) Below, we intend to have a closer look at the local changes which occur in the phase space of mathematical models. We concentrate on invariant sets and their restructurings, which simplify the dynamics by gradually transforming the non-synchronized chaotic attractor into the coherent attractor of the phase-synchronized state and, further, into a set which corresponds to the state of lag synchronization. To follow the evolution of the attracting set under the increase of the coupling, we trace the fate of unstable periodic orbits (UPOs) embedded into the attractor. A universal and powerful tool for the exploration of chaotic dynamics [Cvi91], unstable periodic orbits proved to be especially efficient in the context of synchronization [Rul96]. Interpretation in terms of UPOs helped to understand the onset of phase synchronization in the case of a chaotic system perturbed by an external periodic
2.2 Periodic orbits in the absence of coupling
29
force [POR+ 97, PZR+ 97], i.e. in the case of unidirectionally coupled periodic and chaotic oscillators. Below, we extend this approach to a system of two bidirectionally coupled non-identical chaotic oscillators. This situation is more complicated, since now each of the participating subsystems possesses an infinite set of UPOs. In the following section we briefly characterize the properties of the unstable periodic orbits which exist in both subsystems in the absence of coupling. In section 2.3 we describe the changes in the structure of the attractor of the coupled system which occur a the coupling strength is increased. We interpret the onset of phase synchronization in terms of phase locking on unstable tori, and argue that transition to lag synchronization should be preceded by extinction of most of the unstable periodic orbits. In sections 2.4 and 2.5 respectively, these qualitative arguments are supported by numerical results which illustrate the role of UPOs in the intermittent bursts close to thresholds of both phase and lag synchronization.
2.2
Periodic orbits in the absence of coupling
As an example, we consider the system of two coupled R¨ossler oscillators under the same set of parameter values for which lag synchronization was reported for the first time [RPK97] (note that two terms ω1,2 have been introduced with respect to Eq. (1.12)) in order to get two oscillators with different frequencies):
x˙ 1,2 = −ω1,2 y1,2 − z1,2 + ε(x2,1 − x1,2 ) y˙ 1,2 = ω1,2 x1,2 − ay1,2
(2.1)
z˙1,2 = f + z1,2 (x1,2 − c) Below, only the coupling strength ε is treated as an active parameter; the remaining parameters have fixed values a = 0.165, f = 0.2, c = 10, ω1,2 = ω0 ± ∆ (ω0 = 0.97, ∆ = 0.02). Besides the original paper [RPK97], scenarios of onset of lag synchronization in equations (2.1) under these parameter values have been discussed in subsequent publications [SBV+ 99, BV00, ZWL02]. In each of the subsystems, taken alone, this combination of parameters ensures chaotic oscillations (Fig. 2.1(a)). Projected onto the xy plane of the corresponding subsystem, these oscillations look like rotations around
30
Role of Unstable Periodic Orbits in Phase and Lag Synchronization ...
y
10
(a)
0 -10 -20 -10
0 x
10
20
15 (c)
(b)
z
-xn+1
0.015 0.01
10 5
-15
-10 x
-5
5
10 -xn
15
Figure 2.1: R¨ ossler oscillator at ω = ω1 = 0.99: (a) projection of phase portrait; solid line: location of the Poincar´e plane, (b) trace of the attractor on the Poincar´e plane, (c) one-dimensional return mapping.
the origin; this allows us to introduce phase geometrically, as a lift of the angular coordinate in this plane: φ1,2 = arctan
y1,2 π + sign(x1,2 ) x1,2 2
(2.2)
The mean frequency of the chaotic oscillations is then defined as the mean angular velocity: Ω(1,2) = hdφ1,2 /dti. Difference in the values of the parameters ω1,2 makes the mean frequencies of uncoupled oscillators slightly different: at ε = 0, they are Ω(1) = 1.01926 . . . and Ω(2) = 0.97081 . . ., respectively. As a result, the phases of the oscillators drift apart; in order to enforce phase synchronization, the coupling should be able to suppress this drift by adjusting the rotation rates. In order to understand the role of unstable phase orbits in the phase space of the coupled system, it is helpful to start with the classification of such orbits in the absence of coupling. In its partial subspace, each of the 3-dimensional flows induces the return map on an appropriate Poincar´e surface (it is convenient to use for this purpose the trajectories which in the i-th system intersect the surface yi = 0 “from above”). This two-dimensional mapping is, of course, invertible; however, due to the
2.2 Periodic orbits in the absence of coupling
31
strong transversal contraction, the trace of the attractor on the Poincar´e surface is graphically almost indistinguishable from a one-dimensional curve (Fig. 2.1(b)). Parameterizing this curve (e.g. by the value of the coordinate x), we arrive at the non-invertible one-dimensional map shown in Fig. 2.1(c). Since the latter turns out to be unimodal, its dynamics is completely determined by the symbolic “itinerary” [CE80]: the sequence RLL... in which the j-th symbol is R if the j-th iteration of the extremum lies to the right from this extremum, and L otherwise. According to numerical estimates, for ω = ω1 = 0.99 the itinerary is RLLLLRLLL..., and for ω = ω2 = 0.95 it becomes RLLLLLRLL... The starting segments of the two symbolic strings coincide, the first discrepancy occurs in the 6th symbol; therefore, the number of unstable periodic orbits which make l turns around the origin is the same in both subsystems, if l does not exceed 5. The number of orbits with length l ≥ 6 is larger in the second subsystem. Comparison of the initial 25 symbols with the symbolic itinerary of the logistic mapping xi+1 = A xi (1 − xi ) shows that flows with ω = ω1 = 0.99 and ω = ω2 = 0.95 correspond to maps with A = 3.9977031 . . . and A = 3.9904857 . . ., respectively. For our purpose, we need to modify some conventional characteristics. When we consider the flow near a long periodic orbit, the duration of each single revolution (turn) in the phase space appears to be of little importance: what matters for phase dynamics is the mean duration of the turn, i.e. the overall period of the orbit divided by the number of turns in this orbit. Below, we refer to the number of turns as to the (symbolic) length of the orbit [PZR+ 97]. Since the time between consecutive returns onto the Poincar´e plane depends on the position on this plane, the periods of all periodic solutions are –in general– different. It is convenient to characterize periodic orbits in terms of “individual frequencies” Ωi ; these are not the usual inverse values of the corresponding overall periods, but mean frequencies per one turn in the phase space: for an orbit with period T consisting of l turns, Ωi ≡ 2πl/T . Fig. 2.2 presents the distributions of individual frequencies for periodic solutions for both subsystems in the absence of coupling. Since usually the orbits with relatively short periods are sufficient for an adequate description of the whole picture [HO96], we restrict ourselves to orbits with length l ≤ 10; this yields 164 UPOs at ω = 0.99 and 196 UPOs at ω = 0.95. As shown in Fig. 2.2, two frequency bands are separated by a gap. For (1) ω = 0.99 the individual frequencies belong to the interval between Ωmax = (1) 1.035519 . . . (orbit with length 1) and Ωmin = 1.014042 . . . (one of the orbits
32
Role of Unstable Periodic Orbits in Phase and Lag Synchronization ...
1.04
Ωi
1.02
1
0.98
0.96 2
4
6
8
10
symbolic length Figure 2.2: Frequencies of unstable periodic orbits embedded into the attractors of the R¨ ossler equations. Circles: ω = 0.99; crosses: ω = 0.95. (2)
with length 5). For ω = 0.95 the values are distributed between Ωmax = (2) 0.9927899 . . . (the same orbit with length 1) and Ωmin = 0.9790416 . . . (one of the orbits with length 6). Besides periodic orbits, the R¨ossler equations possess a saddle-focus fixed point located close to the origin. Although this point does not belong to the chaotic attractor, it is not irrelevant: under coupling, it interacts with periodic orbits of the complementary subsystem and contributes to the general scenario. To unify notation, we refer to this fixed point as the “length-0 orbit”
2.3
Attractors of a coupled system: role of unstable tori in synchronization transitions
Formally, at ε = 0 (absence of coupling) the attractor in the joint phase space of the two systems contains a countable set of degenerate invariant 2tori: direct products of each periodic orbit from the 1st subsystem with each periodic orbit from the 2nd one. Again, for the purpose of comparison of phase evolution in both subsystems, it is convenient to redefine the usual
2.3 Attractors of a coupled system: role of unstable tori ...
33
notion of the rotation number on such tori: let the mean times of one revolution around the torus for the projections onto two subsystems be, respectively, τ1 and τ2 . Then the rotation number is introduced as the ratio ρ = τ1 /τ2 . If the equality ρ = 1 holds, within a sufficiently long time the projections of trajectories make an equal number of rotations in the subspaces of subsystems: the torus is “phase locked”. Generalization of this interpretation for other rational values of ρ is straightforward. Obviously, at ε = 0, the rotation number is ω1 /ω2 where ω1 and ω2 are the individual frequencies (per one rotation, as discussed above) of the two periodic orbits which form the torus. As soon as the infinitesimal coupling between the subsystems is introduced, the degeneracy of tori is removed. The UPOs shown in Fig. 2.2 produce 164 × 196 = 32144 tori whose rotation numbers (in the above (2) (1) (2) (1) sense) lie between Ωmin /Ωmax = 0.92737 . . . and Ωmax /Ωmin = 0.97904 . . .. In general, each torus persists in a certain range of ε, and its rotation number ρ is a devil’s staircase-like function of ε: intervals of values of ε correspond to rational values of ρ. Since the periodic orbits in the subsystems are unstable, the tori are also unstable: for small values of ε, a trajectory on the toroidal surface has at least two positive Lyapunov exponents. The boundaries of “locking intervals” of ε for each torus are marked by tangent bifurcations of periodic orbits. Such a bifurcation creates/destroys on the surface of the torus two closed trajectories, one stable (with respect to disturbances within the surface), the other one unstable. Since the motion along the torus is parameterized by the phases of subsystems, below we refer to these orbits as “phase stable” and “phase unstable” [PZR+ 97], respectively. The following argument demonstrates that on each torus the phasestable and phase-unstable orbits are not necessarily unique. Let the torus originate from the direct product of two periodic orbits: an UPO from the first subsystem with length l and an UPO from the second subsystem with length m. Then the main locking (1 : 1 in our notation) assumes that the phase curve is closed after n turns, n being the least common multiple of l and m. Let us take the projection of the periodic orbit onto the subspace of the first subsystem, and select some particular point on it (e.g., the highest of the n main maxima for one of the variables). By translating forwards and backwards the partial projection onto the other subsystem, we get n configurations in which one of the n maxima of the second variable is close to the selected point. Fig. 2.3 shows such “appropriate for the
34
Role of Unstable Periodic Orbits in Phase and Lag Synchronization ...
x1,2
10
-10 0
5
10 t
10 x1,2
Figure 2.3: Configurations favorable for the locking on the torus originated from the direct product of UPOs of length 2: (a) “inphase”; (b) “out-of-phase”. Solid curves: x1 (t); dashed curves: x2 (t).
(a)
0
(b)
0 -10 0
5
10 t
locking” configurations for the torus generated by two orbits of length 2; in this case, l = m = n = 2. In general, this implies that we should expect to observe on the surface of a single torus up to n coexisting pairs of phase stable and phase unstable periodic orbits. [Naturally, the argument is not rigorous; in principle, not all of n possible configurations should necessarily be exhausted; on the other hand, the existence of additional lockings cannot be totally excluded as well]. A locking interval in the parameter space ranges from the birth of the first couple of curves with the prescribed locking ratio, to the death of the last such couple. Uniqueness of the rotation number forbids the coexistence on the same torus of periodic orbits with different locking ratios. In the course of time a chaotic trajectory repeatedly visits the neighborhoods of unstable tori; in each of them it spends some time winding along the surface, until being repelled towards some other unstable torus. During the time T spent in the vicinity of the torus with rotation number ρ, the increment of the phase difference φ1 − φ2 between the subsystems is ∆φ ≈ 2π(T /τ1 − T /τ2 ) = 2π(1 − ρ)/τ1 . Hence, unless ρ = 1, the passage close to a torus results in a phase drift. On the other hand, if the torus is locked in the ratio 1 : 1, a passage of a chaotic trajectory along one of the phase-stable UPOs on the toroidal surface leads neither to a phase gain nor to a phase loss. Therefore we can expect that in the phase synchronized state all of the tori embedded into the chaotic attractor are locked and have the same frequency ratio. From this point of view, in the course of the transition to phase synchronization, each of the tori present at ε = 0 should either reach the main locking state or disappear, from the attractor or from the whole phase space. Note that, even if a single torus within the attractor remains not locked, the ergodic nature of chaotic dynamics will ensure that from time to time the trajectory will approach this torus close enough to make the system exhibit a phase jump.
2.3 Attractors of a coupled system: role of unstable tori ...
35
Now we proceed to lag synchronization. Let us start by ordering the (k) UPOs in uncoupled subsystems into two sequences {Ui }, k = 1, 2; i = 1, 2, . . .. The ordering can be done by means of criteria which take into account the symbolic length and topology (expressed e.g. by symbolic itinerary) of the orbits. This induces labeling among the tori of the coupled (1) system: the torus Tij originates from the interaction of the orbit Ui from (2) the first subsystem and the orbit Uj from the second one. As discussed above, for the values of ε beyond the threshold of phase synchronization all the tori inside the attractor should have the same rotation number 1, hence they should possess periodic orbits. In fact, at finite values of ε neither smoothness nor even the very existence of a 2-dimensional toroidal surface can be guaranteed, but this circumstance appears to be of little importance: in the synchronized state the decisive role is played not by the entire torus or its global remnants, but by relatively small segments near closed phase-stable and phase-unstable orbits. The torus may break up, but periodic orbits persist. Therefore, in our discussion below the symbol Tij denotes not so much the actual two-dimensional torus, but rather the set of (possibly several) periodic orbits corresponding to the (1) (2) locking 1:1 on this torus. If Ui and Uj have symbolic lengths l and m, then their symbolic labels A(1) = RL . . . and A(2) = RL . . . consist, respectively, of l and m letters. Let n be the least common multiple of l and m. Symbolic labels B (1) and B (2) for projections of Tij respectively onto the first and second subsystem consist of n symbols: B (1) is n/l times repeated A(1) , and B (2) is n/m times repeated A(2) . It can be shown, that, unless A(1) = A(2) , the labels B (1) and B (2) can neither coincide, nor be obtained from each other by cyclic permutation of symbols. The symbolic label determines the topology of the periodic orbit; in particular, it prescribes the order in which the smaller and larger turns alternate. Therefore, if symbolic labels for projections are different, there is no way to bring one of these projections very close to the other by time shift: for all values of such shift the time-averaged difference between these projections will neither vanish nor become very small. According to this argument, only those Tij for which two generating UPOs have the same length and topology can persist in the attractor of lag-synchronized state. Presence of the “non-diagonal” Tij is incompatible with lag synchronization. Therefore all such tori and associated closed orbits should, in the course of increase of ε, either disappear, or leave the attractor. Further, Tij with identical symbolic labels may contain several phasestable periodic orbits. However, only the passage close to the “in-phase”
36
Role of Unstable Periodic Orbits in Phase and Lag Synchronization ...
orbit would allow for lag synchronization with small (compared to the mean duration of one turn) value of lag. For “out-of-phase” configurations, which are obtained from the “in-phase” ones by cyclic permutation of maxima, the appropriate time shift would be close to several durations (lengths of the shift) of the turn. Apparently, only the “in-phase” orbits contribute to the motion in the lag-synchronized state. For example, the UPO in Fig. 2.3(a) can participate in the lag-synchronized dynamics, whereas the UPO in Fig. 2.3(b) is obviously unsuitable for this purpose and, hence, should not be contained in the attractor. Thus, we expect that the onset of lag synchronization should be preceded by extinction of most unstable periodic orbits which populate the attractor at the onset of phase synchronization. In fact, a set of two oscillators in the state of lag synchronization behaves almost the same way as one of them taken separately; in this sense, the complexity of lag synchronization is relatively low. If the above interpretation is correct, intermittency of respective characteristics, observed below the threshold values of the coupling strength both for phase [LKL98] and lag synchronization [BV00, ZWL02] should be caused by the passage of chaotic trajectories close to the last obstructing invariant sets. In the first case these sets are the last non-locked tori, and in the second case they are either the last remaining UPOs from “nondiagonal” Tij or the “out-of-phase” UPOs. For completeness, it should be mentioned that there are certain UPOs which do not emerge from tori, but, instead, exist already at zero coupling. At ε = 0 they are just direct products of steady state (fixed point) on one side, and an UPO on the other side. Obviously, such orbits are also incompatible with lag synchronization, and should disappear in the course of increase of ε. In the following sections we test these qualitative conjectures about the onset mechanisms of phase and lag synchronization against the numerical data obtained by integration of Eq.(2.1); UPOs have been computed through combination of the Schmelcher-Diakonos[SD98] and NewtonRaphson methods.
2.4
Phase synchronization
Phase synchronization in Eq. (2.1) is observed beyond the threshold value ε = εps . For ε > εps , the difference of phases between two oscillators remains confined within a narrow interval for t → ∞; below this threshold
37
2.4 Phase synchronization
−4
frequency
6
10
pj
10
4
10
−5
10
−6
10
−7
0.037
0.038
0.039
ε
0.04
0.041
10
20
22
24
(εps−ε)−1/2
26
28
Figure 2.4: Mean time hTpj i between phase slips and frequency f of slips vs. coupling strength ε.
it grows unboundedly. According to our computations, εps ≈ 0.0416 (this is somewhat higher than the value 0.036 reported in [RPK97]). In fact, already at ε ≥ 0.036 the phases of two oscillators stay synchronized for most of the time: the plot of phase difference as a function of time reminds a staircase in which long nearly horizontal segments are interrupted by relatively short transitions. Such transitions (phase slips) are not instantaneous; usually, it takes several dozens of turns in the phase space, to increase the phase difference by 2π. However, compared to the average duration of the synchronized segment, phase slips are fast: as seen in Fig. 2.4, when ε approaches εps , such duration grows from hundreds of turns through tens of thousands to millions and further on. The value 0.0416 is the highest value of ε at which we were able to observe a phase jump (only one event within ∼ 109 turns of the chaotic orbit). In the case of chaotic oscillators driven by an external periodic force, the transition to phase synchronization manifests itself in the phase space as a kind of repeller-attractor collision [POR+ 97, PZR+ 97, ROH98]: the local bifurcation (tangent bifurcation in which a phase-stable and a phaseunstable UPOs are born), is simultaneously the global event: disappearance of the last channel for phase diffusion. Rare violations of synchronization below the threshold were named “eyelet intermittency”, since escapes from the phase-locked state were due to the very accurate hitting of a vicinity of the last non-locked torus. The same mechanism is at work in our case just below εps : of infinitely many tori Tij embedded into the chaotic attractor, almost all are locked in the ratio 1 : 1. Only the passages near several remaining non-locked (or locked in other ratios) tori can contribute to gains/losses of phase
38
Role of Unstable Periodic Orbits in Phase and Lag Synchronization ...
difference. Since the tori are unstable, the chaotic trajectories are mostly kicked out from their neighborhoods before producing a noticeable phase difference. Only the trajectories which come very close to the non-locked tori, stay long enough in their vicinities in order to gain a phase slip. The frequency f of such events depends on the distribution of the invariant measure on the attractor. Assuming, for simplicity, that this measure is uniform, the same scaling law for f as in [POR+ 97] can be obtained: √ f (ε) ∼ exp(−1/ εps − ε). This qualitative dependence is well corroborated by our numerical data (cf. Fig. 2.4(b)). Fig. 2.5 presents the “tree” of the periodic orbits of length 1 and 2 as a function of the coupling strength ε. The vertical “amplitude” coordinate on this plot is fictitious: it plays the role of appropriately re-scaled and shifted coordinate values (if actual values of coordinates were used, most of the branches would overlap, strongly hampering understanding of the bifurcation sequences). Tangent bifurcations are marked with asterisks (*), and perioddoublings are denoted by small circles (◦). The notation m − m0 stands for the locking on the torus which is the direct product of the length-m and the length-m0 orbits of the first and second oscillators, respectively. Thus, the 2-2 torus undergoes two lockings: at the moment of birth of corresponding UPOs, phase lags between both oscillators are respectively ∼ π/2 and ∼ 2π + π/2; as ε grows, the values of these lags decrease. In accordance with the above classification, we call these orbits in- and outof-phase lockings. It may be seen in Fig. 2.5 that the tangent bifurcations which create orbits of length 1 and 2, occur in a small interval around ε = 0.04, i.e. close to the approximate threshold of phase synchronization. At slightly higher values (ε > 0.05) we detect period-doubling bifurcations. The presence of period-doublings, as well as of Hopf bifurcations on other branches (see below) indicates that the smoothness of the corresponding toroidal surfaces is already lost. We remind that label 0 denotes orbits which are born from the direct products of the steady solution with periodic solutions. The plot shows that, as expected, such orbits disappear relatively early: the branch 2-0 joins the branch 1-0 in the course of the inverse period-doubling bifurcation. The branch 1-0, in its turn, annihilates at ε = 0.0767361 with one of the branches born on the torus 1-1. As a further illustration, in Fig. 2.6 we show solution curves and bifurcation points for orbits of length 3. This case is richer, insofar as each isolated oscillator contains two UPOs of this length (cf. Fig. 2.2);
39
2.4 Phase synchronization
1
1−2
stable window of length 5
1−1 2−0 1−0
2−2 out
2−1
2−2 in 2
0
0.05
0.1
ε
0.15
0.2
0.25
Figure 2.5: Bifurcation diagram showing UPOs of length 1 and 2. Notation: ∗ saddle-node bifurcations, ◦ - period-doubling bifurcations. Solid, dashed, and dotted lines: orbits unstable in 1, 2 and 3 directions, respectively.
40
Role of Unstable Periodic Orbits in Phase and Lag Synchronization ...
they are labeled 3a and 3b. Since every orbit possesses three maxima of x1,2 , on each of the 4 emerging tori there can be up to three pairs of UPOs: along with in-phase orbits, there are two out-of-phase configurations, with phase lags ∼ 2π and ∼ 4π, respectively. Now, besides tangent and perioddoubling bifurcations, Hopf bifurcations (denoted by ) are also identified. In fact, it appears that Hopf bifurcations substitute some expected lockings. It should be noted that in this case all tangent bifurcations which create UPOs, occur at ε < 0.04. in−phase
(a)
2π−out−of−phase
(b)
3a 3a−3b
3a−3b
3a−3a
3b
3b−3b
3b−3b
0
0.02
3b−3a
0.04
0.06
0.08
0.1
ε
0.12
4π−out−of−phase
0.14
0.16
0.18
0.2
0.033
0.034
0.035
0.036
0.037
0.038
ε
0.039
0.04
0.041
0.042
0.043
(c)
3a−3b
3a−3a
Figure 2.6: Bifurcation diagram for UPOs of length 3. Notation:
- Hopf bifurcations; others as in Fig. 2.5.
3b−3b
0.03
0.035
0.04
0.045
0.05
ε
0.055
0.06
0.065
0.07
Figure 4c, D. Pazo et al. (2002)
A remarkable feature here are the isolas in Fig. 2.6(b),(c): each family of out-of-phase lockings is not connected to families of periodic solutions and exists only in the relatively small interval of values of ε. As seen in Fig. 2.6(a), for sufficiently high values of ε of all the UPOs of length 3, only two in-phase orbits survive. Several further families of UPOs are not shown on these plots. When ε is increased, the orbits of the type 0 − 2n disappear one-by-one in the inverse period-doubling cascade, and finally the last of them, the UPO 0-1,
2.4 Phase synchronization
41
shrinks and merges with the fixed point of the system (in our notation, 0-0) in the inverse Hopf bifurcation. The orbits 3a-0 and 3b-0 coalesce in a saddle-node bifurcation, as well as the orbits 0-3a and 0-3b. The tori 3a-1 and 3b-1 annihilate each other in the same way as the 2π-out-of-phase lockings, 3a-3b and 3a-3a in Fig. 2.6(b). On the other hand, we failed to locate numerically the 1-3a and 1-3b lockings; it seems that both tori also collide and disappear in a saddle-node bifurcation (or they get locked but their UPOs survive in a very narrow range of ε). Calculations for UPOs of other lengths have shown qualitatively similar pictures, with tangent bifurcations around ε ≈ 0.04 and short-lived out-ofphase lockings. We have also performed numerical experiments in order to verify the conjecture that phase jumps occur when the trajectory approaches a nonlocked torus. Since we are presently unable to locate numerically in the phase space the two-dimensional unstable tori, sometimes it is difficult to assign the jump to the passage near a particular torus. Nevertheless, in certain cases it was possible to identify a configuration which provoked a phase jump. In such situations, at the beginning of the jump the segments of trajectories of the first and the second oscillators resemble closed orbits. An example is shown in Fig. 2.7, where the passage of the system close to a 1-3 torus can be recognized. In general, the farther from 1 is the rotation number ρ on the degenerate torus at ε = 0, the higher should be the magnitude of coupling required for the locking. In the frequency distribution of Fig. 2.2, the highest individual frequency belongs to the orbit of length 1; the tori, built with the participation of this UPO, require relatively strong coupling in order to get locked. In accordance with this, many of the phase jumps close to the threshold of phase synchronization are preceded by an approach of the first oscillator to the orbit of length 1. Notably, the locking on the torus 1-1 occurs at the relatively high value ε = 0.0424585, which is above the empirically determined threshold εps = 0.0416. This means that either this torus does not belong to the attractor, or the close passages happen so seldom, that one should observe the system for times higher than 109 mean rotation periods (our longest runs) in order to experience such jumps. We cannot point out which torus is the last one to be locked. Among the relatively short orbits, the closest to εps locking appears to be the tangent bifurcation, which creates orbits of length 4 at εps = 0.0414302.
42
Role of Unstable Periodic Orbits in Phase and Lag Synchronization ... 10
20
∆φ, ∆φ−2π
x1
8
0
6 −20 4
x2
2 0 4.22
4.2305
4.231 5
4.225
4.23 Time (t.u.)
4.235
4.24 5
x 10
x 10
0
−20
20
5
10
4.23
4.2305 Time (t.u.)
4.231 5
x 10
z2
10
z1
4.23
20
0 20
0 20 20
0 y1
0 −20
−20
x1
20
0 y2
0 −20
−20
x2
Figure 2.7: Occurrence of a phase jump for ε = 0.0403. The upper left figure shows the phase difference (and the phase difference less 2π as a reference) between the two subsystems. The time lap corresponding to the passing through an eyelet is indicated by a segment and an arrow. The dynamics of each subsystem during the crossing is plotted in two time series and two snapshots.
2.5
Lag synchronization
The lag synchronized state in Eq. (2.1) was found to exist above the critical value of the coupling strength εls ≈ 0.14[RPK97]. In this state, the dynamics of both oscillators is very similar to the one that they exhibit being isolated, but now they are related by a time lag: x1 (t) ≈ x2 (t + τ0 ). The transition from phase synchronization to lag synchronization was shown to be preceded by a intermittent region where lag synchronization was interrupted by bursts [RPK97]. Since the R¨ossler oscillator is approximately isochronous, the time lag is practically equivalent to the phase lag. In Fig. 2.8(a) the value of the mean phase difference h∆φi between both oscillators is shown, as well as the corridor formed by this difference ± its standard deviation σ. For ε > 0.14 this corridor is rather narrow (albeit non-zero); when ε is decreased below 0.14, the deviation rapidly grows. However, the minimal and maximal values for deviations of
43
2.5 Lag synchronization
< ∆φ>, +σ, −σ
0.5
0.4
(a)
0.3
0.2
0.1 0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
0.8
(∆φ)
max,min
0.6
104 105 106 7 10
(b)
0.4
turns turns turns turns
0.2 0 −0.2 0.1
0.12
0.14
0.16
0.18
ε
0.2
0.22
0.24
0.26
Figure 2.8: (a) Mean phase difference between subsystems, and the bounds set by standard deviation. (b) Maximum and minimum phase difference computed for trajectories with different number of turns.
phase difference from its mean value remain non-small also beyond ε = 0.14 (Fig. 2.8(b)). This is a typical feature of intermittency. By increasing computing time, we were able to detect larger deviations from h∆φi at higher values of ε; the plot shows dependencies estimated from chaotic orbits of different length. What is the role played by UPOs in this intermittent transition to lag synchronization? We begin the discussion with the observation that growth of the coupling strength reduces the volume of phase space occupied by the attractor. Evolution of the system towards this state is illustrated by return maps for one coordinate, recovered from the intersection of the attractor with the Poincar´e plane y1 = 0 (Fig. 2.9). As ε is increased, the initially diffuse cloud becomes more structured, with more and more points settling onto the “one-dimensional” backbone. For ε ≈ εls , the mapping is reminiscent of Fig.2.1(c) (however, there remains a small proportion of points which lie at a distance from the parabola-like
44
Role of Unstable Periodic Orbits in Phase and Lag Synchronization ...
15
(a) -x1(n+1)
-x1(n+1)
15 10 5
5
10
(b)
10 5
15
5
(n)
-x1
15
(c)
10 5 5
10 -x1(n)
15
-x1
-x1(n+1)
-x1(n+1)
15
10 (n)
15
(d)
10 5 5
10
15
-x1(n)
Figure 2.9: Return maps for the variable x1 on the Poincar´e plane y1 = 0: (a) ε = 0.03; (b) ε = 0.08; (c) ε = 0.12; (d) ε = 0.14.
curve). Such behavior implies that the system must possess a set of UPOs similar to that of an isolated R¨ossler oscillator; according to Fig. 2.2, for ε > εls there should be one UPO of length 1, one UPO of length 2, and two UPOs of length 3. Characteristics of unstable periodic orbits for ε slightly beyond εls are shown in Fig.2.10. According to Fig.2.10(a), correspondence with an isolated oscillator is not yet reached: the full system possesses two UPOs of length 2, as well as four orbits of length 3 and four orbits of length 4, whereas the description based on the unimodal mapping prescribes one orbit of length 2, not more than two orbits of length 3 and an odd number (1 or 3) of length-4 UPOs. Upon further increase of ε, the “superfluous” orbits eventually disappear: two orbits of length 3 annihilate each other throught the tangent bifurcation at ε = 0.154856; then at ε = 0.15694, a period-doubling bifurcation unifies the orbit of length 4 with the “superfluous” orbit of length 2, and finally the branch of the latter UPO (which will be separately discussed below) joins the branch of the length-1 orbit at ε = 0.23892. The frequencies of UPOs are distributed over the narrow range (notably, the state of phase synchronization does not necessarily assume that all these frequencies coincide). To characterize the time shift between the subsystems, we use the value of the phase lag between them at the moment
45
2.5 Lag synchronization
1.01
(b)
(a)
0.4
1.005 τ
Ωi
0.3
1
0.2
0.995
0.1
0.99
0 1
2 3 4 5 Orbit length
1
2 3 4 5 Orbit length
Figure 2.10: Periodic orbits at ε = 0.15. (a) individual frequencies Ωi ; (b) phase lags ∆φ on turns of periodic orbits.
of intersection of the Poincar´e plane y2 = 0. Since we are interested in instantaneous values, each length-l UPO delivers l values of ∆φ. As seen in Fig.2.10(b), most of the values of the phase lag belong to the narrow range between 0.27 and 0.3; however, large deviations from this range are also present. Notably, most of these deviations belong to the “superfluous” orbits. As understood from Fig.2.8(b), noticeable outbursts of phase difference are very rare; this means that a chaotic trajectory only seldom visits the neighborhoods of these UPOs; accordingly, their contribution into the dynamics is relatively small. Growth of ε beyond the values shown in Fig. 2.9 leads to further condensation of the points of the return map onto the one-dimensional backbone; the proportion of deviations becomes smaller. It appears that in the space phase there exists a pattern (at the moment we know too little about its properties to call it an “invariant manifold”), which is responsible for the lag structure and on which dynamics is adequately represented by a unimodal map. This pattern is locally attracting almost everywhere, except for certain “spots”; a chaotic trajectory which hits such a spot, makes a short departure from the pattern and disturbs the lag synchronism. Note that at large values of ε the UPOs have only one unstable direction
46
Role of Unstable Periodic Orbits in Phase and Lag Synchronization ... 0.05
ε = 0.2 0
1 −0.05
0.5(x1− x2)
Figure 2.11: Poincar´e section of the attractor for ε = 0.2; five regions with the largest density of points, corresponding to the “ghost” of the stable window of length-5, can be distinguished. The location of the “transversally unstable” length-1 orbit is indicated with a circle.
3
2
4 −0.1
−0.15
5
−0.2
−0.25
−14
−12
−10
−8
0.5 (x1+ x2)
−6
−4
−2
(one characteristic multiplier outside the unit circle); this corresponds to the instability of all periodic points of the unimodal mapping. When ε is gradually decreased, the first orbit to become unstable in a second direction is the one of length 1 (ε = 0.23892). This bifurcation was reported in [SBV+ 99] where synchronization transitions for different mismatches between ω1 and ω2 were studied. At this critical point, the length-1 periodic orbit embedded into the “lag attractor” undergoes the period-doubling bifurcation. As a result, an orbit of length 2 is created. Tracing this new orbit down to small values of ε, we observe that it ends up as a phasestable orbit on the torus formed by the length-1 and length-2 UPOs of the decoupled subsystems; the corresponding bifurcations are shown in Fig. 2.5. The configuration of this orbit (two approximately equal maxima in the projection onto one subsystem versus two unequal maxima in the second subsystem) is obviously incompatible with the requirements of the lagsynchronized state. Thereby, the loss of perfect lag synchronization occurs because one of the orbits becomes unstable in the direction “transversal” to the lag pattern; in this sense, this is a kind of a bubbling-type transition. The existence of a window of stable length-5 oscillations above ε(5) = 0.23103 (see Fig. 2.5) was not noted in previous works. The stable periodic orbit is born at ε(5) in the saddle-node bifurcation. Below this value, a typical type-I intermittency behavior is observed, and the distribution of invariant measure on the attractor is very non-uniform: the periodic orbit leaves a “ghost”, the density of imaging points is rather high in five corresponding regions of the Poincar´e section, and the length-1 UPO, which lies aside from these regions, is very seldom visited. Probably, this is the reason why the intermittent behavior above ε = 0.145 was not observed earlier. Another interesting feature of the transition from phase to lag
2.6 Discussion
47
synchronization was reported in [BV00]. The criterion for this transition, proposed in [RPK97], requires the minimum of the “similarity function” S 2 (τ ) = h(x2 (t)−x1 (t−τ ))2 i/(hx21 (t)ihx22 (t)i)1/2 to vanish (or nearly vanish) for some τ0 ; naturally, τ0 is the lag duration. In [BV00] it was noticed that besides the main minimum at τ0 /T 1, the similarity function has secondary minima at τ = τ0 + mT , where m = 1, 2, ..., and T is close to the mean duration of one turn in the phase space. When perfect lag synchronization is lost, the magnitudes of the secondary minima of S 2 decrease. It turns out that intermittent violations of lag synchronization consist of jumps from the main lag configuration [x1 (t) = x2 (t + τ0 )] to configurations of the kind x1 (t) = x2 (t + τ0 + mT ). According to [BV00], during the jump stage the system seems to approach a periodic orbit. This observation confirms the above conjecture that the intermittency which precedes the onset of lag synchronization, is caused by passages near the out-of-phase UPOs. Our numerical data shed more light on the nature of these jumps and allow us to identify the orbits responsible for the intermittency. According to Fig. 2.5, among the orbits belonging to the out-of-phase locking of two UPOs of length 2, the least unstable one (the orbit which has only one unstable direction) exists for 0.1246 < ε < 0.1426. Temporal evolution of x1 (t) and x2 (t) for this orbit is shown in Fig. 2.12(b); the phase shift is close to the duration of one turn. Figure 2.12(c,d) shows x1,2 - and y1,2 -projections of this UPO embedded into the attractor. We observe that part of this orbit is “transversal” with respect to the bulk of the attractor. In the course of the intermittent bursts, chaotic trajectories which leave the bulk region, move along this UPO. During this motion the dynamics of both oscillators gets approximately correlated, and the lag between them corresponds to the time shift seen in Fig. 2.12(b): τ ≈ τ0 +T .
2.6
Discussion
Our results show that transition to phase synchronization and onset of lag synchronization between two coupled chaotic oscillators are accompanied by profound changes in the structure of the attracting set. Unstable periodic orbits serve as mediators in these processes: when the coupling strength is increased, they should first appear in the phase space in order to enforce the entrainment of phases, and then, most of them should again disappear, in order to leave in the attractor only suitable patterns for lag synchronization. Absence of necessary UPOs in the first case, and presence of “non-suitable” orbits in the second case are the reasons for
48
Role of Unstable Periodic Orbits in Phase and Lag Synchronization ...
1 15 0
x
Re(λi)
−10 (a) 0.05
ε
0.1
0.15
0
20
15
15
10
10
5
5
0
0
−5
−5
−10 (c)
−10 −10
0 x
1
10
ε=0.14
−15
y2
x2
1
−5
−3
−15 −20
x
0
−2
−4
2
5
−1
(b)
x
10
5 Time (t.u.)
10
(d)
−15 20
−20 −20
−10
0 y
10
20
1
Figure 2.12: Role of the out-of-phase UPO of length 2 in intermittent lag synchronization. (a) Eigenvalues of the second iteration of the Poincar´e map at the UPO as functions of ε. Because of the strong transversal contraction of the R¨ ossler oscillator, two of the eigenvalues are very close to zero and are not depicted. (b) Time series of x1 (black line) and x2 (gray line) over one period; (c,d) Dots: chaotic orbit on the “lag attractor” in the region of intermittent lag synchronization (ε = 0.14); solid line: out-of-phase UPO.
2.6 Discussion
49
intermittency. Before the onset of phase synchronization such intermittency is caused by passages near the 2-tori which are not yet locked, or locked in ratios different from 1:1; in the latter case such intermittency is, in fact, a certain form of “other” short-lived synchronization. The intermittency preceding the onset of lag synchronization is due to the passages near the periodic orbits, in which two oscillators are locked “out-of-phase”; such passages make the system exhibit momentary exotic lag configurations. We feel that unstable periodic orbits are an appropriate tool for the analysis of intricate details of these transitions; further numerical advances would probably require the technique for the calculation of unstable 2-tori. The studied system of two coupled oscillators is nonhyperbolic, at least in the parameter region around the onset of phase synchronization. As seen in Fig. 2.5, increase of ε leads to the decrease in the dimension of the unstable manifolds of UPOs. As a result, over large intervals of ε we observe coexistence of UPOs with 2-, 3- and 4-dimensional unstable manifolds. In general, this phenomenon, known under the name of ‘unstable dimension variability’, has important implications for dynamics itself as well as for the validity and applicability of numerical algorithms [BS00]; its significance in the context of synchronization is yet to be analyzed.
50
Role of Unstable Periodic Orbits in Phase and Lag Synchronization ...
Chapter 3
Transition to High-Dimensional Chaos through a Global Bifurcation Abstract. We study a novel transition to high-dimensional (d ≈ 4) chaotic behavior in a system composed of three unidirectionally coupled Lorenz oscillators. The transition involves a global explosion that creates a highdimensional chaotic set, formed by an infinite number of unstable tori. The chaotic set becomes attracting after a boundary crisis, and exhibits multistability in a range of parameters, coexisting with two (stable) symmetry-related threedimensional tori (the only attractors before the crisis). These two tori disappear after a saddle-node bifurcation in which they annihilate with two threedimensional unstable tori.
3.1
Introduction
One of the most fundamental problems in the study of nonlinear dynamical systems is the characterization of the routes through which these systems undergo a transition to chaotic behavior. In low dimensional dissipative dynamical systems, chaos quite often appears through a few well characterized routes (or scenarios) [Eck81]: (a) the period-doubling cascade [Fei78]; (b) the intermittency route [PM80]; (c) the route involving quasiperiodic tori [NRT78, CY78]; (d) the crisis route [GOY83a]; (see also [BPV86, Ott93] for a survey). Another possibility is that chaos sets in through a global connection to a fixed point, as is the case of the Lorenz system [Spa82], and of Shil’nikov chaos [AAIS89, Hom93]. Low-dimensional chaos, occurring in at least three-dimensional dissipative flows, can be characterized in terms of the Lyapunov spectrum
52
Transition to High-Dimensional Chaos through a Global Bifurcation
by one positive and one null Lyapunov exponent, while the dissipative nature of the flow implies the existence of a third negative exponent such as the sum of all the exponents is negative. We will represent the Lyapunov spectrum as: (+, 0, −), while regarding the dimension of these attractors, say d, in principle it can be a typical dimension, such as the capacity, D0 , information, D1 , or the correlation, D2 , dimensions. Here we use the Lyapunov spectrum to estimate D1 through the Kaplan-Yorke formula: D1 ∼ 2+ [KY79a]1 , while one could make an analogous reasoning for maps. More recently, some studies have been devoted to the study of highdimensional chaos, that here we shall define as chaotic behavior with d > 3. It is clear that in order to have high-dimensional chaos one needs to increase the dimensionality of phase space, at least by one. An obvious possibility to transit to high-dimensional chaos is to take as starting point a low-dimensional chaotic attractor. This possibility has been usually considered in the context of desynchronization between coupled chaotic oscillators. A possible scenario, and probably the most studied one [HL99, KMP00] involves generating a hyperchaotic attractor, i.e., a chaotic attractor with two (or more) unstable directions (associated to positive Lyapunov exponents) from a low-dimensional chaotic attractor. Thus, in the simplest case this route implies (+, 0, −, −) → (+, +, 0, −), and, accordingly, D1 increases from 2+ to 3+ . Another possibility is to have a chaotic attractor with two null Lyapunov exponents. Although this situation is not generic, it has been shown [MGP+ 97, SMP00, Yan00] to occur in unidirectionally coupled chaotic systems through a symmetric Hopf bifurcation2 [MPL+ 97] (namely a Hopf bifurcation in the k = 1 mode). The transition can be characterized as: (+, 0, −, −, −) → (+, 0, 0, −, −), and thus, D1 increases quite abruptly from 2+ to 4+ , because the fourth exponent is smaller in absolute value than the positive one. Less obvious is the possibility of a direct transition to highdimensional chaos without an intermediate low-dimensional chaotic attractor. Restricting to autonomous ordinary differential equations, the P According to this formula, actually a conjecture, D1 = K + K j=1 (λj /|λK+1 |), being P K the largest integer such that K λ ≥ 0, where the λ are the Lyapunov exponents j j=1 j + ordered from larger to smaller. With, e.g., 2 we shall indicate the order of the integer contribution to the dimension. 2 One can claim that, close enough to the onset of the bifurcation, two Lyapunov exponents are zero, or, at least, very close to zero. In practical terms the situation is analogous to what happens when studying chaotic phase synchronization in two coupled R¨ ossler oscillators, where two Lyapunov exponents remain quite close to zero for a finite parameter range [RPK96]. 1
3.1 Introduction
53
transition to high-dimensional chaos has been found to be associated to quasiperiodicity. Thus, the works by Feudel et al. [FJK93], and by Yang [Yan00], observe a three-frequency quasiperiodic window before the system passes from two-frequency quasiperiodicity to high-dimensional chaos. A more geometrical view is contained in the route (to highdimensional chaos) from a two-dimensional torus reported by Moon [Moo97]. It comprises a global bifurcation whose structure amounts to adding one dimension to each building block of the Lorenz attractor. In the present thesis we shall describe a route to chaos whose probably most novel aspect is that it implies the sudden creation (i.e. without mediating low-dimensional chaos) of a high-dimensional chaotic attractor with D1 > 4. Our approach combines the computation of the Lyapunov exponents (as Refs. [FJK93, Yan00]) with an investigation on the global bifurcations (as Ref. [Moo97]) giving rise to the chaotic attractor. The chaotic set is first created through a double heteroclinic connection between cycles. This set, that is visible by the existence of transient chaos, becomes attracting through a boundary crisis. At this crisis the chaotic attractor touches its basin boundary that is constituted by two symmetry related unstable three-dimensional tori T3 (and their stable manifolds). The resulting chaotic attractor has a single unstable (chaotic) direction and two neutral directions, as indicated by its Lyapunov spectrum: (+, 0, 0, −, −, · · · ). This high-dimensional attractor is known as chaotic rotating wave (CRW) [MGP+ 97, SMP00]. The CRW may be created in a ring of three coupled chaotic oscillators in which the synchronized chaotic state (low-dimensional chaos) is destabilized by a k = 1 Fourier spatial mode3 . Our system of coupled Lorenz oscillators also exhibits a periodic rotating wave (PRW) [MPL+ 97, SM98, MPM98] (this route was found experimentally in Ref. [SM99, S´ an99]). The PRW is found for stronger coupling and it is the starting point of our study. It consists roughly in a dynamics where the synchronous k = 0 mode is in a steady state. In addition to its fundamental interest, the synchronization and the selection of particular phase relations is a fundamental topic in a biological context, ranging from neurology and brain function [RGL+ 99] to animal locomotion [GSBC99]. In this latter case, circular geometries of coupled cells are used for modeling central pattern generators [CS93, CS94] because the system’s symmetries provide the different phase patterns observed for different gaits. 3
If the ring were larger, other modes could become unstable [PMPP00].
54
Transition to High-Dimensional Chaos through a Global Bifurcation
The plan of this chapter is as follows. In Sec. 3.2 the system is described and an overall picture of the route to chaos is presented. Section 3.3 provides an analytical approximation to the state known as periodic rotating wave. Section 3.4 discusses at some detail the creation and the characterization of the two- and three-frequency quasiperiodic attractors found in the system. Section 3.5 presents the numerical evidences that have been used to understand the complex route to chaotic behavior presented by the system. Section 3.6 presents a characterization of the route to high-dimensional chaos through a return map similar to that of the Lorenz system. The goal of Sec. 3.7 is precisely to present and discuss at some length the route to chaos exhibited by the system. Finally, Secs. 3.8 and 3.9 present some further remarks on this work and a summary of the main results, respectively.
3.2
System and overall picture
As already advanced in the Introduction, the goal of the present chapter is to study the transition to high-dimensional chaos in the 9-dimensional system formed by three Lorenz [Lor63] systems coupled according to the synchronization method introduced in [GM95]. The evolution equations for the ring can be written in the form [MPL+ 97], x˙j = σ(yj − xj ) y˙j = R xj − yj − xj zj j = 1, . . . , N = 3 , (3.1) z˙j = xj yj − b zj where xj = xj−1 for j 6= 1, introduces the coupling. The periodic boundary conditions used in this work imply that x1 = x3 , while the following parameters are taken as in [SM99]: b = 3, R ∈ [29, 40] and σ ∈ [18, 22]. The study of this system in this region of parameters has been suggested by the results of the experimental study of three coupled Lorenz oscillators corresponding to these parameters [SM99]. This system exhibits synchronous chaos for R < Rsc ≈ 32.82. At Rsc the system exhibits a Hopf instability but from a chaotic state, yielding a behavior that was first found in rings of unidirectionally coupled Chua’s oscillators, and called a Chaotic Rotating Wave (CRW) [MGP+ 97]. The Hopf bifurcation exhibited by the system is called symmetric, as it is originated from the cyclic (Z3 ) symmetry of the ring. The CRW behavior is characterized by the fact that it is formed by the combination of two different dynamical behaviors, at least close to the onset of
55
3.2 System and overall picture
the symmetric bifurcation: the Lorenz waveform, characteristic of the synchronization manifold existing for R < Rsc (when the system exhibits chaotic synchronization) and the oscillation created by the symmetric Hopf bifurcation (that occurs in the subspace which is transverse to the synchronization manifold). The oscillation created in the symmetric Hopf bifurcation is characterized by a behavior in which neighboring oscillators differ by a phase of 2π/3 (2π/N in the general N -oscillator case), and because of this picture this behavior is the analog of a traveling rotating wave in a discrete system. This picture is valid close to onset, R & Rsc ; when increasing R one observes that this behavior changes, and for R > 35.26 the behavior of the system becomes periodic, but with a waveform still characteristic of the subspace transverse to the synchronization manifold, k = 1 Fourier mode. Two of such periodic behaviors, called Periodic Rotating Wave (PRW) in Ref. [MPL+ 97], are found depending on the initial conditions in the range R ∈ [35.26, 39.25] (approximately). At R = Rpitch ≈ 39.25 both solutions merge giving rise, through a pitchfork bifurcation, to a centered stable symmetric periodic behavior. Now, let us consider a useful representation in the study and characterization of these discrete rotating waves: the use of the corresponding (discrete) Fourier spatial modes [OS89, HCP94, MPL+ 97]. These modes are defined as follows, N 2πi(j − 1)k 1 X xj exp , (3.2) Xk = N N j=1
where N = 3, as already indicated, and i is the imaginary unit. The evolution equations in terms of these modes are as follows, X˙ 0 = σ(Y0 − X0 ) Y˙0 = R X0 − Y0 − X0 Z0 − X1 Z1∗ − X1∗ Z1 ∗ ∗ ˙ Z0 = X0 Y0 − b Z0 + X1 Y1 + X1 Y1 (3.3) X˙ 1 = σ(Y1 − X1 ) ˜ X1 − Y1 − X0 Z1 − X1 Z0 − X ∗ Z ∗ Y˙1 = R 1 1 ∗ ∗ Z˙1 = X0 Y1 + X1 Y0 − b Z1 + X1 Y1 ˜ = R exp(2πi/3). with R
3.2.1
Lyapunov exponents and attractors
In previous studies [MGP+ 97], it was demonstrated that the computation of the transverse Lyapunov exponents to the synchronization
56
Transition to High-Dimensional Chaos through a Global Bifurcation
Lyap. spectr. (−, −, −, −, · · · ) (+, 0, −, −, · · · ) (+, 0, 0, −, · · · ) (+, +, 0, −, · · · ) (0, 0, 0, −, · · · ) (0, 0, −, −, · · · ) (0, −, −, −, · · · )
Attractor Fixed Point Synchronous Chaos Chaotic Rotating Wave I & III Chaotic Rotating Wave II 3-Torus 2-Torus Periodic Rotating Wave
Table 3.1: Correspondence between Lyapunov spectra and attractors. Fourth to ninth Lyapunov exponents are negative in all cases.
manifold allowed to know (approximately) the region of the parameter space where synchronized chaos was stable. Nonetheless, if one wishes to know more about the spatio-temporal structures emerging from desynchronization, it is worthwhile to compute the Lyapunov exponents (LEs) for the whole set of variables (nine in our case). Since Lyapunov exponents are related to the exponentially fast divergence, or convergence of nearby orbits, they can be used to identify attractor types. With this aim we computed the LEs . The method used for the calculation has been the one developed by Benettin et al. [BGGS80] and Shimada and Nagashima [SN79], and described in [WSSV85]. For the orthonormalization process we have used a modified Gram-Schmidt method. The integration of the system of differential equations and the copies of the linear system have been done by means of an adaptive stepsize algorithm based in a fifth order Runge-Kutta method [PTVF92]. The results presented here in Fig. 3.1 were obtained computing a trajectory of 8 × 104 t.u., after a transient of 105 t.u. The results are presented in Fig. 3.1, where the zones on the plane Rσ (b fixed to b = 3) with different Lyapunov spectra, and consequently with different attractors can be observed (see Table 3.1). Thus, we find a region (denoted FP in Fig. 3.1) where all the LEs are negative, which indicates that p oscillators “collapse” to the same fixed point p the three C± = [± b(R − 1), ± b(R − 1), R − 1]. Also, there exists a region with synchronous chaos (SC) where the Lyapunov spectrum contains one positive and one vanishing exponents. For larger values of the parameter R, the synchronized state becomes unstable through a supercritical blowout bifurcation. This means that as R is increased, more and more unstable periodic orbits embedded into the
57
3.2 System and overall picture
22
21.5
FP
I
III
SC
21
2
CRW
20.5
σ
II
T
20
19.5
PRW
19
18.5
18 29
30
31
32
33
34
35
36
37
38
R Figure 3.1: Regions of the (R, σ) plane where the marked states are achieved by a system composed of three Lorenz oscillators. Notation: FP, fixed point; SC, synchronous chaos; CRW, chaotic rotating wave (subregion II exhibits a second LE above zero – i.e. hyperchaos–, we took 10−3 as a cut-off to consider λ2 positive); T2 , two-frequency quasiperiodicity; PRW, periodic rotating wave. Symbols “x” and “+” indicate the loci of RBC and RH (see Eq. (1.16)), respectively.
synchronous attractor become transversely unstable, in such a way that above a value of R the synchronized state becomes unstable on average. It is important to emphasize that we are in a case of local riddling, since the instabilities of the UPOs occur through supercritical Hopf bifurcations. This oscillatory instability gives rise to the previously mentioned structure called Chaotic Rotating Wave (CRW). This state was first found for a ring of Chua’s circuits [MPL+ 97, MGP+ 97, SMP00] and later for a ring of Lorenz oscillators [SM99]. Also, for coupled maps in a ring, an analogous behavior was found in [YP99]. The CRW is characterized by a fast oscillation, associated with a 2π/N phase difference between neighboring oscillators (where N is the number of units in the ring), superimposed to the chaotic motion. We distinguish three regions within this state depending on the LEs. In regions I and III, there exist one positive and
58
Transition to High-Dimensional Chaos through a Global Bifurcation
two vanishing Lyapunov exponents (since the degeneracy of this exponent is not theoretically proved, it is plausible that one exponent has very small magnitude and is numerically indistinguishable from zero). In region II there are two (clearly) positive and one vanishing LEs. At larger values of R we observe a small region that exhibits twofrequency quasiperiodicity (T2 ). The Lyapunov spectrum does not contain any positive exponent, hence chaos has disappeared; instead, there are two null Lyapunov exponents which correspond to the two incommensurate frequencies of the quasiperiodic regime. Finally, for large R we find periodic dynamics (PRW) and, accordingly, only one Lyapunov exponent is zero, whereas the rest are negative. It is interesting to notice the wedge-shaped region in Fig. 3.1 where, apparently, a fixed point state replaces the CRW dynamics. This is due to the fact that inside a stripe-shaped region of the plane (R, σ) the Lorenz system exhibits tri-stability: two fixed points (C± ) and the chaotic attractor. The fixed points become unstable through respective subcritical Hopf bifurcations at RH ; whereas, as explained by Yorke and Yorke [YY79], the chaotic attractor is born (or dies) at a boundary crisis at some RBC < RH (σ fixed). The value of RBC cannot be found analytically, but it is easy to obtain it numerically, because at this value there exist two heteroclinic connections between the fixed point at the origin and both unstable cycles surrounding C+ and C− . The loci of RBC and RH on the (R, σ) plane are depicted in Fig. 3.1 with “×” and “+” symbols, respectively. Between both lines the Lorenz oscillator is tri-stable. The basin of attraction of the (synchronized) chaos is much larger than the basins of C± . For this reason, at the right of RBC synchronized chaos, instead of fixed points, is the most typical behavior. Nonetheless, when synchronized chaos undergoes the instability that should lead to the CRW, the existence of tri-stability affects strongly the dynamics. The CRW is no longer an attractor; and instead after a chaotic transient, all the oscillators decay to the same fixed point (C+ or C− ). This explains the existence of the mentioned wedge-like (see Fig. 3.1) region where the system exhibits exclusively a fixed point behavior.
3.2.2
Behaviors along the line σ = 20
We shall not focus here on the transition from synchronous chaos to CRW; instead, we are more interested on the transitions from PRW to CRW, i.e. we go ‘from order to chaos’. The reader should notice that by CRW we refer to a high-dimensional chaotic attractor characterized
59
3.2 System and overall picture 20
Rbc
Rh1
15
10
X0
5
0
−5
−10
−15
−20 −4 10
Rh2 −3
10
Rpitch −2
−1
10
10
0
10
1
10
R−35.093 Figure 3.2: Bifurcation diagram. Points represent intersections with the Poinacar´e section Im(X1 ) = 0, Im(X˙ 1 ) > 0. The logarithmic scale in the x-axis has been adopted to better resolve dynamics existing in quite different interval ranges of R.
by an oscillation with a 2π/3 phase shift between neighboring oscillators superimposed to an underlying chaotic behavior. Taking a Poincar´e surface of section (Im(X1 ) = 0, Im(X˙ 1 ) > 0), and plotting the coordinates X0 of the intersections we obtain Fig. 3.2. Going from right to left, we first distinguish a pitchfork bifurcation (R = Rpitch ) that mediates the transition from a centered PRW to a pair of symmetry related PRWs. At R = Rh1 the PRWs undergo a Hopf bifurcation giving rise to two symmetry related two-frequency quasiperiodic attractors (the flow fills densely the surface of a torus: T2 ). At a lower value of R = Rh2 , two three-frequency quasiperiodic attractors (T3 ) are born at a secondary Hopf bifurcation. Finally, Rbc marks the point where the CRW appears and the trajectory visits positive as well as negatives values of X0 . The time series for the different behaviors studied in this work are shown in Fig. 3.3. Note that the third frequency, that appears as the T3 is born, manifests as a very slow modulation of the size of the former T2 (the time scale has been broadened in Fig. 3.3(c) to appreciate this).
60
Transition to High-Dimensional Chaos through a Global Bifurcation Time (t.u.) 0
0.5
Time (t.u.)
1
1.5
2
0 3
−5
2
x ,x ,x
−10
1
1.5
2 b)
−5 −10
1
x1, x2, x3
a)
0.5
−15 −15
3 2
x ,x ,x
−10 −20
0
1
x1, x2, x3
d)
20
c)
0
−20 0
100
200 Time (t.u.)
300
0
100
200 300 Time (t.u.)
400
500
Figure 3.3: Time series of the system Eq. (3.1) with σ = 20 and b = 3: (a) R = 35.5 Periodic Rotating Wave (b) R = 35.2 T2 (c) R = 35.095 T3 (d) R = 35.093 Chaotic Rotating Wave. Note the different time scale for each panel.
More precisely, a view of the parameter region that will be considered in this work can be found in Fig. 3.4, in which the Lyapunov spectrum corresponding to the four largest Lyapunov exponents is presented. In Fig. 3.4(b), seen from right to left, the two symmetry-related waves (limit cycle attractors) exhibit a (supercritical) Hopf bifurcation when R is decreased, namely at R = Rh1 ≈ 35.26, yielding two symmetry-related twofrequency quasiperiodic attractors (two vanishing Lyapunov exponents). Diminishing R further, the system exhibits another (supercritical) Hopf bifurcation at R = Rh2 ≈ 35.0955, that yields two symmetry-related threefrequency quasiperiodic attractors (three vanishing LEs). Decreasing R furthermore the system exhibits a boundary crisis, at R = Rbc ≈ 35.09384, in which the chaotic attractor is born (or destroyed when seen from the opposite side). The chaotic attractor is characterized by two vanishing (at least, very approximately) Lyapunov exponents, and a single positive LE that is larger than the absolute value of the fourth LE. This implies, according to the Kaplan-Yorke conjecture, an information dimension D1 > 4. As we shall see below (see Sec. 3.5.3) this dimension is genuine. Take into account that the K-Y conjecture cannot be stated oversimply; for example, when synchronous chaos exists, a small Lyapunov exponent may indicate a transverse contraction to the synchronization manifold, and as long as this direction does not participate in the stretching-and-folding mechanism (associated to chaos) such a Lyapunov exponent should not be considered when calculating the dimension. The presence of 2-D and 3-D quasiperiodic attractors may lead to
61
3.3 The centered periodic rotating wave: analytical solution
1
0.15 33
34
(a)
35
λ1 λ2 λ3 λ4
0
35.094
(b)
35.096
T3
T2
−0.03
0.5
−3 −4 x 10
λ
i
0.05
λi
35.098 0
0.1
0
FP
SC
PRW
"Pitch."
PRW
0 CRW
CRW
T2
−0.05 −0.5 R1
30
32
34
36
R
38
40
−0.1 35
35.1
35.2
35.3
35.4
R
Figure 3.4: (a) The four largest Lyapunov exponents of a ring of three unidirectionally coupled Lorenz systems. The inset shows from second to fourth LEs in the interval with CRW dynamics. It may be seen the the second LE becomes slightly positive which indicates the existence of hyperchaos. (b) Detailed figure of the transition from PRW to CRW; the existence of three-frequency quasiperiodicity is confirmed in the inset where three vanishing LEs exist.
think that chaos appears through a quasiperiodicity transition to chaos (see Section 3.4). However, it will be shown below that the chaotic attractor appears at a boundary crisis, and coexists with the two T3 attractors until the latter are destroyed as each of them collides with a twin unstable T3 , at R = Rsn = 35.09367. We shall show (from Section 3.5) that, indeed, the system exhibits a global bifurcation that implies the sudden creation of an infinite number of unstable 3-D tori. Thus, chaos is created through a global connection in which the reflection symmetry of the system seems to play a fundamental role, analogously to what happens for the Lorenz system [Spa82]. Although symmetry plays an analogous role as in Lorenz model, the higher number of dimensions available in phase space implies a more complex route to (high-dimensional) chaos in this system. A schematic diagram of the whole set of bifurcations linking the PRW and synchronous chaos is shown in Fig. 3.5. As mentioned above, in the interval of R where the CRW is found, the shape of the attractor changes. Anyway, in this work we are not interested in the transitions between different types of CRWs.
3.3
The centered periodic rotating wave: analytical solution
Inspection of the dynamics observed under simulation of the Eqs. (3.1,3.3) suggests to try the following ansatz for the centered periodic
62
Transition to High-Dimensional Chaos through a Global Bifurcation
boundary crisis
“Hopf blowout” SYNCHRONOUS CHAOS
“saddle-node bif.” TE S CO
R=
32.82
Rsc
EX
35.09367
Rsn
I
NC
‘heteroclinic explosion’
CHAOTIC TRANSIENT
H.-D. CHAOTIC ATTRACTOR (CRW) E
Hopf bif.
(2)T 35.09384
Rbc
3
35.0955
Rh2
pitchfork bif.
Hopf bif.
(2)T
2
35.11
Rexpl
(2)T 35.26
Rh1
1
1
T
(PRW)
39.25
Rpitch
Figure 3.5: Diagram representing schematically the transitions from synchronous chaos (left) to a PRW (right) as R is increased.
rotating wave:
X0 = Y0 = 0
(3.4)
Z0 = const.
(3.5)
X1 = AX eiωt
(3.6)
Y1 = AY ei(ωt+φY )
(3.7)
Z1 = AZ ei(−2ωt+φZ )
(3.8)
where time invariance lets us set φX = 0. We first examine the effect of X1 , Y1 , Z1 on the dynamics of the zero mode, to check whether it is compatible with a steady state for the mode k = 0. We observe that the equations for the mode k = 0 in Eq. (3.3) are identical to the equations for a Lorenz oscillator with two forcing terms K and L, K = X1 Z1∗ + X1∗ Z1
(3.9)
L = X1 Y1∗ + X1∗ Y1 .
(3.10)
Substituting our ansatz (Eqs. (3.6-3.8)) we obtain: K = 2AX AZ cos(3ωt − φZ )
(3.11)
L = 2AX AY cos(φY ).
(3.12)
The term L is constant in time, whilst the contribution of K consists in a oscillation with frequency 3ω and zero mean value. The value of ω is experimentally found to be appreciably larger than the natural frequency of the Lorenz oscillator. Therefore, we take K = 0 when working with the equations for the mode k = 0. In this way, to take X0 = Y0 = 0 and Z0 = const = L/b is a good approximation. Note also that the oscillating
63
3.3 The centered periodic rotating wave: analytical solution
form proposed for the mode k = 1 is a solution of the differential equations (if X0 = Y0 = 0, Z0 = const.). Not only the functional form, but the numerical values may be computed taking the ansatz above as starting point. We substitute our ansatz in the equations (recall that X0 and Y0 vanish) obtaining: iωAX iωAY eiφY −2iωAZ eiφZ Z0 =
L b
= σ(AY eiφY − AX )
(3.13)
˜ − Z0 − AZ e−iφZ )AX − AY eiφY = (R
(3.14)
= AX AY e−iφY − bAZ eiφZ
(3.15)
2AX AY cos(φY ) . b
=
(3.16)
Then, we have three complex equations and a real one with seven unknowns: (AX , AY , AZ , φY , φZ , ω, Z0 ). From the first and third equations we obtain the following relations: iω )AX σ
(3.17)
1 − iω σ = A2 , b − 2iω X
(3.18)
AY eiφY = (1 + iφZ
AZ e
which allow to express Z0 (see Eq. (3.16)) as a function of AX : Z0 =
2A2X . b
(3.19)
Finally, substituting Eqs. (3.17, 3.18, 3.19) into Eq. (3.14) we get, discarding the trivial solution AX = 0, a complex equation with two unknowns (AX and ω): 2 1 + iω/σ iω 2 ˜ + AX = R − 1 + (1 + iω). (3.20) b b + 2iω σ We get one equation for the real part, 2 b + 2ω 2 /σ ω2 R 2 + 2 A = − − 1, X b b + 4ω 2 σ 2
(3.21)
and another for the imaginary part: (b/σ − 2) ω 2 A = b2 + 4ω 2 X
√
1 3R −ω 1+ . 2 σ
(3.22)
Assuming ω 2 > σ 2 b2 = 9, Eqs. (3.21,3.22) become highly simplified:
64
Transition to High-Dimensional Chaos through a Global Bifurcation
2 8 + A2X σ b b − 2 A2X σ
ω2 R − −1 σ 2 "√ # 3R 1 ∼ −ω 1+ . = 4ω 2 σ ∼ = 4
(3.23) (3.24)
Casting the value of A2X as a function of ω 2 thanks to Eq. (3.23) and introducing the result in Eq. (3.24) we obtain a quadratic equation for ω: p −β + β 2 − 4αγ 2 αω + βω + γ = 0 → ω = (3.25) 2α where the + sign is taken because we are assuming a high frequency (recall that we are considering ω 2 b2 ), and: α = 1+ √
1 1− − σ 1+
b 2σ 4σ b
= 1.017 . . .
3R 2 (R + 2)(2 − b/σ) . 4 (1/σ + 4/b)
(3.26)
β = −
(3.27)
γ =
(3.28)
If we consider that 4αγ/β 2 ∝ 1/R 1, then, we may approximate ω from Eq. (3.25) by: γ 1 ω ≈ −β − 2 . (3.29) α β In Fig. 3.6, we compare the numerically observed frequencies with the ones obtained from Eq. (3.25) as a function of R. The theoretical values of ω provide, using Eq. (3.24), the values of AX , so we show in Fig. 3.6 the maximum and minimum values of AX as a function of R. Also it must be remarked that X0 and Y0 do not depart too much from zero. Thus for R = 40, X0 and Y0 remain respectively within ±0.05 and ±0.3. The numerical results agree with the analytical ones for R > Rpitch ≈ 39.25. Below this point the centered solution renders unstable and two symmetry-related cyclic solutions appear. For these solutions X0 , Y0 are no longer close to zero, and the analysis of the new solutions becomes much more convoluted because cross-terms arise, yielding other components of the Fourier series of the mode k = 1.
65
3.4 Transition to quasiperiodic behavior 55
12
50
11 10
ω
AX
45 9
40 8 35 30
7 40
45
50
55
6
60
40
45
R
50
55
60
R
34.5
7.5
AX
ω
34
33.5
Amax X Amin X
7
33
32.5
39
39.5
40
R
40.5
6.5
39
39.5
40
40.5
R
Figure 3.6: Numerical results for the frequency ω (left panels) and the max. and the min. values of the amplitude of the coordinate X1 (right panels) as function of R. The lower panels show zooms at the pitchfork bifurcation Rpitch . Solid lines correspond to theoretical predictions (see text).
3.4
Transition to quasiperiodic behavior
As described above, the (symmetric) periodic rotating wave, that is the starting point of our analysis, exhibits a transition to quasiperiodic behavior T3 , mediated by a pitchfork bifurcation, in which two (asymmetric) periodic rotating waves are born; the latter, in turn, exhibit two consecutive Hopf bifurcations. A small note is in place about the stability of this type of unusual attractors, as in many places in the literature it is stated that these attractors are intrinsically unstable. The question is: which is the behavior that one expects to find typically in a system where n frequencies have been generated (n ≥ 2): chaotic or (quasi)-periodic? According to the Newhouse-Ruelle-Takens (NRT) Theorem [NRT78], if one has a flow on a torus Tn = Rn /Zn (n ≥ 3) every neighborhood of the flow contains a vector field with a strange attractor. (A weaker version of the Theorem was proved by Ruelle and Takens [RT71], stating basically the same result for n ≥ 4). It is clear that even in the n = 2
66
Transition to High-Dimensional Chaos through a Global Bifurcation
case, not covered by the NRT Theorem, a torus can be destroyed following one of the routes proposed by Afraimovich and Shil’nikov [AAIS89]. Thus, the destruction of the torus due to the interaction of resonances can lead to chaotic behavior, as in the route first studied by Curry and Yorke [CY78]. Quasiperiodic motion, living on a torus, is quite fragile, because resonances constitute a fat fractal that densely fills parameter space (although their joint Lebesgue measure may be small) [Ott93]. In the simplest setup, corresponding to the analysis of the circle map [Arn65, Sch88, Ott93] and valid for the analysis of nonautonomous systems, rational values for the winding number, that have zero measure when the nonlinear parameter is zero, are the seeds for finite measure regions (so-called Arnold tongues) in which the winding number exhibits a certain rational value p/q (q > p). The study of autonomous systems, in which a T2 torus, is born in a Neimark-Sacker bifurcation [Kuz95], is a bit more involved. First of all because the regions exhibiting T1 and T2 behavior (or, in general, Tn and Tn+1 ) are separated by a line (the Neimark-Sacker bifurcation line) that needs two parameters to be defined. In addition, the winding number changes along this line as a function of the system parameters. Again, points in this line for which the winding number is rational, p/q, exhibit a resonance, and as one goes into the T2 region one gets a resonance horn emanating from each rational; in general this is a multidimensional object in parameter space. For q ≤ 4, the so called ‘hard resonances’, the structure is more involved in general, and has to be worked out in a case by case basis, i.e. for specific values of p and q. The boundaries of the resonances are characterized by the creation of a pair of orbits (one stable and the other unstable) in a saddle-node bifurcation on the surface of the torus as one enters the resonance horn. In our case we have not detected resonances in the T2 regime. This may occur due to the fact that along R, the winding number does not cross hard resonances, the largest denominator is q = 11 (corresponding to the winding number 4/11), so the resonance horns may be quite small. Perhaps the different nature of both frequencies (one is spatial with the phase shifts along the ring whereas the other is related to a uniform oscillation) is responsible of the weak interaction between them. Concerning the existence of a 3D-torus –which in principle would be forbidden by the NRT theorem– it must be stressed that numerical studies [GOY83a, Bat88], and also a number of experiments [GB80, LC89, AR00], give support to the existence of three-frequency (and even fourfrequency, and, perhaps, higher-frequency) quasiperiodic attractors. The key in understanding the prevalence of these high-frequency quasiperiodic
3.4 Transition to quasiperiodic behavior
67
Figure 3.7: Plot of a Poincar´e cross section for the system (3.1) with σ = 20 and b = 3: (left) R = 35.1, for which the system exhibits two–frequency quasiperiodic behavior (one– frequency periodic in the Poincar´e section); (right) R = 35.095, for which the system exhibits three–frequency quasiperiodic behavior (two–frequency quasiperiodic in the Poincar´e section). The representation has been carried out by using the (complex) mode representation of the system (3.3), and the three axes correspond to the x coordinate and are the uniform (k = 0) mode, and the real and imaginary parts of the k = 1 mode. The Poincar´e cross section has been defined by the condition: Z0 = 34 ≈ R − 1 with Z˙0 > 0.
attractors is to understand that the existence of robust strange attractors in its neighborhood does not mean that the measure of quasiperiodicity in a range of parameters is necessarily small [Ott93, Eck81, Ash98]. In some cases, such as in the work of Feudel et al. [FJK93, ASFK94, FSKA96] convincing arguments have been advanced regarding the robustness of 3D-tori in systems with conserved quantities. In their study of a model of a solar dynamo [FJK93] these authors showed that in a range of parameters one of the variables in their analysis was cyclic and thus, could be decoupled from the equations. Thus, such a system could be considered as a three-frequency torus consisting of a generic two-frequency torus plus an extra frequency arising from this conservation law. It is in this sense that the conclusions of the NRT Theorem [NRT78] should not apply to this case. In our case it can be numerically checked that the system exhibits three-frequency quasiperiodic behavior by looking at the Poincar´e cross section (see Fig. 3.7) of the torus in phase space (and also
68
Transition to High-Dimensional Chaos through a Global Bifurcation
by a careful computation of the Lyapunov spectrum). Also, Ref. [Yan00] reports the existence of three-frequency quasiperiodicity in a ring of coupled Lorenz oscillators. We believe that the key point is the very small frequency associated to the secondary Hopf bifurcation, that is about two orders of magnitude smaller than the other two. It appears as a very slow modulation of the 2D-torus (see Fig. 3.3(c)) so the relative winding numbers have large denominators. A more physical point of view is the possibility of applying some kind of adiabatic principle to understand the very weak interaction of this very low frequency with the other two. In fact, a case of a third very low frequency associated to a 3D-torus has also been observed in Refs. [WN87, LM00]. Let us finish this section by pointing out that when quasiperiodic motion is present in a dissipative dynamical system one is, at first sight, inclined to think (with good sense) that the system will exhibit chaos due to global connections originating in the Arnold tongues (because of to its above mentioned fragility) that typically lead to wrinkling and corrugation of the torus. Instead, as it will be shown in subsequent sections, this system is an example in which chaos does not arise in this way, but rather in a global connection that is quite linked to the reflection symmetry exhibited by the system. See Ref. [Moo97] for a further example of a route to chaos in a system exhibiting quasiperiodic behavior through a global connection, and not due to resonant interactions of the torus (that imply themselves global connections, but of a different kind).
3.5
Numerical evidences of the route to chaos exhibited by the system
In this section we are going to analyze the route through which a chaotic attractor is born in this system. As this has some similarities to the classical route to chaos for an isolated Lorenz system, we are going to draw some useful analogies with this system, although it is warned that the analogy is not complete (otherwise, we would not have a new route to chaos). As it can be found in textbooks [Ott93], the Lorenz system exhibits a route to chaos through a double homoclinic connection of the fixed point at the origin, that fixing σ and b occurs for a particular value R = RHOM 4 . As may be seen in Fig. 1.6, this double homoclinic orbit has the shape of the butterfly (taking into account how the unstable directions reenter the 4
For σ = 10 and b = 8/3: RHOM ≈ 13.926, RBC ≈ 24.06, and RH ≈ 24.74.
3.5 Numerical evidences of the route to chaos exhibited by the system
69
saddle point at the origin, being this dictated by the reflection symmetry), and this implies [AAIS89] that a chaotic set is born for R > RHOM (a homoclinic explosion [YY79, Spa82]). The closure of this set is formed by the infinite number of unstable periodic orbits that can be classified according to their symbolic sequences of turns around the right (R) and the left (L) fixed points (C+ and C− ) [Spa82]. The existence of these UPOs reflects the dramatic change undergone by the stable manifold of the fixed point that allows initial conditions at one side of the phase space jump to the other side (before falling to one of the two symmetry related stable fixed points C+ and C− ), being these jumps impossible for R < RHOM . For a larger value of R, R = RBC , the chaotic set becomes stable in a boundary crisis, that occurs precisely when the two shortest symmetry related length1 unstable periodic orbits (R and L) located at each lobe, and that at R = RHOM coincide with the two homoclinic orbits, shrink, such that the chaotic set has a tangency with these two orbits [Spa82]. At this value of R there exists a double heteroclinic connection between the equilibrium at the origin and the mentioned length-1 UPOs. For RBC < R < RH these two orbits (and their respective tubular stable manifolds) form the basin boundaries between the chaotic attractor and the two stable asymmetric fixed points (C± ), and, consequently, the system exhibits multistability. These two fixed points loose their stability in a collision (a subcritical Hopf bifurcation) with the two mentioned length-1 orbits, that occurs when these shrink to a point coinciding with (C+ , C− ), that become unstable from this point. First of all, and in analogy to our analysis in the Section devoted to the analysis of the quasiperiodic motions, we shall reduce the dimensionality of the problem by eliminating the fast frequency, that involves a phase shift by 2π/3, as it leads to a conserved quantity (this time lag) and consequently a null Lyapunov exponent. In the mode representation of Eq. (3.3) this amounts to perform a cut through the Poincar´e section Im(X1 ) = 0. So, in visualizing objects cycles will become fixed points, T2 -tori cycles and T3 -tori will become T2 -tori, although we shall refer to these objects referring to the complete phase space, and not to this cut. Now, we shall shed light on the nature of the transition by performing a number of numerical experiments and establishing analogies with the route to chaos of the Lorenz system.
3.5.1
Coexistence between 3D-torus and CRW
The first important remark about our system of three coupled Lorenz oscillators is that the two T3 attractors are not directly involved in the birth
70
Transition to High-Dimensional Chaos through a Global Bifurcation
R = 35.093 + r × 10−5 0 −0.5 −4
λi (×10 )
slave locking
−1
−1.5 −2 λ1 λ2 λ3 λ
−2.5 −3
4
−3.5
67
70
73
76
79
82
r Figure 3.8: Blowout of the four largest Lyapunov exponents for the T3 attractors in the region in which they coexist with the chaotic attractor. The fourth Lyapunov exponent approaches zero in a square-root fashion.
of the high-dimensional chaotic attractor. Indeed, one can easily perform the following numerical experiment for values of R close to the critical value at which the (high-dimensional) chaotic attractor is born (more precisely for Rsn < R < Rbc ). If one chooses an initial condition that corresponds to a T3 attractor obtained in the nearby range Rbc < R < Rh2 , a stable T3 is obtained, although not so large perturbations in this initial condition yield chaotic behavior. This evidence implies that the system exhibits multistability between the high-dimensional chaotic and the T3 attractors in the range Rsn < R < Rbc , with the latter attractor having a relatively small basin of attraction. This remark is important, because it implies that the highdimensional chaotic attractor is not created through some route involving the T3 attractors, e.g., through the some kind of merging of these attractors. A detailed view of the behavior of the four largest Lyapunov exponents
3.5 Numerical evidences of the route to chaos exhibited by the system
71
Figure 3.9: Schematic of a ‘slave locking’. A portion of a section of the tori (stable and unstable) are shown with bold lines. The tori are not differentiable at the foci.
is represented in Fig. 3.8. It can be seen that close to R = Rsn = 35.09367 the fourth Lyapunov exponent exhibits a square-root profile as expected for a saddle-node bifurcation. This suggests that the stable T3 is approaching an unstable T3 . Also one can appreciate quite clear lockings5 where the third and fourth Lyapunov exponents become equal. As R approaches Rsn the existence of a smooth invariant 3D-torus cannot be guaranteed, because if one locking occurs the strength of the normal contraction can be of the same order than the rate of attraction of 2D-tori on the 3D-torus. Thus, in Fig. 3.9 a possible scenario for the interaction of the tori, that would explain the lockings observed in Fig. 3.8 is presented. Because of the extremely small transversal stability of the T3 (represented by the fourth exponent), for R & Rsn , the transversal direction ‘slaves’ the tangential one (represented by the third exponent). Then, we observe two identical non-vanishing exponents that indicate the existence of a stable focus-type T2 on the surface of the T3 . The T3 continues to exist but is non-differentiable at the stable T2 located on its (hyper)surface. The participation of the unstable 3D-torus in the final annihilation of the stable one is not trivial (the existence of the unstable T3 is corroborated by Sec. 3.5.4). As we said before, the fast spatial frequency can be considered as a kind of forcing and interacts very weakly with the other two. Then we can argue that we may take use of the routes leading to the destruction of a 2D-torus. It can be found in [AAIS89] that a (stable) torus may be annihilated by another (saddle) torus after a locking of both that leads to a saddle-node bifurcation between twin cycles on both tori. 5
Recall that the classical theory for two-frequency quasiperiodicity tell us that when a two-torus locks a pair of stable-unstable orbits are born on its surface through a saddlenode bifurcation. According to this, one of the Lyapunov exponents becomes slightly negative, indicating the small attraction along the surface of the torus of the stable cycle (the new attractor of the system). Generically, when a parameter varies, the torus visits some (formally infinity) Arnold tongues where its rotation number is rational, as expected from the existence of a stable periodic orbit on its surface. Analogous resonances appear for 3D-tori.
72
Transition to High-Dimensional Chaos through a Global Bifurcation
If the first saddle-node bifurcation involves the stable cycle no attractor remains in the neighborhood of the former T2 ; in the other case the T2 is destroyed but the stable orbit continues to exist as an attractor (which probably undergoes the saddle-node bifurcation a bit later). We speculate that one of these routes, with the proper substitution of cycles and two-tori by two- and three-tori respectively, occurs in our system. Unfortunately, because of the precision of our computations we are not able to resolve this fine structure.
3.5.2
Heteroclinic explosion
The second important remark is that one can find a value of R = Rexpl ≈ 35.11 that defines a clearcut transition in the way in which transients approach the attractors (that for R ∼ Rexpl are two T2 -tori). For R > Rexpl the basin of attraction of each T2 is quite simple. But below, R < Rexpl trajectories may tend asymptotically to one of the T2 after visiting the neighborhood of the other torus. Following the analogy with the Lorenz system, we conjecture that precisely at R = Rexpl a global bifurcation occurs, and past this value (R < Rexpl ) an infinite number of unstable objects are created. To check this we stabilized the symmetric PRW (an unstable fixed point in the Poincar´e map) by a Newton-Raphson method and observed the fate of the trajectories starting from (approximately) that solution. In Fig. 3.10 the evolution of the trajectories for two values of R above and below Rexpl is shown. After approaching one the asymmetric PRWs (its location is marked with black circles) the trajectory jumps or not to the other side. The result obtained for Rexpl is the same if one takes as starting point one of the asymmetric PRWs.
3.5.3
Four-dimensional branched manifold
The third important numerical remark refers to the Lyapunov spectrum (Table 3.2) of the different attractors in the multistability region (of course, and due to symmetry, the Lyapunov spectra of the two T3 attractors are identical). As one can see, fifth to ninth LEs are quite similar, indicating that they are approximately embedded in the same four-dimensional space, part of the total phase space. Thus, we state that, as the dimension of the chaotic attractor is about four, the dynamics can be simplified to the study of a fourdimensional branched manifold. A theoretical justification of this statement
73
3.5 Numerical evidences of the route to chaos exhibited by the system 20
X0
15
10
5 R=35.12 0
0
50
100
150
100
150
200
250
300
350
400
200
250
300
350
400
20
X
0
10
0
−10 R=35.10 −20
0
50
Time(t.u.)
Figure 3.10: Numerical experiment showing trajectories starting in an initial condition in the symmetric unstable PRW: (a) for R = 35.12, a condition before the explosion (b) for R = 35.10 just past the explosion.
is impossible, but one can argue along the lines of (a higher-dimensional generalization of) the Birman-Williams Theorem [Gil98]. The theorem states that –for a hyperbolic low-dimensional chaotic attractors– under identification of points in phase space with the same future, the strange attractor projects down to a two-dimensional branched manifold. Under such projection, no orbit cross through each other and their topological organization is invariant. The B-W theorem requires the strange attractor to be hyperbolic, and this is clearly not fulfilled even for the Lorenz system alone (mostly due to the fact that the fixed point at the origin is part of the chaotic set), nonetheless this theorem has demonstrated to be a useful tool to manage usual (i.e. non-hyperbolic) chaotic attractors. In our case we can argue that our high-dimensional chaotic attractor has, according to Kaplan-Yorke conjecture [KY79a], a value of the information dimension D1 that is slightly larger than 4. And the mentioned projection (or “deflation” of the attractor) occurs when the dimension approaches 4, considering that the negative Lyapunov exponent that does
74
Transition to High-Dimensional Chaos through a Global Bifurcation
λi λ5 λ6 λ7 λ8 λ9
CA −5.2547 −5.2548 −18.6124 −18.6124 −24.2734
T3 −5.2027 −5.2027 −18.6519 −18.6519 −24.2904
Table 3.2: The five smallest Lyapunov exponents for the two attractors (CA: chaotic, and T3 : three-frequency quasiperiodic) coexisting at R = 35.0938.
not contribute to the integer part of D1 is strongly dissipative. That is, see Table 3.2, λ5 → −∞ (see [Gil98] for a presentation of this argument), D1 = 4 +
λ1 + λ 4 →4 |λ5 |
(3.30)
where, physically, one is increasing the dissipation without bound. In order to check the validity of the conjecture of Kaplan-Yorke for our attractor, we have also measured the correlation dimension D2 with the algorithm of Grassberger and Procaccia (see Fig. 3.11), obtaining a value close to 4 (recall that D1 ≥ D2 ). A more correct picture of the attractor amounts to consider that this 4-dimensional manifold is a leave, and that many of these thin leaves make up the attractor (4+ ). Of course that this reduced 4-dimensional picture can be only considered to be a more or less faithful representation of the system when it is orbiting in one of the subspaces created by the symmetry hyperplane. In the epochs in which a trajectory jumps to the other side (subspace), which means reinjections through an extra dimension, it is when the existence of branching is needed in order to the trajectory not to intersect itself. In our case the ‘tear point’ is the symmetric PRW. This is in complete analogy with what happens with the Lorenz system, where the attractor can be understood as a template composed of a branched twodimensional manifold with a tear point at the origin. Rotations around one of the lobes are roughly planar, but reinjections between the two (planar) lobes, that form themselves an angle, involve the third dimension. An important remark concerning this 4-dimensional picture is that a T3 torus has “enough dimension” to divide R4 , in two regions (just the same as a cycle divides R2 ). Then in the regime with coexistence, between chaos and three-frequency quasiperiodicity, it is the pair of unstable 3D-tori what defines the basin boundary of the chaotic attractor. In the Lorenz system
75
3.5 Numerical evidences of the route to chaos exhibited by the system
6
log10(C(ε))
4 2 0
−2 −4 −2
−1
0
1
log10(ε) Figure 3.11: Determination of the correlation dimension for R = 35.05. We used the algorithm of Grassberger and Procaccia [GP83] with the attractor sampled with 4 × 106 points. We measured the average of points C() inside of a ball of radius centered at one point of the attractor. The average was made over 104 points, except for < 10−1 where 105 points were used. The fitting slope is D2 = 3.96 ± 0.05. To be compared with the information dimension obtained from the Lyapunov exponents according to the Kaplan-Yorke conjecture: D1 ≈ 4.005.
the length-1 unstable periodic orbits divide the approximately 2D-manifold in two regions, where the chaotic and the fixed point attractors live; and also acts as the basin boundary. In the multistability region, trajectories spiral in one lobe away from the fixed point (say C+ ) because of the repulsive effect of the UPO surrounding that fixed point. After some turns the trajectory jumps to the other lobe by using the third dimension (i.e. thanks to the branching). If the trajectory approaches close enough to C− surpassing the ‘barrier’ constituted by the length-1 UPO, it is captured by this fixed point and no more jumps occur (i.e. the chaotic transient finishes). Analogously, the unstable T3 -torus acts as a dividing hypersurface for trajectories in the 4-dimensional branched manifold.
76
Transition to High-Dimensional Chaos through a Global Bifurcation
7
log10()
6 5 4 3 2 −5
−4.5
−4
−3.5
−3
−2.5
log10(R−Rbc) Figure 3.12: Log-log representation of the average chaotic transient as a function of the distance to the critical point Rbc . Each point is an average over 100 realizations. Data fit to a straight line of slope γ = −1.53 ± 0.06.
3.5.4
Boundary crisis and power law of chaotic transients
The fourth remark regards numerical studies for R & Rbc . We have measured the average time of the chaotic transients (hτ i) when approaching R = Rbc . We observe that these transients diverge satisfying a power law (hτ i ∼ (R − Rbc )γ ) for an asymptotic value R = Rbc = 35.093838, as expected for a boundary crisis [GORY87]. Inspired in the boundary crisis occurring in the Lorenz system [KY79b, GORY87, Spa82], we took initial conditions in one of the asymmetric (already unstable) Periodic Rotating Waves for different values of R. When R is clearly above Rbc (but below Rexpl ) the trajectory jumps to the Tk (k = 2, 3 depending on R) attractor located at the other side of phase space and no chaotic transient is observed. However, for R just above Rbc the result of using a point in this orbit as initial condition may be seen in Figure 3.13 (the PRW is a fixed point in the figure, as it has been
77
3.6 Description in terms of a return map
20
20 (b)
X0
(a) 10
10
0
0
−10
−20
−10
0
500
1000 Time (t.u.)
1500
1.4
−20
1.45 4
x 10
Figure 3.13: Numerical experiment showing a trajectory (only points at the Poincar´e section are shown) starting in an initial condition at one of the asymmetric unstable PRWs for R = 35.09389 & Rbc . The unstable 3D-torus, larger, is seen initially, but finally (part (b)) the system decays to the smaller stable 3D-torus. Note that the minimum width along time is larger for the stable 3D-torus as expected for a torus located inside another (see the discussion about the branched manifold).
stroboscopically cut through the Poincar´e hyperplane Im(X1 ) = 0). The trajectory seems to approach a T3 before ‘falling’ to the T3 attractor. Thus, we state that is the unstable T3 which constitutes the basin boundary of the chaotic attractor and the object involved in a global connection that marks the birth (or the death, depending on the viewpoint) of the chaotic attractor. Notice also, that the stable and the unstable T3 -tori have quite similar sizes. Ultimately, at Rsn , the multistability region has an end when the two T3 -tori disappear in the saddle-node bifurcation (previously discussed in Sec. 3.5.1).
3.6
Description in terms of a return map
Lorenz described a nice technique for reducing the complexity of the solutions of the Lorenz equations. By recording the successive peaks of the variable z(t), he reduced the dynamics of the Lorenz systems to a one dimensional map. Denoting the nth maximum of z(t) by Mn , he plotted successive pairs (Mn , Mn+1 ) of maxima, finding that points lay (very approximately) along a Λ-shaped curve. In this way, the dynamics is reduced to the “Lorenz map”: Mn+1 = Λ(Mn ).
78
Transition to High-Dimensional Chaos through a Global Bifurcation R = 35.094, TRANSIENT CHAOS
R = 35.0937, COEXISTENCE
47
47
46 chaotic transient →
46 chaotic attractor → Nj+1
48
Nj+1
48
3 45 unstable T →
45
44
44 3
←T
43 43
44
← T3
43 45 Nj
46
47
48
43
44
45 Nj
46
47
48
Figure 3.14: Return map of the maxima of the variable Z0 satisfying to be larger than their adjacent maxima (see text). The Upper plot shows a regime of transient chaos; after a transient the trajectory decays to the stable T3 (following the dotted arrows). As R is decreased the stable and the unstable T3 get closer. Beyond some point (R < Rbc ) orbits inside the chaotic set do not escape, which means that the chaotic set has become an attractor. The lower plot shows the attractors occurring for R into the coexistence interval.
In our case we may try to reduce the dynamics by means of a return map. The fast dynamics concerning the k = 1 mode is approximately filtered when considering the k = 0 mode. Then, the T3 attractor is seen in the k = 0 framework as a T2 plus some small oscillating component. A return map of the variable Z0 would reduce the dimension of the attractor by one, and giving, as a result an (approximately) one dimensional curve. But in analogy to the Lorenz system we would like to get a reduction of the chaotic attractor to a one dimensional curve (notice that the chaotic attractor has got a dimension larger in approximately one unit to the threetorus). Therefore, we must take a secondary return map to reduce the T3 to a fixed point. Considering the set of maxima of Z0 (t), {Mn }, we took the subset of maxima whose neighboring maxima were smaller: {Nj } = {Mn , Mn > Mn±1 }. A plot of successive pairs (Nj , Nj+1 ) gives a fixed point for the T3 , approximately because of the residual fast dynamics of the k = 1 mode. The results for two values of the parameter R are shown in Fig. 3.14. The chaotic attractor exhibits a rough Λ-shaped structure as occurs with the Lorenz map. Probably, the existence of the residual fast component makes the attractor to deviate significantly from one-dimensionality. In the light of the paper by Yorke and Yorke [YY79] who studied the transition to sustained chaotic behavior in the Lorenz model with the Lorenz map, it is found that our results are consistent with a chaotic attractor of dimension around four, and a boundary crisis mediated by an unstable three-torus.
79
3.7 Route to chaos: theoretical analysis
a)
b)
c)
d)
e)
f)
g)
h)
Figure 3.15: Three-dimensional representation of the proposed heteroclinic route to create the high-dimensional chaotic attractor. Black and gray points correspond to stable and unstable fixed points (cycles in the global phase space), respectively.
3.7
Route to chaos: theoretical analysis
Condensing all the information obtained from numerical experiments in the previous sections, we suggest the route to high-dimensional chaos represented in Fig. 3.15 (where the usual cross section through the fast rotating wave is understood). The high dimension of our attractor makes somewhat convoluted a geometric visualization. As we explained above, we postulate a chaotic attractor whose structure may be simplified in terms of a 4-dimensional branched manifold, and therefore the Poincar´e section reduces the attractor to a 3-dimensional branched manifold. Figure 3.15 represents a projection onto R3 , hence some (apparently) forbidden intersections between trajectories appear because of the branching. As occurs with the Lorenz attractor when it is projected on R2 (say x − z), the intersections between trajectories coming from different lobes, and also of them with the z axis (that belongs to the stable manifold of the origin) are unavoidable. Recall that it is the moment of the jump when the extra dimension is needed, and this is provided by the definition of a branched manifold. Remember that the shown route runs for descending values of R. Summing up our previous numerical findings: the centered PRW (a) becomes unstable through a pitchfork bifurcation (a→b) and two symmetry related PRWs appear (b). At a supercritical Hopf bifurcation (b→c) the
80
Transition to High-Dimensional Chaos through a Global Bifurcation
T2 appears. When R is slightly decreased the 2D-torus becomes focustype (the leading Lyapunov exponent becomes degenerate as may be seen in Fig. 3.4). Hence the unstable manifold of the asymmetric PRW forms a whirlpool when approaching the T2 (d). At Rhet a double heteroclinic connection between the asymmetric PRWs and the symmetric one occurs (e). At this point the chaotic set is created, that includes a dense set of unstable 3D-tori. In (f) the two simplest 3D-tori are represented with dotted lines, because of the heteroclinic birth one of the frequencies of these tori is very small. Note that the plot shows that the unstable manifold of one of the asymmetric PRWs intersects the stable manifold of the symmetric PRW which in principle violates the theorem of existence and uniqueness. This occurs, as we said above, because we are projecting the Poincar´e section onto R3 (our particular ‘flatland’). Twin secondary Hopf bifurcations (f→g) render unstable the 2D-tori and give rise to two stable T3 (g). When R is further decreased the unstable manifold of the asymmetric PRWs do not asymptotically tend to the stable T3 (h), and the chaotic set becomes attracting. A two-dimensional cut of the schematic shown in Fig. 3.15 is depicted in Fig. 3.16. A one-to-one relation between subplots of both figures does not exist. Thus, Fig. 3.16(h) corresponds to R = Rbc , whereas (i) corresponds to R = Rsn . It is to be stressed that the existence of a stable T3 is not a fundamental part of the transition to high-dimensional chaos. Just a focus-type T2 is needed such that the unstable manifold of the asymmetric PRWs forms a whirlpool. In fact, Ref. [SNN95] shows that such a whirlpool structure is related to the existence of a torus that becomes a heteroclinic connection between saddle-foci, in the vicinity of a codimension-two point. In this way, regarding the chaotic attractor, no fundamental change occurred if the unstable T3 shrank to collide with the stable T2 in a subcritical Hopf bifurcation. This picture would be more similar to the transition in the Lorenz system where C± become unstable through a subcritical Hopf bifurcation.
3.8
Further remarks
In this section we want to address some subtle aspects concerning the transition to chaos shown in this chapter. Analyzing bifurcations occurring in high-dimensional systems may become a quite convoluted task. Thus, when describing the disappearance
81
3.8 Further remarks
a)
f)
b)
c)
g)
d)
h)
e)
i)
Figure 3.16: Two-dimensional representation of the proposed heteroclinic route to create the high-dimensional chaotic attractor. It represents a vertical cut of Fig. 3.15. Periodic orbits (that correspond to 2D-tori in the real phase space) are represented by dots surrounded by small circumferences.
of the T3 attractors in Sec. 3.4 it was noted that the existence of an attractor-saddle collision between 3D-tori is expected to be nongeneric. Therefore, making an intense zoom at R ∼ Rsn would reveal the exact mechanism leading to the disappearance of the 3D-tori. A possible route was pointed out there. Nonetheless, from a practical point of view, it can be assumed that the T3 attractor disappears in a saddle-node bifurcation as long as the final step occurs in a extremely small interval of the parameter R. A similar situation occurs when studying codimension-m (m > 1) bifurcations6 . When increasing the highest order n, of the terms constituting the normal form (or n-jet) or adding a non symmetric term, substructures in the bifurcation set emanating from the codimension-m point can be resolved. For example, the effect of adding non-axisymmetric terms to the normal form of the saddle-node Hopf (codimension-two) bifurcation has been studied by Kirk [Kir91]. The first thing that must be noted is that Fig. 3.15 is representing a map (instead of a continuous dynamical system) because a Poincar´e section of the fast dynamics involving the spatial mode is assumed. Global connections between hyperbolic cycles have been studied mostly in a mathematical framework (excluding period-doubling and quasiperiodicity 6
The codimension is the number of constraints imposed on the control parameters to reach the critical point.
82
Transition to High-Dimensional Chaos through a Global Bifurcation
routes to chaos that have been also studied intensively from experiments) by means of diffeomorphims in the plane (see [GH83] and references therein). But an important aspect of the route shown here is that the fast spatial rotating wave is conserved along the transition. The weak interaction of this frequency is manifest by the absence of visible lockings of the T2 attractors (R ∈ [Rh2 , Rh1 ]) and for the presence of an additional vanishing Lyapunov exponent in the high-dimensional chaotic region. Thus, the fast spatial frequency is somehow ‘orthogonal’ along the transition and therefore phase space may be understood as a direct product of this frequency with the transition shown in Fig. 3.15 that can be assumed not to differ too much from a continuous system. Let us consider further which is the meaning of the Lyapunov spectrum of the chaotic region. According to the route described by Figs. 3.15, 3.16 an infinity of unstable T3 is created at the heteroclinic explosion (recall that only the two simplest 3D-tori are shown). Hence, we could expect to have a chaotic attractor with three vanishing Lyapunov exponents. Why only two are observed? One could be tempted to think that due to the non-hyperbolicity of quasiperiodicity it happens that a finite ratio of the 3D-tori set corresponds to locked tori (probably with large denominators), contributing to shift the vanishing Lyapunov exponents from zero. Nonetheless, it may be seen in the paper by Rosenblum et al. [RPK96] that two coupled R¨ossler oscillators exhibit (very approximately) a pair of null exponents in a appreciable range of the coupling strength (below phase synchronization). This occurs in spite of the finite ratio of tori that (according to theoretical arguments) are locked. Of course, the large denominators of the lockings cannot induce a significant shift on the exponents, but it is important to regard this, mainly from a fundamental perspective. However we believe that the absence of a third vanishing Lyapunov exponent is due to the following. It must be noted that the heteroclinic connection between the symmetric and the asymmetric PRWs is structurally unstable, so for R < Rh1 it should disappear. Nonetheless, we have numerically observed (see Fig. 3.10) that the connection (approximately) persists. Also, the axial symmetry of Fig. 3.15 has not a theoretical justification. Hence we expect that, a perturbation on the mechanism shown in Fig. 3.15 will destroy its nice simplicity. Inspired on previous works [Gas93, Kir91] dealing with the effect of non-symmetric terms on codimensiontwo bifurcations, we believe that homoclinic connections will replace
3.9 Discussion
83
heteroclinic connections. A double (‘figure-eight’) homoclinic of the symmetric PRW as well as homoclinic connections of the asymmetric PRWs will occur. Symmetric and asymmetric PRWs are both saddlefoci so it is important to know their saddle indexes7 in order to find out whether (Shil’nikov) chaos will arise in a neighborhood of homoclinicity. Thus, computing the eigenvalues from the (8-dimensional) return map (the same mapping that allowed us to stabilize these orbits), and finding the logarithms of their modules, we get the saddle indexes for the symmetric and the asymmetric PRWs. Thus, for the asymmetric PRW we get under time-reversal a saddle-index larger than one, which means that when the real time is recovered there exist a single orbit of repeller type. For the symmetric PRW the saddle index is smaller than 1/2, which means that homoclinic chaos will occur. The existence of homoclinic chaos seems to be nothing new, because the creation of infinite 3D-tori is substituted by an infinity of 2D-tori, which will produce homoclinic chaos of the type first reported years ago in Ref. [ACT81] plus an extra superimposed fast spatial wave. But it is important to emphasize that as long as the exact mechanism is very related to that shown in Fig. 3.15 the first negative Lyapunov exponent is very close to zero which makes the information dimension to be larger than four (or larger than three if the spatial oscillation is considered extra and/or trivial) The next question is how the two simplest unstable 3D-tori appear if the route is not exactly as is shown in Fig. 3.15. A possibility is that they appear though a saddle-node bifurcations between two unstable 2-tori, but we have no way to find out this. The reader may become disappointed with the observations of this section, because the exact route of the system is not clarified. However, this is typical of the kind of attractors that are called quasi-chaotic attractors due to the presence of structurally unstable homoclinic orbits. As observed in [SNN95] a complete description of a quasi-attractor in unattainable due to infinity many non-controlled bifurcations of various types.
3.9
Discussion
In this chapter we have studied by numerical and theoretical arguments the transition to high-dimensional chaos, in a model of three coupled 7 For a saddle-focus with three eigenvalues λs = ρ ± iω (ρ < 0), λu > 0 (as the symmetric PRW), the saddle index is defined to be δ = −ρ/λu , and chaos will occur for δ < 1. Time reversal must be applied to study the asymmetric PRW.
84
Transition to High-Dimensional Chaos through a Global Bifurcation
Lorenz oscillators. The transition from a periodic rotating wave to a chaotic rotating wave has been investigated. The structure of the global bifurcations between cycles, underlying the creation of the chaotic set, is such that a chaotic attractor with dimension d ≈ 4 emerges. The transition is not mediated by low-dimensional chaos. Also it must be noted that even if the fast rotating wave that is present all along the transition is omitted, we have studied the creation of an attractor with dimension D1 & 3. As occurs with the Lorenz system, the existence of a reflection (Z2 ) symmetry seems to play a fundamental role. The high-dimensionality of the chaotic attractor is not reflecting hyperchaos. Far from it, there is only one positive Lyapunov exponents but high-dimensionality is possible due to the existence of two vanishing and one slightly negative LEs. Hence, according to the Kaplan-Yorke conjecture the information dimension is above four. We have also measured, with the algorithm of Grassberger and Procaccia, the correlation dimension obtaining a value very close to four. The degeneracy of the null LE make us think that a set of unstable tori is embedded in the attractor. We have focused on giving a geometric view of the bifurcations occurring in the 9-dimensional phase space of the system. Although the precise sequence of bifurcations is probably resistant to analysis, we have been able to give a geometric view of the transitions that explains the emergence of the chaotic set, through a ‘heteroclinic explosion’, and its conversion into an attractor. This step occurs through a boundary crisis when the chaotic attractor collides with its basin boundary formed by two unstable 3D-tori. In consequence a power law for the mean length of the chaotic transients is observed.
Chapter 4
Onset of Traveling Fronts in an Array of Coupled Symmetric Bistable Units Abstract. A symmetry breaking mechanism is shown to occur in an array composed of symmetric bistable Lorenz units coupled through a nearest neighbor scheme. When the coupling is increased, we observe the route: standing → oscillating → traveling front. In some circumstances, this route can be described in terms of a gluing of two cycles on the plane. In this case, the asymptotic behavior of the velocity of the front is found straightforward. However, it may also happen that the gluing bifurcation involves a saddle-focus fixed point. If so, front dynamics may become quite complex displaying several oscillating and propagating regimes, including chaotic (Shil’nikov-type) front propagation. These phenomenology occurs for different couplings as well as for other discrete bistable media. The case of the discrete FitzHugh-Nagumo model shows some peculiarities that are analyzed. Thus, we suggest that there exist two universal forms of symmetry breaking leading to traveling fronts. These two routes would correspond to the two cases of a Takens-Bogdanov codimension-two bifurcation occurring at the continuum limit.
4.1
Introduction
Several areas of science use model equations of the reactiondiffusion type. Moreover, besides its special interest in some fields as chemistry [Tur52, Fif79] and biology [Mur89], reaction-diffusion equations are considered as abstract models for pattern formation [CH93]. Real systems are frequently composed of discrete elements, where material models and biological cells are just two examples. Therefore, it is natural to deal with the discrete space version of the reaction-diffusion equation. In this point, it must be emphasized that discreteness may
86 Onset of Traveling Fronts in an Array of Coupled Symmetric Bistable Units manifest itself in a strong way, such that some phenomena are not simple analogies of the continuous ones. Here, we focus on a discrete reaction-diffusion model such that its local dynamics (reaction term) is bistable, with two stable fixed points. Particularly important examples of bistable dynamics are found in optics [BCR81, Fir88], chemical systems [LE92], and biology [Mur89, McK70, Fit61]. The main interest of these systems lies in the behavior of the fronts that constitute the boundaries of the domains of both (stable) states. Intuitively, the front moves from the most to the less stable state, enlarging a domain and shrinking the other. In fact, great attention has been devoted to the phenomenon of ‘propagation failure’ or ‘pinning’ [Kee87, EN93, PPC92, MPG+ 95, MS95, KMB00, CB01], because it is usual that in discrete systems some threshold must be surpassed to achieve propagation. In principle, one could expect that propagation does not succeed in systems with symmetric bistability. Nonetheless, it was demonstrated time ago [IMN89] that such possibility exists. Hagberg and Meron [HM94] studied a continuous bistable reaction-diffusion system, the FitzHughNagumo model, showing that, in the symmetric case, propagation occurred after a symmetry breaking front bifurcation. In this scenario, the front breaks its symmetry through a pitchfork bifurcation, at the same time that propagation is initiated. Also, an analogous bifurcation was found for the complex Ginzburg-Landau equation [CLHL90] with the name of nonequilibrium Ising-Bloch bifurcation. In this chapter, we report the existence of front propagation in discrete symmetric bistable media. The transition is exclusive for discrete systems and it is possible thanks to the multi-variable nature of the local dynamics. One variable systems are not able to break symmetry [Kee87] even if the function that describes the local dynamics is not continuous [F´at98]. It is shown that the mechanism leading to propagation is not a pitchfork bifurcation. Contrastingly, it consists in a Hopf bifurcation followed by a global bifurcation, that is equivalent to a gluing bifurcation of cycles. In correspondence, the velocity of the front shows a logarithmic dependence with the coupling strength close to the onset. We also explain under which circumstances chaotic motion of the front is to be expected, which makes the transition between oscillation and propagation more convoluted. The occurrence of this route for different couplings and different systems (including the FitzHugh-Nagumo model) is investigated.
87
4.2 The system
4.2
The system
We consider a discrete reaction-diffusion equation in one dimension: r˙ j = f (rj ) +
D Γ(rj+1 + rj−1 − 2rj ), 2
(4.1)
where D accounts for the coupling strength between neighbors, and the coupling matrix Γ says which variables get coupled. In what concerns the local bistable dynamics, we deal with the well-known Lorenz oscillator [Lor63] as has been made in previous works [JW00, BPP00, Par01]. It exhibits several characteristic behaviors depending on its internal parameters [Spa82]: monostability, bistability, limit cycle, ‘butterfly chaos’, ‘noisy periodicity’, etc. We consider the range where the system contains two stable symmetry related fixed points (bistability). In what concerns the type of coupling matrix, several situations seem to be quite natural. These are the cases where all the elements of Γ are zero except one (that we take equal to one). The case Γ = I has been studied very recently [BPP00, Par01]. Three situations are of special interest as long as they present front propagation Γ = γkl = δk2 δl2 , γkl = δk2 δl1 and γkl = δk1 δl2 . The last possibility is studied in the next chapter, and we focus here in the first case. Thus, the differential equations describing the temporal evolution of the array read:
x˙ j
= σ(yj − xj )
y˙ j
= rxj − xj zj − yj +
z˙j
= xj yj − bzj
D (yj+1 + yj−1 − 2yj ) 2 j = 1, . . . , N
(4.2)
The standard values are chosen for the σ and b parameters: σ = 10 andpb = 8/3. p Then, the Lorenz system contains two stable foci C± = (± b(r − 1), ± b(r − 1), r−1) for the parameter r in the range (1.35, rH ); at r = rH ≈ 24.74, C± become unstable through a subcritical Hopf bifurcation. Also, it is important to note that there exists a saddle fixed point located at the origin for r > 1. A fourth-order Runge-Kutta method [PTVF92], with time step 0.01, was used to integrate Eq. (4.2). A step-like initial condition is imposed to the system. Results on front propagation do not depend on the boundary conditions provided that the array is large enough. Thus, if one wishes the system evolve for long time in the regime of traveling front, one should
88 Onset of Traveling Fronts in an Array of Coupled Symmetric Bistable Units move the boundary whilst the front propagates. Also, one may imposed to the system periodic boundary conditions, with two fronts diametrically opposed as initial conditions. If the initial condition satisfies xj = −xj+N/2 , yj = −yj+N/2 , zj = zj+N/2 , both fronts propagate always in the same direction, avoiding its mutual annihilation.
4.3
Standing, oscillating and traveling fronts
Three main front dynamics are found when varying the parameter r (r < rH ), and the coupling strength D (D > 0). We distinguish among standing (static), oscillating and traveling fronts. In Fig. 4.1, a centered step-like initial condition is imposed to the system with free ends and r = 14; two different non-static regimes are achieved depending on the value of the coupling strength D. Note that the Lorenz model is symmetric and the coupling that appears in Eq. (4.2) destroys this local symmetry but preserves the global reflection (or Z2 ) symmetry:
(x1 , y1 , z1 , ..., xN , yN , zN ) −→ (−x1 , −y1 , z1 , ..., −xN , −yN , zN )
(4.3)
Therefore, for N large enough, both senses of propagation are equally likely. A diagram of the different regions on the r − D plane is shown in Fig. 4.2. It may be seen that static fronts are found if D and/or r are small, whereas traveling fronts appear for large D and r. Also, it is significative that oscillating fronts always exist between the standing and the traveling fronts regions. The line (Dos ), that defines the boundary between the standing and the oscillating regions, approaches D = 0 as r → rH , since the nature of the fixed points (C± ) deeply influences the dynamics of the front. This is not very surprising, but indicates that oscillating fronts will be usually found in bistable systems with stable foci, rather than nodes. Somehow, the stability properties of the fixed points are transmitted to the front. Also, it is significative that Dth (the threshold for traveling fronts) seems to diverge asymptotically at r = r∞ ≈ 13.5. Instead of visualizing the system as in Fig. 4.1, we show now which is the behavior of the different units when the coupling is increased from zero. It is clear that for D = 0 the step-like solution that we were considering consists of rj = C+ , j = 1, . . . , 25 and rj = C− , j = 26, . . . , 50 (N = 50). As the coupling increases this solution can be smoothly continued. But there exist two constrains for the stationary solutions, that come from Eq. (4.2):
89
4.3 Standing, oscillating and traveling fronts
D = 39
5 0 −5 0
xi
xi
5 0 −5 0
D = 40
10
20
30
40 i
10
20
time 50
30
40 i
time 50
Figure 4.1: Spatio-temporal evolution of the front for two different values of the coupling strength D and r = 14. A step-like initial condition is imposed for an open array of N = 50 oscillators. When the coupling surpasses the critical value D = Dth ≈ 39.63 Figure 1 the front propagates through the array shifting the oscillators from C+ to C− . The time D. Pazo and V. Perez−Muñuzuri (2002) interval shown is 100 time units.
60 I − static front II − oscillating front 50
III − travelling front
40
D
III 30
Dth
20
II 10
I 0 12
Dos 14
16
18
20
22
24
r Figure 2 are distinguished as a function of r and D. Figure 4.2: Three main types of dynamics V. Perez Muñuzuri (Chaos, The boundaries among them D. arePazo theand critical lines Dos and Dth . 2002)
90 Onset of Traveling Fronts in an Array of Coupled Symmetric Bistable Units
xj yj ∀j ∈ {1, 2, ..., N } (4.4) b Hence, all the units lie on a parabola that passes through C± and the origin. However, when Dos is reached the front undergoes a Hopf bifurcation, and all the units start to oscillate leaving the mentioned parabola. The projection of the parabola onto the plane (x, y) is a straight line, the bisectrix of the first and third quadrants. However, to better visualize oscillations out of the parabola, in Fig. 4.3 we performed a 45◦ rotation of the reference system. Obviously those units closer to the front oscillate with larger amplitude that those located far from the front that are almost insensitive to the bifurcation. When Dth is reached a multiple collision occurs; the orbits of neighboring units collide creating two “channels” going from C± to C∓ . Like in the symmetric FitzHugh-Nagumo model studied by Hagberg and Meron [HM94], the traveling front is not symmetric. However, in our case the mechanism leading to propagation is not a pitchfork bifurcation, instead, a Hopf bifurcation that creates the oscillating front is the precursor of the traveling front. The description provided by Fig. 4.3 is illustrative, but it is incomplete if one does not realize that besides the static solution that loses its stability through a Hopf bifurcation, there must exist another (unstable) stationary state that mediates the multiple collision of cycles. Therefore, one must search the stationary solutions monotonic in {xj } and {yj }. It is not difficult to find out that only two monotonic solutions, called stable and unstable dislocations, exist (discarding spatial translations). They are the continuation of the solutions in the uncoupled limit: (. . . , C+ , C+ , C− , C− , . . .) and (. . . , C+ , C+ , 0, C− , C− , . . .). We shall refer to them as A- and B-state, respectively, and Fig. 4.4 shows a sketch of them. It is the B-state which mediates the transition at Dth , and it will be shown below that the stability properties of this solution will determine important features of the onset of the wavefronts. xj = yj ,
4.4
zj =
Gluing of cycles
In this section we demonstrate how the transitions presented in the previous section may be described in the context of a gluing bifurcation, in which two limit cycles become a two-lobed cycle by involving an intermediate saddle point. The gluing bifurcation is usual in systems with Z2 symmetry [LM00, RFH+ 95, HFP+ 98], because provided the existence of this symmetry it becomes a codimension-one bifurcation.
91
4.4 Gluing of cycles 0.04 (−xi+yi)/21/2
D=8 0.02 0
C−
i=24
i=25
i=26
i=27
6
8
C+
−0.02 −0.04
−8
−6
−4
−2
0
2
4
0.04 (−xi+yi)/21/2
D = 39 0.02 0
i=25
i=26
i=24
i=27
C
C
−
+
−0.02 −0.04
−8
−6
−4
−2
0
2
4
6
8
0.04 (−xi+yi)/21/2
D = 40 0.02 0
C
C−
+
−0.02 −0.04
−8
−6
−4
−2
0
(xi+yi)/21/2
2
4
6
8
√ √ Figure 3, D. Pazo & V. Perez−Muñuzuri (2002) Figure 4.3: Projections onto the reference system ((xi + yi )/ 2, (−xi + yi )/ 2) of all the oscillators of the array with step-like configuration and r = 14. From top to bottom, standing, oscillating, and traveling cases are shown. In the traveling case, the line is the trajectory followed by each oscillator going from C+ to C− as the front advances.
Figure 4.4: A sketch of the two monotonically decreasing stationary solutions for D > 0. a) A-state, continuation of the state (. . . , C+ , C+ , C− , C− , . . .) at D = 0. b) B-state, continuation of the state (. . . , C+ , C+ , 0, C− , C− , . . .) at D = 0.
4.4.1
Cylindrical coordinates
Figure 4.3 shows some kind of collective motion when the front oscillates. Then, the whole set of variables (xj , yj , zj ) is discarded, and instead, we chose to build a reduced phase space to study the dynamics. Notice that if the medium is infinite, there exists a perfect symmetry under translation along the array. With this in mind, we define two auxiliary
92 Onset of Traveling Fronts in an Array of Coupled Symmetric Bistable Units variables ξ and η, that have sense for large enough N and when the front is far enough from the boundaries: N
ξ =
1 X √ xj + yj 2 j=1
η =
1 X √ −xj + yj . 2 j=1
p mod 2 2b(r − 1)
(4.5)
N
(4.6)
Variable ξ accounts for the propagation of the front and is not p bounded by default, then it must be defined within the range ∆ξ = [2 b(r − 1) + p p √ 2 b(r − 1)]/ 2 = 2 2b(r − 1), that is the magnitude that ξ increases or decreases when the front moves one position along the array. On the other hand, η is defined so that it is bounded under front propagation, notice that it is defined from Eq. (4.4). If the medium consists of a number N even of units, the static solutions are located in this cyclic phase space in:
A:ξ = η=0 p B : ξ = ± 2b(r − 1), η = 0
(4.7) (4.8)
If N is odd both solutions exchange their coordinates. In what follows, we only take N even, unless it is specified. The transition shown in Fig. 4.3 is now shown in the cylindrical phase space in Fig. 4.5. Notice that when the front is static, we have an equilibrium point at ξ = η = 0 (A-state). When the front oscillates a limit cycle exists, and finally when the coupling goes beyond Dth we find two symmetry related limit cycles, that turn around the cylindrical phase space in opposite directions. The situation is quite similar to a pendulum where libration corresponds now to oscillation, and rotation of the pendulum corresponds to propagation. Thus, at D = Dth there exist two homoclinic loops that connect the B-state with itself (called separatrices for the pendulum). This bifurcation is called gluing bifurcation because it involves the collision of two cycles to create another; although in our case it could be more appropriate to talk of a inverse gluing or a splitting bifurcation. Nonetheless, it is better to visualize our gluing bifurcation in R2 . A possible transformation consists, topologically, in what follows: First of all, cut the cylinder with two planes perpendicular to its axis in such a way
93
4.4 Gluing of cycles D=8
D = 39
D = 40
0.1
0.1
0.05
0.05
0.05
0
0
0
−0.05
−0.05
−0.05
η
0.1
−0.1
−5
0
ξ
−0.1
5
−5
0
ξ
5
−0.1
−5
0
ξ
5
Figure 4.5: Evolution on the reduced cylindrical phase space spanned by ξ and η for three different values of D and r = 14; from left to right: standing, oscillating, and traveling front. At D = Dth ≈ 39.63 a double homoclinic connection to the B-state arises. Figure 5 D. Pazo and V. Perez−Muñuzuri (2002)
that the saddle point and the heteroclinic orbits stand between both planes. Then compress both circumferences that limit the “cylinder” inbetween to a point. Thus, we have got an object topologically equal to a sphere. These two steps could be substituted by making the section of the cylinder equal to zero at η = ±∞ and then making a transformation that makes our object finite. The last step is to make a hole at ξ = η = 0 and deform what remains into a plane. In Fig. 4.6 the sketch of a gluing bifurcation equivalent to the one shown in the cylindrical phase space (Fig. 4.5) is depicted. Notice that for D < Dth there exist an orbit that approaches twice per period close to the saddle point (the B-solution), whereas for D > Dth two symmetry related cycles, corresponding to both senses of propagation, coexist.
4.4.2
Velocity of the front as a function of D − Dth
The collision (and subsequent destruction) of a periodic orbit with a saddle, called saddle-loop bifurcation, is characterized by a logarithmic lengthening of the period of the cycle [Gle94, Str94]. For the scenario depicted in Fig. 4.6, we expect to find the following dependencies for the periods (T1,2 ) in a neighborhood of Dth : 2 ln(Dth − D) λu 1 = a2 − ln(D − Dth ), λu
T1 = a1 − 1 = T2 c
(4.9) (4.10)
94 Onset of Traveling Fronts in an Array of Coupled Symmetric Bistable Units
λu λs
D < D th
D = D th
D > D th
Figure 4.6: Schematic of a (inverse) gluing bifurcation on R2 . For D < Dth the trajectory approaches twice per cycle the saddle point and the front oscillates. Beyond the critical value D = Dth two symmetry related orbits, corresponding to both senses of propagation of the front, appear. λu and λs stand for the eigenvalues of the saddle fixed point at homoclinicity.
where λu is the unstable eigenvalue of the saddle fixed point. Taking into account that one turn around the cylinder is equivalent to the movement of the front in one cell, it is clear that T2 is the inverse of the velocity of the front (c). In the equation for the period T1 a factor 2 appears because the orbit approaches twice per cycle to the neighborhood of the saddle point. Moreover, it is expected that the ‘fast dynamics’ (the motion far from the saddle) contained in variables a1,2 will be approximately the same at both sides of the transition, and then, a1 ≈ 2a2 . We show here the results for r = 16 in Fig. 4.7. Notice that the velocity of the front grows quite abruptly from zero at D = Dth . The derivative of c = c(D), near Dth is, according to Eq. (4.10): 1 dc = dD λu (D − Dth )
4.4.3
a2 −
1 λu
1 ln(D − Dth )
!2
D→Dth
−→ ∞.
(4.11)
Quantitative analysis
In order to verify, not only qualitatively, but quantitatively the tendency of T1 and c when D → Dth we computed numerically the value of the eigenvalues of the B-state at D = Dth for finite N . To preserve the
95
4.4 Gluing of cycles
7
1/c T1/2
7
a)
6
6
5
5
4
4
3
3
2 18.5
19
19.5
20
D
20.5
21
21.5
2 −8
b)
~ −λ−1 u
1/c T1/2 −6
−4
−2
ln | D−Dth |
0
Figure 4.7: 1/c (solid line) and T1 /2 (squares) as a function of D − Dth (a) and ln(|D − Dth |) (b) for r = 16. The behavior agrees with that predicted by Eqs. (4.9) and r=16 (4.10). The critical coupling is found to be Dth = 20.13952. In (b) a straight line −1 with slope −λu is depicted, being λu = 1.7959 the unstable eigenvalue of the B-state computed numerically for an array of N = 21 units.
symmetry, when computingFigure the 7eigenvalues, the number of units N was D. Pazo and V. Perez−Muñuzuri (2002) chosen to be odd, with the central unit located at the origin. The jacobian matrix (J) for a discrete reaction-diffusion model (see Eq. (4.1)) exhibits the following tridiagonal structure: H1 D 0 ··· 0 0 D H2 D · · · 0 0 0 D H3 · · · 0 0 J= . (4.12) .. .. . . .. .. , . . . . . . . 0 0 0 · · · HN −1 D 0 0 0 ··· D HN which, according to Eq. (4.2) has got the components: −σ σ 0 0 0 0 Hj = r − zj −1 − D −xj , D = 0 D/2 0 , yj xj −b 0 0 0
(4.13)
recall that x(N +1)/2 = y(N +1)/2 = 0. Therefore, −b is always an eigenvalue with trivial eigenvector (0, 0, ..., 0, (0, 0, 1), 0, ..., 0)T . Using the program matlab we obtained the eigenvalues for an array of N = 21 units, that we expect to be large enough to inform us about the properties of a front in an infinite medium, as long as the front is a very localized structure. Figure 4.8 shows the results for three values of the parameters r and D. Fig. 4.8 (a) shows the eigenvalues (λi ) for D = 0 and r = 14, which
96 Onset of Traveling Fronts in an Array of Coupled Symmetric Bistable Units r=14
Im(λi)
r = 16, D = Dth
b)
a) 5
c)
5
5 λu
−b
0
0
−5 −20
r=16
r = 14, D = Dth
r = 14, D = 0
λs
−5 −10 Re(λi)
0
−10
λ
s
λ
u
0 * λs
−5 −5 Re(λi)
0
−10
−5 Re(λi)
0
Figure 4.8: Eigenvalues of the B-state for different values of D and r.
illustrates the meaning of the different eigenvalues. There are three simple Figure 8 eigenvalues corresponding oscillator located in the center of the D. Pazoto andthe V. Perez−Muñuzuri (2002) array whose coordinates are x(N +1)/2 = y(N +1)/2 = z(N +1)/2 = 0, recall that we are considering a B-state. Moreover, there are three (N − 1)degenerate eigenvalues, corresponding to the oscillators with coordinates C± . Eigenvalues are depicted with circles of different sizes to better observe degeneracy. When D grows above zero degeneracy is lost. In Fig. 4.8 (b), we graph the eigenvalues for D = Dth ≈ 39.63 (only those eigenvalues satisfying Re(λi ) > −10 are shown). The unstable eigenvalue corresponds to the fixed point at the origin of a Lorenz oscillator. There is also a slightly negative eigenvalue (λs ) that is the leading eigenvalue among the whole spectrum of negative eigenvalues. A different scenario is found for r = 16, as shown in Fig. 4.8 (c). In this case, it is not so clear which stable eigenvalue must be considered as the leading one in the gluing process. There are several eigenvalues making up two lines at both sides of the real axis in the complex plain, but these are not relevant, as long as they are the eigenvalues associated with the stability properties of each domain at both sides of the front where quasi-homogeneous domains around C± exist. As well, if the eigenvalue located at −b is considered, its eigenvector becomes trivial and it cannot be expected to participate in the gluing bifurcation. Then, the solitary eigenvalue denoted by λs (= ρ + iω) is the one (with its complex conjugate λ?s ) that defines the leading (two-dimensional) stable manifold. In fact, by increasing continuously r along Dth , the eigenvalue λs could be followed directly from that obtained for r = 14 (see Figs. 4.8 (b,c)). According to the statements exposed above, the graph of η vs. ξ, for D close to Dth (r) and
97
4.5 Exotic front dynamics 0.8
0.6
0.4 0.02 0.2
0.01 η
η
Figure 4.9: Projection onto (ξ,η) at r = 16, D = 20.136 < Dth ≈ 20.1395. In the inset, the spiraling approach of the trajectory to the Bstate (filled circle) may be observed.
BBB
0 0 −0.01
−0.2
8.8
8.9 ξ
9
−0.4
−0.6
−0.8 −10
−8
−6
−4
−2
0
ξ
2
4
6
8
10
Figure 9 D. Pazo and V. Perez−Muñuzuri (2002)
r = 16, shows some spiraling1 when the trajectory approaches the B-state. Figure 4.9 shows this effect that confirms that now the gluing is mediated by a saddle-focus. The cyclic definition of ξ has been relaxed in order to get a better observation of the spiral trajectory near the B-state. Nonetheless, if one looks at the asymptotic properties of the oscillation period of the front and its velocity, no difference with the case of the planar connection is found. So, Fig. 4.7 already confirmed the Eqs. (4.9) and (4.10), and the slopes agree with the numerical value: λu ≈ 1.7959. A sketch of the gluing bifurcation in the saddle-focus case is shown in Fig. 4.10. Homoclinic orbits, at D = Dth are denoted by Γ0 and Γ1 . The process of gluing is drawn in analogy to Fig. 4.6. Although, from both figures, one could think that the complex and the real cases are almost equivalent, there exist very fundamental differences between both cases. Thus, the transition from oscillating to traveling fronts may become quite convoluted in the complex case. This is observed for larger values of r and is the subject of the next section.
4.5
Exotic front dynamics
In Figure 4.11 the value of the imaginary part of the leading stable eigenvalues (ω), along the line D = Dth , is shown as a function of r with a dashed line. It is found that above r = rsf ≈ 15.45 the leading stable eigenvalues are complex conjugates, and therefore, the gluing bifurcation occurs mediated by a saddle-focus point. Many works have been devoted to the problem of saddle-loop bifurcations involving a saddle-focus fixed point (see e.g. [Str94, Gle94, 1 One should define a new extra variable ζ to work in a hyper-cylindrical phase space. P One could define, for instance, ζ = xj yj − bzj . Nonetheless, we shall continue to work with ξ and η only, although keeping in mind that we are projecting the third variable.
98 Onset of Traveling Fronts in an Array of Coupled Symmetric Bistable Units
Γ0
Γ1
D < D th
D = D th
D > D th
Figure 4.10: Schematic of a gluing bifurcation mediated by a saddle-focus. homoclinic trajectories Γ0 ,1 exist at D = Dth .
Two
6
5
ω, δ
4
3
2
1
r 0 13
r
sf
14
15
1
16
17
18
19
20
21
22
23
24
r Figure 11 Figure 4.11: Dashed line: Imaginary part of the stable leading eigenvalue ω of the D. Pazo and V. Perez−Muñuzuri (2002) B-state. Solid line: saddle index δ. For r > rsf , ω > 0; and for r > r1 , δ < 1. Both magnitudes are computed as a function of r, along the curve D = Dth (r).
4.5 Exotic front dynamics
99
GH83, Wig88] and references therein). A deep presentation of this problem is out of the scope of this thesis, so we just recall some of the most relevant results. For the case of a single homoclinic orbit, occurring at a critical value of the control parameter (say µ = 0), it was found by Shil’nikov [Shi65] that when the ratio Re(λs ) ρ δ=− =− , (4.14) λu λu called saddle index, is less than one, there are a countable infinity of periodic orbits in a neighborhood of the homoclinic orbit, all of which are saddles (see. Sec. 1.3.5.b). These saddle orbits are created in a sequence of tangent bifurcations at both sides of µ = 0, such that the stable branches (provided δ > 1/2) become unstable through period-doubling bifurcations. This scenario, where infinite orbits, with periods ranging from some finite value to an infinite one, arrange in a wiggly curve around the critical value, is known as Shil’nikov wiggle. And it is, with the Lorenz and the bifocal mechanisms, one of the universal routes to chaos through homoclinicity [GH83, Wig88]. However, the results by Shil’nikov apply in a very small neighborhood of the critical parameter. Therefore, the relationship between the local (theoretical) results and the global (observed) behaviors cannot be stated oversimply. It was shown by Glendinning and Sparrow [GS84] that the difference between the δ < 1 and the δ > 1 approaches to homoclinicity may not be easily observed numerically and may not be relevant from a global point of view. In fact, for the lowest branch (the orbit with the lowest period) that already exist far from the critical parameter, deviations from the behavior predicted by the local analysis are most likely to occur. One may realize the complexity of the problem if one notices that, besides the Shil’nikov wiggle, there exist subsidiary homoclinic connections. Following the nomenclature of Ref. [GS84], subsidiary homoclinic orbits are multiple-pulse homoclinic orbits, i.e. orbits that pass several times near the stationary point without achieving homoclinicity. In case the homoclinic connection is double (two loops, Γ0 and Γ1 ), as occurs in the gluing bifurcation, some differences are appreciated with the case described above. The most important point is that, although for δ > 1 no Smale horseshoe exist as in the single case, the approach of orbits to Γ0 ∪ Γ1 (Fig. 4.10) is chaotic [Hol80, Wig88]. Figure 4.11 shows the value of δ (solid line) as a function of r along the line D = Dth . It is found that at r = r1 ≈ 22.3, δ crosses one. Interested in the complex behaviors that our system could show, we focused
100 Onset of Traveling Fronts in an Array of Coupled Symmetric Bistable Units in two values of r above and below r1 : r = 20 (δ ≈ 1.18) and r = 23 (δ ≈ 0.96). We do not want to carry out a detailed analysis of the solutions found, but just to show that the results have got the “flavor” expected according to previous theoretical studies on bi-homoclinicity to a saddlefocus [Hol80, Gle84, ACT81].
4.5.1
r = 20 (δ > 1)
In the following, we analyze the transition from an oscillating front to a traveling front for r = 20. Fig. 4.12 (a) shows the dependence of the period of oscillation of the front (T1 ) on D. It is found that the period does not grow continuously, and instead, there are some jumps, which give rise to hysteretic cycles (Fig. 4.12(b) shows an inset of Fig. 4.12(a)). These wiggles (note that unstable branches join consecutive stable branches) are not surprising according to [GS84], and are expected to appear when approaching δ = 1. Unfortunately, we have not been able to find oscillations with semi-periods larger than 3.2 t.u. This occurs because as the period grows, and therefore (bi-)homoclinicity is reached, the chaotic transients become larger and larger. The different regimes may be labeled by their representative symbolic sequences of 0’s and 1’s, depending on the number of turns at each side of the saddle point. For example, the uniform propagating regimes are labeled by {0} or {1}, but if a period-doubling occurs the new (stable) regimes are labeled by {02 } or {12 }, respectively. On the other hand, the oscillating regime, described by a two-lobed cycle, is represented by the code {01} (or {10}). For the traveling front region, the dependence of the velocity with D shows some features that were also reported in [GS84], where a system of three ordinary differential equations, that exhibits a single homoclinic connection to a saddle-focus, was studied. In Fig. 4.12 (c,d) the inverse of the velocity is plotted as a function of D. In those intervals of D where no line appear, the front continues to exhibit chaotic motion, characterized by “spontaneous” front reversals, after, at least, 30000 t.u. of transient; but some small ‘windows’ with regular motion can be found. Figure 4.13 shows the dynamics of the front for D = 17.85 during 200 t.u., after some transient. In Fig. 4.13 (a), the variable x of the oscillators of the array is represented in gray scale, whereas Fig. 4.13 (b) shows the phase portrait (ξ, η). The front reverses its propagation several times and displays a quasi-erratic motion. Although the behavior is likely to be a chaotic transient, from a practical point of view, it is indistinguishable of
101
4.5 Exotic front dynamics 3.5
2.6
b)
a)
3
2.4
T /2
2
2.2
1
T1 / 2
2.5
2
1.5 1.8
1 0.5 15
15.5
16
16.5
17
17.5
18
1.6 17.68
17.73
D
17.78
3.2
17.785 2
0 0
0
0
3
3.1 3
2
2.8
d)
3.2
0
2
0
2.6
0 02
1/ c
1/ c
17.79
4
3
17.78
D
2.5
0
2 0
c) 2.4
17.8
17.85
17.9
1.5 17.5
18
D
18.5
19
19.5
20
D
Figure 12, D. Pazo and V. Perez−Muñuzuri (2002)
Figure 4.12: 1/c (solid line) and T1 /2 (squares) as a function of D for r = 20. Fig. (a) shows that the oscillation period does not grow continuously. An inset, showing one of this jumps can be seen in (b). The velocity of the front –Fig. (d) and inset (c)– is not defined in some regions where chaotic states are found. Period-doublings are marked with circles and different states are labeled according to its symbolic code. Because of the symmetry all 0’s may be substituted by 1’s.
“true chaos”. On the other hand, transitions between different regimes exhibiting sustained propagation, occur at period-doubling bifurcations (circles in Figs. 4.12 (c,d)). In addition, we find the well-known period-doubling route to chaos, see the inset of Fig. 4.12 (c). In Fig. 4.14 three periodic orbits and a chaotic one are shown. From right to left, a period-doubling cascade leads to a type of chaos that is characterized by sustained propagation of the front with non-periodic velocity. This scenario, that generates chaos with δ & 1 at homoclinicity, was already found in [GS84].
4.5.2
r = 23 (δ < 1)
For r = 23, δ is less than one, and therefore in a small neighborhood of Dth , one expects to find, according to [Gle84], Shil’nikov wiggles for both,
102 Onset of Traveling Fronts in an Array of Coupled Symmetric Bistable Units
2 C−
1
0
η
0 (i.e. saddle-focus with δ < 1).
4.7 4.7.1
Universality? Other couplings and systems
One may ask whether the symmetry breaking front bifurcation presented here is universal or not; in other words, in what kind of system could one expect to find this transition? We have found that other coupling matrices are able to induce front propagation in an array of coupled Lorenz oscillators. If one considers coupling matrices with all elements zero except one, these two off-diagonal couplings: Γ = γkl = δk1 δl2
(4.15)
Γ = γkl = δk2 δl1
(4.16)
exhibit front propagation through a route that is similar to the one explained here. However, it may happen that the transition occurs in such a way that the standing front loses its symmetry in a small interval. This happens because a pitchfork bifurcation renders the A-state unstable, and when the coupling is increased further, the new (static nonsymmetric) solutions “collide” with the B-state transferring the stability (through a pitchfork bifurcation again). Later, it is the B-state which undergoes a Hopf bifurcation, and finally the oscillating solution touches the A-state creating the traveling solutions. The logarithmic laws, Eqs. (4.9) and (4.10), are also obtained as we shall see in the next chapter.
106 Onset of Traveling Fronts in an Array of Coupled Symmetric Bistable Units We have also checked our results in other two bistable systems: an array whose local dynamics is a truncation of the magnetohydrodynamic partial differential equations of a disc dynamo [CH80], and the FitzHugh-Nagumo model [Fit61]. The dynamo model is similar to the Lorenz system and in some parameter range it exhibits symmetric bistability. The equations of the model are: x˙ = α(y − x) y˙ = zx − y
(4.17)
z˙ = β − xy − κz The system has one unstable fixed √ √ point at P0 = (0, 0, β/κ) and two stable fixed points P± = (± β − κ, ± β − κ, 1) in the range β ∈ (κ, βH ), with βH = 15 for α = 5, κ = 1. We have found a transition like that of the Lorenz systems, for instance for the coupling matrix Γ = γkl = δk2 δl2 . Depending on the value of β the transition is smooth or chaotic. Whereas for β = 6 we find a fine logarithmic profile of the front velocity, for β = 14 the front velocity function is interrupted by a chaotic regime –like that shown in Fig. 4.13– when approaching the threshold. Also, the oscillating dynamics, for β close to βH , is not so simple as that shown in Fig. 4.3. Since in the FitzHugh-Nagumo model the transition from static to traveling front occurs in a different way, we devote a subsection (Sec. 4.7.3) specifically to this system.
4.7.2
Large D
As the coupling D is increased, the front involves a larger amount of oscillators. In some sense, the front becomes more ‘continuous’ (or less steep). This tendency allows us to understand better the behavior of Dos and Dth in the large D region (see Fig. 4.2). When D is large, neighboring oscillators have similar (x, y, z) values. Therefore, once the Dos line is crossed, a small increase of D (in comparison with Dth or Dos ) is needed to achieve the multiple collision of cycles, i.e. the Dth line. As long as D is large, both A- and B-states are in a quasi-continuum and should exhibit similar eigenvalue spectra. Formally we can make A− and B− states coalesce, as intuitively occurs in the infinite diffusion limit, redefining the variable ξ: ξnew = ξ/Dγ , (γ > 0). In this way, the size of the cylinder shrinks to zero at infinity.
107
4.7 Universality?
So as r → r∞ ≈ 13.5, λu and λs of the B-state at Dth should meet the pair of complex conjugate eigenvalues that characterizes the Hopf bifurcation of the A-state at Dos . In short, both lines meet at D = ∞ in a double-zero eigenvalue point. For this reason, oscillations above Dos have a small frequency and δ → 1+ when r → r∞ (see Fig. 4.11). We continue our discussion of the continuum limit with the FitzHugh-Nagumo model in Sec. 4.7.3.b.
4.7.3
The discrete FitzHugh-Nagumo model
The FitzHugh-Nagumo (FHN) model was derived from the HodgkinHuxley model for nerve membranes [Mur89], with the aim of making it analytically tractable. It has become one of the most important reactiondiffusion models. Depending on the parameters that control the local dynamics it may describe an excitable medium, a medium undergoing either a Hopf or Turing bifurcation, and a bistable medium. This last possibility is of our interest, and in spite of being not so often considered as the excitable case it may be relevant in some situations [GC77, ML81] . 4.7.3.a
Transition to traveling front
The discrete FitzHugh-Nagumo model reads: u˙ j
= uj − u3j − vj + D(uj+1 + uj−1 − 2uj )
v˙ j
= (uj − a1 vj − a0 )
j = 1, . . . , N
(4.18)
we take a0 = 0 and a1 = 2 which provide the Z2 symmetry and bistability, respectively. The local dynamics presents a saddle equilibrium point at q the origin, q and two odd symmetric stable fixed points (u± , v± ) = (±
a1 −1 1 a1 , ± a1
a1 −1 a1 ).
As occurs in the continuous version [HM94],
propagation only succeeds for small (O(10−1 )). For very small D, only one stable solution exists, the standing one (A-state). When D increases, two (counterpropagating) traveling solutions coexist with the standing one. Finally, the standing solution undergoes a subcritical Hopf bifurcation and the traveling solutions become the only stable ones. This route is apparently very different from those shown above, since a computation of the eigenvalues corresponding to the B-state reveals that λu > −λs (δ < 1). Therefore, if any gluing bifurcation exists, it involves unstable cycles. Hence, we speculate that the transition is as follows: When D reaches a critical value DSN , two stable traveling solutions (both propagation senses)
108 Onset of Traveling Fronts in an Array of Coupled Symmetric Bistable Units 12
6
4
4
0
DG
0.1
2 DSN
0
4 2
0.2
0.3
2
b)
c)
~1/2
0
0
8
6
~−λ−1 u
6 ( DSN , 1/c0 )
ln(1/c −1/c)
10
8 1/c
12 1/c
a) 10
DG≈0.125
−6
−4 −2 0 ln( D − DG )
−2 DSN=0.10782325
−4
1/c =11.971 0
−15 −10 −5 0 ln( D − DSN )
DG
0.5
1
1.5
D
Figure 4.17: Velocity of the front for the discrete FitzHugh-Nagumo model (a). Internal parameters are a0 =0, a1 = 2, and = 0.1 (see Eq. (4.18)). Data are shown with dots and circles. It may be observed that the traveling solution ceases to exist at D = DSN through a saddle-node bifurcation. Nonetheless, in the interval shown with circles the increase of 1/c looks logarithmic. In Fig. (b), a semilog plot shows that at some interval, the slope of 1/c agrees with the value of the unstable eigenvalue of the B-solution at DG : λu = 0.6607. This suggests that the traveling solution is approaching the (non-complex) saddle point (B-state) when D decreases. However the collision is prohibited, because the traveling solution is stable and the saddle index of the B-solution is approximately 0.7 for the values of D considered. This implies that the gluing bifurcation may occur between unstable cycles only. Hence, the gluing mechanism involves the unstable traveling solutions created at D = DSN . In Fig. (c) a log-log plot shows the dependence of 1/c as a function of D − DSN . The slope near DSN agrees with the expected value 0.5, characteristic of a saddle-node bifurcation.
are born with nonzero velocity (c = c0 6= 0), in two simultaneous saddlenode bifurcations. The unstable traveling solutions that appear in these saddle-node bifurcations become glued when D is slightly increased, at D = DG 2 . In this way, the (unstable) oscillating solution that coalesces with the A-state at the (subcritical) Hopf bifurcation is created. In Fig. 4.17, we present the values of 1/c as a function of D, for = 0.1. One may observe that within a range of values of D, 1/c exhibits a logarithmic profile (Fig. 4.17 (b)) but finally departs from that tendency, and shows a root-square dependence, typical of a saddle-node bifurcation (Fig. 4.17 (c)): 1 1 − ∝ ±(D − DSN )1/2 . (4.19) c c0 In our case, the + (resp. −) sign corresponds to the unstable (resp. stable) solution. DSN is close to DG and that is the reason why the partial 2 Considering initial conditions very close to the B-state for increasing values of D, DG may be estimated as the critical value of D at which the asymptotic state changes from the A-state to a traveling state.
109
4.7 Universality?
0.8
DH DG DSN
0.7
D
0.6 0.5
traveling
0.4 0.3 0.2
coexistence
static
0.1 0.1
0.15
ε
0.2
0.25
Figure 4.18: Diagram showing regions with different front dynamics in the discrete FHN model. At DSN , two pairs (stable-unstable) of counterpropagating traveling solutions appear. At DG , both unstable traveling solutions become ‘glued’ to create an unstable oscillating solution. Finally, at DH this solution coalesces with the stable static solution (A−state) in a subcritical Hopf bifurcation, rendering unstable the static front solution.
logarithmic dependence can be recognized in Fig. 4.17 (b). In fact the saddle index is not very small, δ ∼ 0.7, what makes possible the unstable cycles to disappear at DSN close to DG . 4.7.3.b
The continuum limit
Let us consider consider now the continuous version of the FitzHughNagumo (FHN) model: ∂t u(x, t) = u − u3 − v + ∂xx u ∂t v(x, t) = (u − a1 v − a0 )
(4.20)
The diffusion constant of the variable u can be chosen as one without√loss of generality. This may be done by rescaling the variable x: x → x/ Du
110 Onset of Traveling Fronts in an Array of Coupled Symmetric Bistable Units (take into account that we are considering an infinite medium). We focus, as in the discrete case, on a single front satisfying (u, v) = (u± , v± ) as x → ∓∞. As proved by Hagberg and Meron [HM94], the static front is stable above a critical value of the parameter . At the critical value c = 1/a21 the stationary front renders unstable through a pitchfork bifurcation, that gives rise to two nonsymmetric (but symmetry-related) traveling solutions. The velocity of these solutions grows from zero following a square root law: c ≈ k(c − )1/2 It would be interesting to compare these results with those obtained for discrete case. This is done by changing variables u(x, t) → uj (t) and discretizing the partial derivative such that ∂xx u(x, t)
u(x + h) + u(x − h) − 2u(x) h2 → D(uj+1 + uj−1 − 2uj ). =
lim
h→∞
(4.21)
Hence, the diffusion constant D is a measure of the spatial step, and therefore the continuum limit (h = 0) is at D → ∞. Keeping the arguments exposed above in mind, we swept the parameter space − D for a particular value of a1 , and paying special attention to the limit D → ∞. In Fig. 4.18, the regions in the − D plane with different front solutions are shown, for a1 = 2. The lines DSN,H divide the phase space in three parts: static, traveling, and coexistence of both (that is located always in between). The static front is stable below the line DH and the (counterpropagating) traveling solutions exist above the line DSN . A third line, DG , indicates where the double homoclinic connection, that mediates the gluing of two unstable traveling solutions, occurs. Notice that in Fig. 4.17, we studied the transitions along the line = 0.1. The most relevant feature of Fig. 4.18 is the fact that all the lines diverge at = 1/a21 = 1/4, which means that the continuum is effectively recovered at D = ∞. In fact the coexistence region shrinks as D → ∞, which agrees with the ‘soft’ transition from static to traveling front observed in the continuous version. As it occurred for the array of Lorenz units, the bifurcation lines diverge to infinity. Formally, the point located at D = ∞ –where the transition between static and traveling front occurs– is a double zero-eigenvalue point. This kind of degeneracy is known as Takens-Bogdanov (see the Appendix for an example). Following the reasonings above, we are obliged to compare our results with the unfolding of a TB bifurcation corresponding to the case studied
4.7 Universality?
111
here. Notice that the size of the cylinder is zero at D = ∞ (after the rescaling of ξ proposed in Sec. 4.7.2), so D = ∞ is itself a bifurcation line of codimension one. The two counterpropagating traveling fronts that appear through a pitchfork bifurcation for the continuous version at = 1/a21 , are the limit case of the cyclic solutions observed in our cylinder when its width approaches zero. Unfortunately, we have not been able to find in the literature an unfolding of the TB with periodic boundary conditions in one of its variables; and it seems not to be a straightforward task. Nonetheless, in the book of Guckenheimer and Holmes [GH83] one can find the unfolding of a TB bifurcation with “cubic” symmetry, or symmetry under rotation through π. This unfolding considers variables in R2 , which is different to our case but satisfies the symmetry properties (ξ, η) → (−ξ, −η). Allowing time reversal this unfolding presents only two cases, whose bifurcation lines (emanating from the TB point) strongly remind the scenarios observed for the two systems studied in detail in this Chapter (the array of Lorenz units and the discrete FHN model). Thus, for the unfolding of the symmetric TB, and in the region of parameter space where three static solutions exist3 , one follows one of these two routes: Hopf bif. stable double heteroclinic connect. no one stable stable −−−−−−−−−−−−−−−−−−→ 1. stationary −−−−−→ periodic solution solution solution
two stable Saddle-node coexistence gluing bif. + stable stationary −−−−−−−−−→ periodic + −−−−−−−−−−→ periodic 2. solutions of cycles stationary sols. subcr. Hopf bif. solution
We introduce now some comments about the relations between these routes and the transitions we have observed. Route 1 is related to the route in the array of Lorenz oscillators, whereas route 2 presents strong analogies with the discrete FHN model. First of all notice that as long as these routes occur in R2 , there are three stationary solutions at every moment (appearing at a nearby pitchfork bifurcation). Identifying two of them (that are symmetry-related) is –somehow– a way to make the analogy with our periodic condition (due to the cylindrical geometry) stronger. 3 Since a pitchfork bifurcation line passes through the TB point, there are three solutions at one side of this line and one at the other. The latter is not of our interest because it is formally beyond D = ∞.
112 Onset of Traveling Fronts in an Array of Coupled Symmetric Bistable Units Thus in route 1, the double heteroclinic connection becomes a double homoclinic one. The final state does not reproduce our findings because the two counterpropagating solutions that turn around the cylinder have not analogy, because of the different topology of R2 . Something similar occurs with route 2. A TB bifurcation of this type is presented in the Appendix, so the diagram found there may help to better visualize the transitions. At the beginning of this route there are two stable stationary solutions that are symmetry-related and correspond to our stable front (A−state). A saddle-node bifurcation of cycles produces a stable periodic orbit and, as a consequence, an interval with multistability. This new orbit would correspond to the two counterpropagating solutions of the FHN model. Finally, the stationary solutions become unstable through a subcritical Hopf bifurcation.
4.8
Discussion
It has been shown that an array composed of (symmetric) bistable Lorenz units undergoes a front bifurcation originating two counterpropagating traveling solutions. The mechanism consists in a Hopf bifurcation of the static front, followed by a global bifurcation equivalent to a gluing bifurcation of cycles onto a cylindrical phase space. Accordingly, close to the threshold, the period of oscillation of the front (T1 ) and the speed of the front (c) follow logarithmic laws. A saddle static solution mediates the gluing process, in such a way that the value of its unstable eigenvalue determines the rate of divergence of T1 and c−1 . The transition is typically discrete, and is possible thanks to the multi-variable and non-gradient nature of the local dynamics. Also, it has been demonstrated that the gluing transition may be mediated by a saddle-focus point. In that case, when the value of the saddle index (δ) is below or close to one, the transition becomes much more convoluted. Different oscillating and traveling regimes are observed, including chaotic motion of the front due to Shil’nikov chaos. In Sec. 4.7 we have dealt with other couplings and other bistable systems, obtaining similar results (see also the next Chapter). The case of the discrete FitzHugh-Nagumo model shows a different transition, in which a region of coexistence between static and traveling fronts is found. For the system of Lorenz oscillators the oscillating-front region decreases as the coupling D becomes larger, such that it becomes virtually zero at infinity. Something similar occurs with the FHN model, because the
4.8 Discussion
113
coexistence region shrinks to zero at the infinite coupling limit. Our results indicate that the transition point between static and traveling fronts located at infinity can be considered a codimension-two point where the stable and the unstable dislocation solutions coalesce (because of being at D = ∞). At the same time, they present a double zero eigenvalue (also known as Takens-Bogdanov degeneracy). At present, a normal form does not exist for the Takens-Bogdanov bifurcation in so special a situation as the one considered here. Nonetheless, a comparison with the unfolding for the case when symmetry under rotation through π is imposed let us to make some conjectures. We believe that the two transitions shown here are the only ones to be expected (at least, at large diffusion) when considering one-dimensional reaction-diffusion systems where the local dynamics presents symmetric bistability. Of course, we are restricting to the case where the local dynamics exhibits one (centered) unstable fixed point. This is quite natural if one is considering multi-variable generalizations of the (one-variable) Nagumo equation. Also, it is important to note that any numerical computation introduces a discretization of a system and therefore, one of the two transition types shown here should exist in a small parameter region around the parity front bifurcation point. In our opinion, further investigation is needed, to know a priori what local dynamics and what couplings (besides some trivial considerations) are suitable to achieve propagation in a discrete symmetric bistable medium.
114 Onset of Traveling Fronts in an Array of Coupled Symmetric Bistable Units
Chapter 5
Spatio-Temporal Patterns in an Array of Non-Diagonally Coupled Lorenz Oscillators Abstract. The effects of coupling strength and single-cell dynamics (SCD) on spatio-temporal pattern formation are studied in an array of Lorenz oscillators. Different spatio-temporal structures (stationary patterns, propagating wavefronts, short wavelength bifurcation) arise for bistable SCD and two well differentiated types of spatio-temporal chaos for chaotic SCD (in correspondence with the transition from stationary patterns to propagating fronts). Front propagation in the bistable regime occurs through a route studied in the previous chapter while the short wavelength pattern region emerges through a pitchfork bifurcation.
5.1
Model
Now, we consider an array of Lorenz oscillators coupled though a nearest neighbor scheme with a coupling matrix: 0 1 0 Γ = 0 0 0 (5.1) 0 0 0 We decided to focus with this coupling matrix since in this case, as it will be shown later, the system undergoes a short wavelength bifurcation. Thus, our dynamical equations are: x˙ j y˙ j z˙j
= σ(yj − xj ) + D 2 (yj+1 + yj−1 − 2yj ) = rxj − xj zj − yj j = 1, . . . , N = xj yj − bzj
(5.2)
116 Spatio-Temporal Patterns in an Array of Non-Diagonally Coupled Lorenz ... Off–diagonal diffusive type coupling appears in mechanical models with elastic junctions, like the stick-slip Burridge-Knopoff model (see e.g.[CHGP97] and references therein). Moreover, in terms of an electronic caricature, this coupling represents an active transmission line where cells are coupled through capacitances and inductances. Both periodic and null-flow boundary conditions are used. The parameters σ and b are fixed at the values σ = 10 and b = 8/3. The other relevant parameter besides the coupling strength D is r, that controls the internal state (chaotic or bistable) of the oscillators. Recall that for the dynamics of a single unit there exists a critical value, which for the given values of σ and b is rH ≈ 24.74. For 1 < r rH all the fixed points are unstable and the unit exhibits the well known Lorenz strange attractor. A fourth-order Runge-Kutta method was used to integrate Eq. (5.2). A diagram of the patterns obtained by varying D and r is shown in Fig. 5.1. We analyze the bistable region in Sec. 5.2 and the chaotic one in Sec. 5.3.
5.2
Bistable region
As mentioned above, the single–unit fixed points C± are stable in the range (1, rH ). Therefore, the whole system is multistable in that range1 for D = 0.
5.2.1
Traveling fronts
The transition from standing to traveling fronts is very similar to the one explained in the previous chapter. As the coefficient D is increased from zero, boundaries between domains of both solutions (C+ and C− ) become smoother than the step–like boundary observed for D = 0. That is, some oscillators move to the vicinity of C+ and C− . At a given value of D the boundaries start to oscillate, i.e. they undergo a Hopf bifurcation, but they do not propagate, until finally over a threshold Dth propagation occurs. If the array is open (null-flow boundary condition), the whole system collapses finally to one of the two stable solutions. On the other hand, for the case of a ring (periodic boundary condition) stationary traveling wavefronts solutions can be found. 1
For values of r smaller than, but very close to rH , the chaotic attractor coexists with the fixed points C± , but this is of minor importance here.
5.2 Bistable region
117
Figure 5.1: Phase space of the patterns obtained for an array of coupled Lorenz oscillators. Left regions correspond to r < rH where Lorenz oscillator is bistable, and right regions correspond to r > rH where Lorenz oscillator is chaotic. A representative temporal evolution of a ring is located for each zone. They display in gray scale the x-coordinate vs. time that is running downwards; initial conditions are random. Each region corresponds to a different characteristic dynamics. Region I (light yellow): multistability; Region II (light green): front propagation; Region III (light blue): short wavelength ordering; Region V (dark yellow): spatio-temporal chaos; Region VI (dark green) : spatiotemporal chaos with front propagation; Region VII (dark blue): short wavelength ordering with transient chaos; Region IV (red): spatio-temporal chaos without clusters. Solid lines correspond to theoretical results: DC , DN , Eqs. (5.4, 5.5) respectively.
118 Spatio-Temporal Patterns in an Array of Non-Diagonally Coupled Lorenz ... 8 4
8
8
8
4
4
4
0
0
0
y
C+ 0 C− −4
−4
−4
D=0
y
−8 −8
−4
0
4
−4
8
−8 −8
−4
0
4
D=7.4
D=6.93
D=6 8
−8 −8
−4
0
4
8
−8 −8
8
8
8
8
4
4
4
4
0
0
0
0
−4
−4
−4
D=8 −8 −8
−4
0 x
4
8
−4
0 x
4
0
8
−8 −8
4
8
−4 D=8.8
D=8.6 −8 −8
−4
−4
0 x
4
D=8.9 8
−8 −8
−4
0 x
4
8
Figure 3
Figure 5.2: Transition to propagation of a front from the trivial solution at D = 0 as D etofal. increases. A projection onto theD. x −Pazo y plane the(2000) trajectory followed by the oscillators of the array is shown for r = 8 and different values of the coupling strength D. For 0 < D < 6, 92 the system is odd symmetric in the x − y plane. For D ≈ 6.92 this symmetry is broken through a pitchfork bifurcation; but is recovered again for D ≈ 6.95. At D ≈ 7.55 the system undergoes a supercritical Hopf bifurcation so the front starts to oscillate. A further increase of D enlarge the orbit of each oscillator. Finally, for D ≈ 8.85 all the orbits collide with their neighbor orbits. For a larger D propagation occurs.
In order to get a more precise knowledge of the bifurcation arising in the system we consider a single front and represent in Fig. 5.2 the projection, onto the plane x − y, of the trajectory followed by all the oscillators of the array. For D = 0, oscillators are in a step-like configuration, as D increases, few cells in the neighborhood of the border of the step-like initial condition go to nearby points to C+ and C− . At D ≈ 6.92 the symmetry of the system is broken through a pitchfork bifurcation (note that it does not induce propagation), and is recovered when one of the oscillators reaches the origin at D ≈ 6.95. At D ≈ 7.5 the stationary front solution becomes unstable through a supercritical Hopf bifurcation. This corresponds to the point where the boundary between both domains starts to oscillate. The amplitude of the oscillations grows with D. Finally for D = Dth ≈ 8.85 the orbit of each oscillator collides with the orbits of its
119
5.2 Bistable region 8 7
8 1/c T1 /2
(a)
(b)
7
6
6
5
5
4
4
3
3
2
2
1/c T /2
1 −8
−6
~ −λ−1 u
1
1 −1.5
−1
−0.5
0 0.5 D − Dth
1
1.5
50
50
40
40
−4 −2 ln | D−Dth |
0
Figure 5.3:1/c1/c (solid line) and T1 /2 (squares) as a function of D − Dth (a) and (c)8.841526 and λ−1 ≈ 0.637. (d) /2 for r = 8. D ≈ ln(|D − Dth |)T1(b) u 2 . This collision originates a multiple heteroclinic connection. neighbors 30 30 As we know, if the system is infinite, in terms of global coordinates, the 20 situation can be reduced to a pair 20of symmetry related homoclinic 1/c /2 connections in a cylindrical phase space. Thus, inT1the last picture of Fig. 5.2, 10 propagation occurs the oscillators follow 10 when a trajectory from C+ −1.5 −1 −0.5 0 0.5 1 1.5 −8 −6 −4 −2 0 to C− Dth | D−Dth | close to the set ofD −heteroclinic orbits. The opposite lnsolution (from C− to Figureis 2, odd D. Pazo & V. Perez−Muñuzuri C+ ) is also possible and symmetric with(2001) respect to the one displayed in Fig. 5.2. The front velocity (c), near the onset, obeys a logarithmic law as the one shown in the previous chapter (Eq. (4.10)). The results for r = 8 may be seen in Fig. 5.3 where the oscillation period can also be observed. The abrupt increasing from zero of the velocity of the front can be appreciated in Fig. 5.4. Finally, it must be pointed out that at larger values of r the transition to traveling fronts shows an interval of D with chaotic propagation, due to Shil’nikov chaos, equivalent to the one explained in the previous Chapter.
5.2.2
Short wavelength bifurcation
The trivial homogeneous solutions r1 = r2 = · · · = rN = C± are stable up to a critical value of the coupling parameter D. For this value, a short wavelength bifurcation [HPC95] arises (see Fig. 5.1). This new pattern can coexist with front propagation; now we observed a zig-zag pattern in each domain at both sides of the wavefront. 2
We have measured such distance observing that the distances between neighboring orbits really fall to zero for D = Dth
120 Spatio-Temporal Patterns in an Array of Non-Diagonally Coupled Lorenz ... Figure 4 D. Pazo et al. (2000) 0.8
c (cells/t.u.)
0.6
Figure 5.4: Front propagation speed c as a function of the coupling strength D for r = 8.
0.4
0.2
0
9
9.5
10
10.5
D
1
0
D=1
−2
λ
max
−1
−3
D = 11 −4
−5
0
0.1
0.2
0.3
0.4
0.5
k/N Figure 5.5: Largest eigenvalues at the uniform states in C± as a function of the wave number k for D = 1, 2, ..., 11 and r = 20.
Figure 5 D. Pazó et al. (2000)
121
5.3 Chaotic region
For a ring, the observed critical value fits very well to that calculated (DC ) from a linear stability analysis of the Fourier modes around any of the fixed points C± . The linearized equation for the Fourier modes (χk ) is
χ˙ k = S(k)χk ,
−σ σ − 2D sin2 ( πk 0 N) S(k) = 1 −1 ∓˜ c ±˜ c ±˜ c −b
(5.3)
p where c˜ = b(r − 1). The real part of the most unstable eigenvalue is plotted in Fig. 5.5 as a function of k for different values of D. For k = 0 the stability is independent of D and the fixed point is a focus (because the less attracting eigenvalues are complex conjugates). However, for a large enough value of D shortest wavelength Fourier modes (k ≈ N/2) are governed by the real eigenvalue. The maximum value of this eigenvalue is at k = N/2 (shortest wavelength) and crosses zero at D = DC . At this point, the determinant of the Jacobian matrix S(k = N/2) is zero. This condition gives the value of DC , DC = σ
(r − 1) . (r − 2)
(5.4)
The result for an open array is the same (or the same up to order O( N12 ) depending on the exact form of the boundary condition, see the book of Barnett [Bar96] for details). Above this critical value other modes become unstable and the dynamics turns out to be more complex than the one shown in the top pictures of Fig. 5.1.
5.3
Chaotic region
At r = rH the single–unit fixed points C± become unstable through a subcritical Hopf bifurcation. The behavior of an isolated Lorenz oscillator then becomes chaotic for r ≥ rH . The system does not possess any stable stationary or periodic state. For D = 0 the oscillators are uncoupled and the whole system is highly chaotic. The situation persists for low values of D. Finally, as the coupling increases, the system tends to form clusters around C+ and C− . The development of clusters occurs in a very smooth way, hence a precise limit cannot be assigned to such transition. Anyway, an analytical calculation evidences a tendency to cluster formation as r grows. The latter proceeds through calculating the eigenvalues of the system at the uniform state (all
122 Spatio-Temporal Patterns in an Array of Non-Diagonally Coupled Lorenz ...
D=0
D=1
D=2
D=3
D=4
D=5
D=6
D=7
D=8
D =9
D = 10
D = 11
Figure 5.6: Different behaviors for several couplings (D) for r = 28 (the same initial condition was imposed in all cases). The behavior of the system changes drastically at two values of D. One point separates propagation and no-propagation of fronts, and is located between D = 6 and D = 7. The second point separates the zone where short wavelength bifurcation arises, and is located just above D = σ = 10. In the picture corresponding to D = 11 the system suffers a short wavelength bifurcation with two dislocations (located at the center and at the right-hand-side).
123
5.3 Chaotic region 3
2
z− (r−1)
1
Figure 5.7: Dynamics of one oscillator for D = 8 and r = 25 in a reference framework where phase can be readily computed.
0
−1
−2
−3
−2
−1
0
1
2
(x2+y2)1/2− (2b(r−1))1/2
Figure 7 D. Pazó et al. (2000)
the oscillators are in the same fixed point). We first notice that at D = 0 all Fourier modes are unstable (assuming r > rH ). But as long as D increases enough, there is a Fourier mode that becomes stable. This mode is k = N/2 (the shortest wavelength mode); the fact that this mode becomes stable makes the dynamics of neighboring oscillators not to diverge so strongly as before; this can be considered as the beginning of cluster formation. The value of D corresponding to this point is a function of r, DN =
(σ + 1 + b)b(σ + r) − 2σb(r − 1) . 2(σ + 1 + b(3 − r))
(5.5)
The corresponding curve is shown in Fig. 5.1. On the other hand, the maximum transverse Lyapunov exponents λk≥1 of each mode χk calculated around the chaotic synchronized state (see e.g. [ZHY00]) shows a dependence with D equivalent to the behavior of the Lyapunov exponents at the fixed points, but the mode k = N/2 becomes stable at a larger value of D. The system prefers to form clusters around the fixed points than to form an extended coherent structure (consisting for example of some cells of the array partially synchronized) with the Lorenz chaotic attractor as the basis for each cell dynamics. Since our clusters are defined as longlasting ensemble of cells close to C+ or C− , transverse Lyapunov exponents around the chaotic synchronized state do not give any new information as our clusters are far away from it. At higher values of D, for which traveling waves appeared for r < rH , now the array forms clusters whose boundaries propagate as traveling waves (see Figs. 5.1 and 5.6). Because of the instability of the fixed points, front reversals are observed as well as the spontaneous formation of new clusters through the appearance of two counter-propagating fronts. The formation of new clusters becomes more frequent as r grows. So, for r
124 Spatio-Temporal Patterns in an Array of Non-Diagonally Coupled Lorenz ... just above rH the whole array stays for a long time with all the oscillators turning around one of the fixed points (either C+ or C− ), until two counterpropagating fronts are able to emerge. Recall that independently of the initial conditions, the onset of traveling fronts makes the system collapse around C+ or C− . For this situation, the phase (φ) and amplitude (A) can be properly defined [PROK97] by means of the projection shown in Fig. 5.7. We chose to define phase and amplitude as follows3 : ! p x2 + y 2 − 2b(r − 1) φ = arctan z − (r − 1) 2 p A2 = | x | − b(r − 1) + p
2 p | y | − b(r − 1) + (z − (r − 1))2 .
(5.6)
(5.7)
There are domains where the phase of neighboring oscillators is highly correlated (see Fig. 5.8). The limits of these domains are oscillators whose amplitude is zero or close to zero, and could be considered as defects (in analogy with the Ginzburg-Landau equation phenomenology). The subcritical Hopf bifurcation observed in the Lorenz oscillator does not involve a pair of stable limit cycles (one for C+ and another for C− ). There is not a range of values of r where the system presents multistability between a limit cycle and a fixed point [DP90, NMV96]. Therefore the transition observed in the array cannot be considered simply as a discretized subcritical-G-L transition in C+ or C− [FT90, vSH90]. Finally, it must be pointed out that as r grows, the fixed points C± become more and more unstable. As a consequence, cluster sizes get smaller and eventually it is not possible to distinguish between non–propagating and propagating regions. With respect to the short wavelength bifurcation arising in the bistable region, this structure remains stable beyond rH . The (Hopf) instability of C± does not affect the zig-zag structures bifurcated from the uniform solutions. An analytical calculation of the stability of these patterns is not 3 A stricter way to define φ and A would be to pass to the two-dimensional manifold where the subcritical Hopf bifurcation takes place. In this 2-D manifold a GinzburgLandau equation should be found in such a way that φ and A are strictly defined. Nonetheless, our definitions are one-to-one related with those whose definition came from the G-L equation. Furthermore, it would be intriguing how to cast a definition valid for C+ and C− . Hence, the important features of the dynamics will be readily provided by our operational definitions.
125
5.3 Chaotic region
r = 25
r = 26
r = 27
r = 28
Figure 5.8: The system for D = 8 and r = 25, 26, 27, 28. Each value of r shows three pictures, from left to right: x variable, amplitude and two-color-discretized phase. As long as r increases phase correlation diminishes because of the appearance of more fronts and defects (small amplitude, i.e. white color).
126 Spatio-Temporal Patterns in an Array of Non-Diagonally Coupled Lorenz ... feasible because the solutions depend, both on r and D. We can however argue why the result obtained in Eq. (5.4) can be extended successfully to the chaotic region (Fig. 5.1). In this region the mode k = 0 is unstable, and as r increases further from rH some long wavelength modes (k = 1, 2, · · · ) become also unstable. So in this zone the problem is harder to treat than in the bistable one, because there are more than one unstable modes. The dynamics of the system is determined by the behavior of each Fourier mode. Whereas k = 0 is unstable (independently of D), long wavelength modes stabilize as D increases. On the other hand, short wavelength modes become stable for low D but turn unstable again when D approaches DC . Therefore, the shortest wavelength modes govern the dynamics at high values of D. Due to the chaotic dynamics existing for r > rH the system is eventually close enough to one of the uniform solutions, and then, is shifted to one of the “frozen” zig-zag patterns.
5.4
Discussion
This chapter studies an array of oscillators where wavefront solutions arise. In the bistable region the transition has been demonstrated to be equivalent to that shown in the previous chapter for on-diagonal coupled oscillators. Accordingly, logarithmic profiles for the oscillation period and the velocity of the front are obtained. We have extended our study to the parameter range where the fixed points become unstable (r > rH ). For D < DC , the system exhibits two types of spatio-temporal chaos depending on whether exists or not front propagation. In the propagating region we find two characteristic processes: spontaneous creation of counter-propagating fronts and front reversal. The boundary that separated propagating from non-propagating regions in the bistable case separates now these two kinds of spatio-temporal chaos. We have also found that over a certain coupling the system undergoes a short wavelength bifurcation. This kind of bifurcation is observed in discrete systems only (as well as the onset of propagating fronts by the route explained in this thesis). The preponderance of the short wavelength modes at large D suggests the lack of a continuum limit for the kind of coupling studied in this chapter. We have also observed that this short wavelength pattern inhibits the spatio-temporal chaos beyond rH , giving rise to an ordered pattern. All the necessary conditions to achieve the behaviors described above are still a matter of future research.
Chapter 6
Conclusions and Outlook Interaction between oscillators is an issue of interest for diverse disciplines ranging from biology to engineering. Throughout this work several techniques have been used in order to elucidate different phenomena occurring in this kind of systems: synchronization, transition to highdimensional chaos, wavefronts, ... In the first part of this thesis the problem of the onset of phase and lag synchronization between chaotic oscillators has been addressed. To get a deeper insight on the transitions occurring in the system, we have focused on the invariants inside the attractor. The stabilization of the unstable periodic orbits allowed us to grasp the profound changes occurring in the invariant set when transiting between different states. In particular, the observation of intermittent exotic lag configurations just before perfect lag synchronization settles, was clarified. It is explained by the approach of the trajectory to an UPO, that was born at an out-of-phase locking in the transition to phase synchronization. The second chapter deals with a transition to high-dimensional chaos occurring in a ring of three unidirectionally coupled Lorenz oscillators. Although the system exhibits three-frequency quasiperiodicity, it is a global bifurcation between cycles which creates the chaotic set; and, accordingly, a chaotic transient is observed. The mean chaotic transient diverges following a power law when approaching the point where the chaotic set becomes attracting through a boundary crisis. In this transition, there exists a double heteroclinic connection to a pair of unstable three-frequency quasiperiodic tori. Finally, after a small parameter range of coexistence between chaos and quasiperiodicity, the mentioned pair of unstable 3tori collide simultaneously with the two stable 3-tori through saddle-
128
Conclusions and Outlook
node bifurcations. The (remaining) chaotic attractor has an information dimension, obtained by the Kaplan-Yorke formula, above four. Finally, the robustness and generality of the route to high-dimensional chaos is discussed along one section. The largest part of this thesis is devoted to the onset of traveling fronts in discrete bistable media. We consider reaction-diffusion systems where the local dynamics is bistable and symmetric, so there is not a preferred state. We have observed two characteristic routes leading to traveling fronts. In the first one, the static front becomes unstable through a Hopf bifurcation giving rise to a oscillating front. Finally, a global bifurcation consisting on two homoclinic loops creates two counterpropagating traveling solutions. The velocity of these traveling solutions obeys a logarithmic law with regard to the distance to the critical point, which is visible as an abrupt increase from zero at that point. We have also observed that, in some circumstances, the double homoclinic loop connects a saddle-focus equilibrium. In this case it is possible to find a much more convoluted transition between oscillating and traveling front, such that at some values of the coupling the front propagates in an irregular manner which is a manifestation of the underlying Shi’lnikov-type chaos. For the discrete FitzHugh-Nagumo model the transition is different. Instead of a regime with oscillating front, there exists a region of coexistence between static and traveling fronts. In this route –as well as in the route that involves an oscillating front– the intermediate region shrinks to zero in the infinite coupling limit, that corresponds to the continuum limit. We have seen that the point located at infinity, where the transition occurs, is a double zeroeigenvalue point and therefore the bifurcation lines emerging from that point should be predicted by the normal form of such codimension-two bifurcation. Since our case is geometrically somewhat special, we have not found in the literature the corresponding normal form. Nonetheless, the normal form for the double-zero eigenvalue with symmetry (in the plane) presents two characteristic cases that present strong analogies with the transitions we have observed in discrete bistable media. Therefore we conjecture that only two possible scenarios are expected to be found in the infinite coupling limit when considering the transition to propagation in discrete bistable media. The last part of this thesis studies the transitions occurring in an array of non-diagonal coupled Lorenz oscillators. In this case we considered bistable as well as chaotic Lorenz oscillators. In the first case, there exist the transition to traveling front through an oscillating front; but
129 furthermore, for large coupling the system undergoes a short wavelength bifurcation. In the chaotic case, the transition between oscillating and traveling fronts becomes a well-differentiated transition between two types of spatio-temporal chaos. Also, the short wavelength bifurcation inhibits spatio-temporal chaos giving rise to an ordered pattern.
Outlook Regarding the onset of chaotic phase synchronization, it would be desirable to develop an efficient technique to find unstable tori (something like the ones we have used to find unstable periodic orbits). This would make possible to get a finer description of the onset of chaotic phase synchronization. Most theoretical works are interested in the rigorous proof of the existence (or not) of a horseshoe, under a given global bifurcation. On the other hand, other scientists do not try to give a geometric view of their transitions to chaos and quite often the computation of the Lyapunov exponents is their only interest. It is to be expected that a stronger cooperation between pure mathematicians and more applied scientists will allow to elucidate the general mechanisms leading to high-dimensional chaos. In what concerns the transition to traveling front in bistable media, it is evident that a normal form of the codimension-two point (on a cylinder) should be developed in order to get a stronger proof of universality. Also, the necessary conditions to observe a transition to traveling front should be established.
Appendix A The FitzHugh-Nagumo cell: a case of a Takens-Bogdanov codimension-two point The aim of this Appendix is to describe the parameter space of a cell of the FitzHugh-Nagumo type. This system contains two variables (u, v) that obey the following differential equations:
u˙ = u − u3 − v
(A.1)
v˙ = (u − a1 v − a0 )
(A.2)
The parameter measures the different time scales of both variables and, like in Chapter 4, we restrict here to the case a0 = 0, what makes our system symmetric under reflection (u, v) → (−u, −v). The trivial solution (0, 0) undergoes a pitchfork bifurcation at a1 = 1 giving q rise forqa1 > 1 to two symmetry related solutions (u± , v± ) =
). For a system with two variables the stability of , ± a11 a1a−1 (± a1a−1 1 1 a fixed point may be studied considering the values of trace (tr) and determinant (∆) of the Jacobian matrix J at that point. In our system J takes the form: J=
1 − 3u2 −1 −a1
(A.3)
132
Appendix A
If the determinant is negative the fixed point is a saddle with two eigenvalues λu > 0 and λs < 0 (notice that the trace is equal to λu + λs and therefore depending on the sign of the trace either the stable or the unstable eigenvalue is the largest in absolute value). In Fig. A.1 the locus of ∆0 = 0 (⇒ a1 = 1) is marked with a black dash-dotted line. Accordingly, the solution at the origin undergoes a pitchfork bifurcation at this line which makes that solution to be a saddle at the right of this line, where ∆0 < 0.
Figure A.1: The parameter space of the FHN cell. Black and blue lines are bifurcation lines of the fixed points (0, 0) and (u± , v± ), respectively. The black dash-dotted line (a1 = 1) is the locus of a pitchfork bifurcation. Thus, only one stationary solution exists at the left of this line and three at its right. The red line (G) is the locus of the gluing bifurcation between two unstable cycles. The green line (SN) corresponds to a saddlenode bifurcation of cycles. The tr± =0, G and SN lines are born at the codimension-two point located at (a1 , )=(1,1). Between tr± =0 and SN, a stable limit cycle and two symmetry related stable stationary solutions coexist.
A positive determinant offers a larger amount of characteristic dynamics depending on the values of the trace. Thus, if the trace is positive (resp. negative) the fixed point is unstable (resp. stable). If ∆ < tr2 /4 the equilibrium is a node whereas if ∆ > tr2 /4 it is a focus. Hence, if ∆ > 0 the line tr = 0 marks the transition from a stable focus to an unstable focus
The FitzHugh-Nagumo cell: a case of a Takens-Bogdanov ...
133
(or viceversa). This means that tr=0 corresponds to a Hopf bifurcation provided the system to have nonlinearities (otherwise at this line the system would be conservative and the transition would be mediated by a center). Solid lines in Fig. A.1 are the loci of tr0 = 0 and tr± = 0 with black and blue colors respectively. It is easy to check that the origin has two degenerate vanishing eigenvalues at (a1 , ) = (1, 1). This is a (local) codimension-two point that comprises the Takens-Bogdanov bifurcation; in fact, the pitchfork and the Hopf bifurcation lines meet at this point1 . The Takens-Bogdanov bifurcation may be found in several books [GH83, Wig90, Kuz95]. Nevertheless, the situation we are dealing here, that involves a symmetry under reflection, is to our knowledge only considered in the book of Guckenheimer and Holmes [GH83]. Every system undergoing a Takens-Bogdanov bifurcation may be reduced, by successive changes of variables, to its corresponding normal form (this is a mathematically convoluted although somewhat mechanical at the same time). Depending on the system, the TB with symmetry is found to have two different realizations (allowing time reversal). We observe one of both here, and it is characterized by three bifurcation lines (see Fig. A.1) that emanate from the TB point: a Hopf (tr± = 0, solid blue), a saddle-node of cycles (SN, green), and a double homoclinic connection (G, red) where a gluing bifurcation between unstable cycles occurs. Finally, it is to note that at (a1 , ) = (1.5, 0). there exist another codimension two point. This point is quite special because it is the simultaneous occurrence of a TB bifurcation for the two symmetry related fixed points (u± , v± ).
1 If a0 were considered as a third parameter we could say that we are studying a codimension-three point, and therefore letting a0 to vary around zero would give a picture much more complex (and complete at the same time) than the one studied here.
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List of Publications This thesis gave rise to the following related publications: I. D. Paz´o, I. P. Mari˜ no, V. P´erez-Villar, and V. P´erez-Mu˜ nuzuri, “Transition chaotic phase synchronization through random phase jumps”, Int. J. Bif. and Chaos 10, 2533-2539 (2000). II. D. Paz´o, N. Montejo, and V. P´erez-Mu˜ nuzuri, “Wave fronts and spatiotemporal chaos in an array of coupled Lorenz oscillators”, Phys. Rev. E 63, 066206(1-7) (2001). III. D. Paz´o, E. S´anchez, and M. A. Mat´ıas, “Transition to highdimensional chaos through quasiperiodic motion”, Int. J. Bif. and Chaos 11, 2683-2688 (2001). IV. D. Paz´o and V. P´erez-Mu˜ nuzuri, “Onset of wavefronts in a discrete bistable medium”, Phys. Rev. E 64, 065203[R] (2001). V. D. Paz´o, M. Zaks, and J. Kurths, “Role of unstable periodic orbits in phase and lag synchronization between coupled chaotic oscillators”, Chaos 13, 309-318 (2003). VI. D. Paz´o and V. P´erez-Mu˜ nuzuri, “Traveling fronts in an array of symmetric bistable units”, (submitted to Chaos). VII. D. Paz´o and M. A. Mat´ıas, “Transition to high-dimensional chaos through a global bifurcation” (submitted to Phys. Rev. Lett.). VIII. E. S´anchez, D. Paz´o, and M. A. Mat´ıas, “Experimental study of the transitions between synchronous chaos and a periodic rotating wave” (submitted to Phys. Rev. E).
Index Newhouse-Ruelle-Takens Theorem, 65
ansatz, 61 Birman-Williams theorem, 73 bistability, 86 boundary crisis, 76
pendulum, 92 period-doubling cascade, 101 periodic rotating wave, 55, 62 phase locking, 33
chaotic rotating wave, 54 chaotic transients, 76 cluster formation, 121 continuum limit, 109 cylindrical coordinates, 91
R¨ossler oscillator, 29 reaction-diffusion equation, 85, 87 return map, 77
dynamo model, 106 eyelet intermittency, 37 FitzHugh-Nagumo 107, 131
model,
86,
Ginzburg-Landau equation, 124 gluing bifurcation, 92, 107, 133 sadle focus, 97 heteroclinic explosion, 72 high-dimensional chaos, 52 hyperchaos, 52 Kaplan-Yorke conjecture, 52 logarithmic divergence, 93 logistic map, 31 Lorenz oscillator, 87 Lyapunov spectrum, 56
saddle index, 99 Shil’nikov wiggle, 99 short wavelength bifurcation, 119, 124 slave locking, 71 synchronization, 27 lag, 28, 42 phase, 28, 36 Takens-Bogdanov 110, 133 UPO, 28
bifurcation,