coupled oscillators: chaotic synchronization, high-dimensional

UNIVERSIDADE DE SANTIAGO DE COMPOSTELA

FACULTADE DE FISICA

DEPARTAMENTO DE FISICA DA MATERIA CONDENSADA

GRUPO DE FISICA NON LINEAL

Coupled oscillators:

chaotic synchronization,

high-dimensional chaos and

wavefronts in bistable media

Memoria presentada por DiegoPazo Bueno para optar o grao de Doutoren Ciencias Fısicas pola Universidade deSantiago de Compostela.

Xaneiro, 2003

Dissertation of the Faculty of Physics,

University of Santiago de Compostela, Spain

Diego Pazo 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.

mailto:[email protected]

Vicente Perez Munuzuri, profesor titular da Universidade de Santiagode Compostela,

CERTIFICA

que a presente memoria, titulada “Coupled oscillators: chaotic synchro-nization, high-dimensional chaos and wavefronts in bistable media”, foirealizada por Diego Pazo Bueno baixo a sua direccion, e que concluea Tese que presenta para optar o grao de Doutor en Ciencias Fısicas.

E, para que ası conste, asina a presente en Santiago de Compostela a 9de Xaneiro de 2003.

V. e praceVicente Perez Munuzuri

Asdo.: Diego Pazo Bueno

A mis padres.

A g r a d e c i m i e n t o sc k n o w l e d g m e n t s

Haber acabado esta tesis me hace sentir que estos ultimos anos hanservido para algo. El camino no ha estado exento de sinsabores ni demomentos en los que me he tenido que armar de paciencia. Sin embargo, esjusto acordarse ahora de todo aquello que me han llenado de satisfacciontanto a nivel humano como cientıfico. Primeramente, he conocido a muydiversas personas de lugares aquı y alla, y de todas he aprendido algo.Haber tenido la oportunidad de visitar otros sitios y otras culturas me haenriquecido enormemente. Por otro lado, en no pocas veces, aquello nuevoque he apendido de fısica o matematicas me ha maravillado. ¡Una ecuacionpuede ser bella! Los siguientes parrafos me dan la oportunidad de expresarmi gratitud a todos aquellos a los que se la debo.

Al profesor Vicente Perez Villar, el ‘jefe’ del Grupo de Fısica no Lineal(GFNL), le doy las gracias por haberme dado la oportunidad de trabajar ensu Grupo y por prestarme su ayuda cuando la he precisado. A mi directorVicente Perez Munuzuri le estoy agradecido por guiarme y aconsejarme enmis primeros pasos en el mundo de la ciencia, y por emplear su tiempoen supervisar esta tesis. Tambien agradezco a Alberto Perez Munuzuri y aMoncho Gomez Gesteira, la ambilidad y la simpatıa con la que siempre mehan tratado.

No puedo pasar por alto a la ‘masa obrera’ del Grupo con quienes hecompartido, en muchos casos, nuestra condicion becaria (o precaria) ademasde muchos buenos momentos. Mi primera mencion debe ir hacia aquellasque han compartido conmigo durante mas o menos tiempo su pertenencia al‘grupusculo caotico’: Nieves, Ines y Noelia. Les doy las gracias por ayudarmetantas y tantas veces. Tambien quiero destacar a los companeros con quieneshe convivido en el despacho: Adolfo, Irene y David. Fue una gran suertetener un espacio de trabajo tan agradable y tan lleno de ganas de echar unmano cuando la precise. Por supuesto, tambien estoy en deuda con el restode gente con la que he coincidido en el Grupo: Bea, Carlos, Chus, Edu,Ivan, Jose Manuel, Juan, Pablo, Pedro, Manuel, Maite, Nico y Roi. Todosellos me ayudaron en mas de una ocasion y no recuerdo ni una mala cara.

Tambien merecen una resena aquellos que visitaron el GFNL duranteestos anos pasados. A todos ellos les agradezco su simpatıa. En especial aRoberto Deza le agradezco sus sugerencias en la elaboracion de mis artıculosy de esta tesis.

Por lo que respecta a la gente ajena al Grupo merece una mencionespecial Manuel Matıas con quien llevo colaborando desde hace ya variosanos. Le estoy agradecido por haber aprendido tantas cosas y por suhospitalidad en mis estancias en Salamanca primero, y en Palma deMallorca despues. En este punto, tambien quiero hacer partıcipes de migratitud al resto de miembros del imedea de Palma.

Ich mochte auch Jurgen Kurths and Misha Zaks fur ihre Hilfe undMitarbeitung danken. Ich fand meine deutsche Erfahrung in der PotsdamUniversitat sehr ergiebig und interessant. Mi estancia en Potsdam me obliga aacordarme de mas gente a la que estoy agradecida, principalmente de ErnestMontbrio gracias al cual habituarme a un nuevo paıs me resulto mucho masfacil.

Tambien me gustarıa aprovechar estas lıneas para agradecer la amistadque me une a tanta gente fenomenal en muchos sitios (Leon, Madrid,Pontevedra, ...).

Mi mayor gratitud es para mis padres, por preopuparse por mı yquererme tanto.

Para rematar, desexo agradecer as institucions oficiais o apoioeconomico sen o cal serıa imposible para min ter feita esta tese. Estas son aSecretarıa Xeral de Investigacion e Desenvolvemento da Xunta de Galiciae a Universidade de Santiago de Compostela a traves do seu Vicerrectoradode Investigacion.

Santiago de Compostela, Enero 2003 Diego

Contents

Resumen xix

Summary xxvii

1 El Caos: Fundamentos 11.1 Introduccion . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Bifurcaciones locales . . . . . . . . . . . . . . . . . . . . . . 51.3 Rutas al caos . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.1 Cascada de duplicacion de periodo . . . . . . . . . . 91.3.2 Intermitencia . . . . . . . . . . . . . . . . . . . . . . 111.3.3 Cuasiperiodicidad . . . . . . . . . . . . . . . . . . . 121.3.4 Crisis . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3.5 Bifurcaciones globales . . . . . . . . . . . . . . . . . 16

1.3.5.a Sistema de Lorenz . . . . . . . . . . . . . . 171.3.5.b Caos de Shil’nikov . . . . . . . . . . . . . . 20

1.4 Orbitas periodicas inestables (UPOs) . . . . . . . . . . . . . 201.4.1 Metodo de Newton-Raphson . . . . . . . . . . . . . 21

1.5 Sincronizacion caotica . . . . . . . . . . . . . . . . . . . . . 23

2 Role of Unstable Periodic Orbits in Phase and LagSynchronization between Coupled Chaotic Oscillators 272.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Periodic orbits in the absence of coupling . . . . . . . . . . 292.3 Attractors of a coupled system: role of unstable tori in

synchronization transitions . . . . . . . . . . . . . . . . . . 322.4 Phase synchronization . . . . . . . . . . . . . . . . . . . . . 362.5 Lag synchronization . . . . . . . . . . . . . . . . . . . . . . 422.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

xii CONTENTS

3 Transition to High-Dimensional Chaos through a GlobalBifurcation 513.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2 System and overall picture . . . . . . . . . . . . . . . . . . . 54

3.2.1 Lyapunov exponents and attractors . . . . . . . . . 553.2.2 Behaviors along the line σ = 20 . . . . . . . . . . . . 58

3.3 The centered periodic rotating wave: analytical solution . . 613.4 Transition to quasiperiodic behavior . . . . . . . . . . . . . 653.5 Numerical evidences of the route to chaos exhibited by the

system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.5.1 Coexistence between 3D-torus and CRW . . . . . . . 693.5.2 Heteroclinic explosion . . . . . . . . . . . . . . . . . 723.5.3 Four-dimensional branched manifold . . . . . . . . . 723.5.4 Boundary crisis and power law of chaotic transients 76

3.6 Description in terms of a return map . . . . . . . . . . . . . 773.7 Route to chaos: theoretical analysis . . . . . . . . . . . . . 793.8 Further remarks . . . . . . . . . . . . . . . . . . . . . . . . 803.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4 Onset of Traveling Fronts in an Array of CoupledSymmetric Bistable Units 854.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.2 The system . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.3 Standing, oscillating and traveling fronts . . . . . . . . . . . 884.4 Gluing of cycles . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.4.1 Cylindrical coordinates . . . . . . . . . . . . . . . . 914.4.2 Velocity of the front as a function of D −Dth . . . . 934.4.3 Quantitative analysis . . . . . . . . . . . . . . . . . . 94

4.5 Exotic front dynamics . . . . . . . . . . . . . . . . . . . . . 974.5.1 r = 20 (δ > 1) . . . . . . . . . . . . . . . . . . . . . 1004.5.2 r = 23 (δ < 1) . . . . . . . . . . . . . . . . . . . . . 101

4.6 The effect of parameter mismatch and asymmetry . . . . . 1044.7 Universality? . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.7.1 Other couplings and systems . . . . . . . . . . . . . 1054.7.2 Large D . . . . . . . . . . . . . . . . . . . . . . . . . 1064.7.3 The discrete FitzHugh-Nagumo model . . . . . . . . 107

4.7.3.a Transition to traveling front . . . . . . . . 1074.7.3.b The continuum limit . . . . . . . . . . . . . 109

4.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

CONTENTS xiii

5 Spatio-Temporal Patterns in an Array of Non-DiagonallyCoupled Lorenz Oscillators 1155.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.2 Bistable region . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.2.1 Traveling fronts . . . . . . . . . . . . . . . . . . . . . 1165.2.2 Short wavelength bifurcation . . . . . . . . . . . . . 119

5.3 Chaotic region . . . . . . . . . . . . . . . . . . . . . . . . . 1215.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6 Conclusions and Outlook 127

Appendix A: The FitzHugh-Nagumo cell: a case of a Takens-Bogdanov 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 Rossler 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 Poincare section and return map of a Rossler oscillator . . . . . 30

2.2 Frequencies of UPOs embedded into the Rossler attractor . . . 32

2.3 Configurations favorable for the locking on the torus . . . . . . 34

2.4 Statistics of phase slips . . . . . . . . . . . . . . . . . . . . . . 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 Rossler oscillators at the beginning ofthe 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 Poincare 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 thePoinacare section Im(X1) = 0 . . . . . . . . . . . . . . . . . . . 59

3.3 Time series corresponding to different behaviors . . . . . . . . 60

xvi LIST OF FIGURES

3.4 The four largest Lyapunov exponents of a ring of threeunidirectionally coupled Lorenz systems . . . . . . . . . . . . . 61

3.5 Diagram representing schematically the transitions from syn-chronous chaos to a PRW . . . . . . . . . . . . . . . . . . . . . 62

3.6 Numerical and theoretical results for the frequency ω and themax. and the min. values of the amplitude of the coordinateX1 as function of R. . . . . . . . . . . . . . . . . . . . . . . . . 65

3.7 Poincare 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 initialcondition 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 thecritical 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 belarger than their adjacent maxima . . . . . . . . . . . . . . . . 78

3.15 Three-dimensional representation of the proposed heteroclinicroute to create the high-dimensional chaotic attractor . . . . . 79

3.16 Two-dimensional representation of the proposed heteroclinicroute 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 . . . 89

4.3 Motions of different units for standing, oscillating and travelingregimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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

LIST OF FIGURES xvii

4.12 Velocity and oscillation period for r = 20 . . . . . . . . . . . . 1014.13 Chaotic motion of a front for r = 20 . . . . . . . . . . . . . . . 1024.14 Period-doubling cascade leading to chaos . . . . . . . . . . . . 1024.15 Velocity and oscillation periods for r = 23 . . . . . . . . . . . . 1034.16 Subsidiary oscillations of the front . . . . . . . . . . . . . . . . 1044.17 Velocity vs. D in the FHN model . . . . . . . . . . . . . . . . . 1084.18 Diagram showing regions with different front dynamics in the

FHN model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.1 Phase space, for the off-diagonal coupling, of the patterns in the

plane r −D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.2 Projection onto the x−y plane of the trajectory followed by the

oscillators of the array . . . . . . . . . . . . . . . . . . . . . . . 1185.3 Logarithmic law of the front velocity for the off-diagonal coupling 1195.4 Velocity of the front for r = 8 . . . . . . . . . . . . . . . . . . . 1205.5 Largest eigenvalues at the uniform states in C± as a function of

the wave number k . . . . . . . . . . . . . . . . . . . . . . . . . 1205.6 Different behaviors for several couplings (D) for r = 28 . . . . 1225.7 Reference framework where phase can be readily computed . . 1235.8 x variable, amplitude and two-color-discretized phase for D = 8

and r = 25, 26, 27, 28 . . . . . . . . . . . . . . . . . . . . . . . . 125A.1 Regions of the parameter space of the FHN cell . . . . . . . . . 132

Resumen

Contexto del trabajo

La cooperacion entre matematicos, fısicos, ingenieros, biologos ycientıficos de otras areas del conocimiento ha dado lugar a lo queactualmente se conoce como inter-disciplinearidad. En este campo, laciencia no lineal –o simplemente la no-linealidad– se ha convertido enel punto de encuentro que proporciona un lenguaje comun, facilitandoel entendimiento entre las diferentes disciplinas. A la luz de las mismasherramientas matematicas, se explican las relaciones inesperadas entrefenomenos con una fısica (subyacente) distinta. Junto con los antiguosmetodos estadısticos, la teorıa de bifurcaciones ha resultado ser la piedraangular que unifica en el mismo marco matematico los resultados dediversidad de experimentos. Probablemente el fenomeno de la dinamicano lineal mas fascinante, y el mas popular, es el caos.

La imposibilidad de predicciones a largo plazo es la consecuencia delcaos. Esto fue primeramente reconocido por Poincare, que quedo asombradocon su complejidad geometrica. En la primera mitad del siglo XX, elcaos en los sistemas conservativos fue el principal tema de interes teorico(Birkhoff, Kolmogorov, Arnol’d, Moser, ...). Al mismo tiempo, hubo uncreciente interes en los experimentos con osciladores no lineales disipativos(Van der Pol, Appleton...). El artıculo de Lorenz de 1963 revelo lanaturaleza omnipresente del caos y la utilidad de los ordenadores, ya quelos experimentos numericos pasaron a ser facilmente asequibles.

De esta forma, desde los anos setenta la dinamica caotica ha sido objetode significativo interes para una amplia comunidad de cientıficos. Las ideasde Cantor sobre conjuntos fractales fueron recuperadas por Mandelbrot,la conexion entre caos y turbulencia fue apuntada por Ruelle y Takens, yFeigenbaum introdujo los conceptos de universalidad y renormalizacion enel contexto de los sistemas dinamicos.

xx Resumen

Hoy en dıa, las transiciones que originan el caos de baja dimensiona partir del movimiento regular (orden) estan bien caracterizadas; y lacorriente principal de investigacion se ha desplazado hacia otros temascomo el caos de alta dimension, la sincronizacion caotica, la formacion deestructuras, y las aplicaciones del caos a la biologıa y a otras ciencias.

Resumen de la memoria

El capıtulo 1 pretende dar una vision personal del mundo del caos. Esel unico que no esta escrito en ingles (el idioma hegemonico de la ciencia yde la fısica en particular) en tanto en cuanto esta pensado como un manualpara aquellos hispanohablantes que llegan al mundo del caos por primeravez. El capıtulo consiste en una introduccion a los conceptos fundamentalesdel caos ası como de las areas relacionadas con el del analisis matematico.Se presta una especial atencion a las ideas mas simples de la teorıa debifurcaciones. Esto proporciona las herramientas basicas para entender losdiferentes escenarios que llevan del movimiento regular (estatico, periodicoo cuasiperiodico) al caos. Tambien se incluye una breve descripcion de losprincipales tipos de sincronizacion caotica.

En el capıtulo 2 se trata el problema de las transiciones a lassincronizaciones de fase y de retardo [PZK03] entre dos osciladores caoticosno identicos acoplados. Como modelo de oscilador caotico con fase coherentese toma el oscilador de Rossler. Para explicar las transiciones observadasse recurre a la ayuda de las ‘orbitas periodicas inestables’ (UPOs) que seconoce se encuentran inmersas en todo oscilador caotico de baja dimension.Se muestra que el comienzo de la sincronizacion de fase corresponde a laaparicion de infinitas UPOs (con enganches 1:1) en la superficie de lostoros invariantes que existen en el lımite de ausencia de acoplamientoentre ambos osciladores. Debido al surgimiento no simultaneo de estasUPOs el sistema exhibe –para un rango del acoplamiento– sincronizacionde fase intermitente. En este estado, largos intervalos de tiempo donde lososciladores tienen sus fases sincronizadas son interrumpidos por saltos de 2πen sus fases relativas. Estos saltos ocurren cada vez con menos frecuencia,a la vez que el parametro de acoplamiento se acerca al parametro crıticopor encima del cual se establece una sincronizacion de fase perfecta. Lossaltos obedecen una ley de escala que se ha denominado ‘intermitencia derendija’. De acuerdo con esto, parece que los saltos en la fase ocurren cuandola trayectoria pasa cerca de aquellos toros que permanecen desenganchados(o estan enganchados con un numero de rotacion diferente de uno). Por

Resumen xxi

otro lado, una descripcion de la transicion a la sincronizacion de retardopor medio de las UPOs, deberıa clarificar la aparente reduccion en lacomplejidad del atractor. Se observa que la mayorıa de las UPOs quehicieron posible la aparicion de la sincronizacion de fase no estan presentescuando se alcanza la sincronizacion de retardo. Mas exactamente, lasUPOs con una estructura no adecuada para la sincronizacion de retardodesaparecen en diversas bifurcaciones. Aquellas UPOs fuera-de-fase quesobreviven para valores altos del parametro de acoplamiento dan lugara un fenomeno conocido como ‘sincronizacion de fase intermitente’. Ası,el sistema exhibe, por momentos, configuraciones con un retardo exoticomientras que permanece sincronizado con un pequeno retardo la mayorparte del tiempo. Finalmente, para un acoplamiento suficientemente grandetodas las UPOs estan asociadas a (aproximadamente) el mismo retardo, yla complejidad del atractor es la misma que la de un unico oscilador.

El capıtulo 3 se dedica a una novedosa transicion al caos de altadimension [PSM01, PM, SPM] en un sistema compuesto de tres osciladoresde Lorenz. Los osciladores estan acoplados unidireccionalmente, y de estamanera forman una geometrıa circular con simetrıa cıclica. En un planodefinido por dos parametros se identifican las regiones con diferentescomportamientos. Nos concentramos en una lınea que pasa a traves delas diversas regiones con la meta de describir las transiciones entre ellas.Para los valores mas pequenos del parametro de control R tenemos caossincronizado. En este estado los tres osciladores siguen la misma trayectoriadentro del atractor de Lorenz. Por otro lado, para valores altos de Rexiste un movimiento periodico conocido como ‘onda periodica rotante’(PRW) donde cada oscilador sigue la misma orbita periodica pero con laparticularidad de que existe una diferencia de fase de 2π/3 entre ellos. Seentiende mejor el sistema cuando las ecuaciones diferenciales ordinariasque describen la dinamica del anillo se expresan en terminos de dosmodos de Fourier discretos k = 0, 1. Ası, existe una solucion analıticamuy aproximada de la PRW, que consiste en un estado estatico para elmodo uniforme k = 0, mas una solucion sinusoidal para el modo espacialk = 1. Por lo que concierne al caos sincronizado, este estado correspondea un movimiento caotico del modo k = 0, y un modo k = 1 nulo. Laperdida del caos sincronizado, lleva a un atractor caotico de alta dimensionconocido como onda caotica rotante (CRW) donde el modo k = 1 pasa aser distinto de cero y oscilante. Entonces la dinamica es caotica al tiempoque se observa, superpuesta al modo k = 0, una dinamica oscilatoria condesplazamientos en la fase de 2π/3 entre las unidades adyacentes.

xxii Resumen

Cuando R disminuye, la PRW sufre una bifurcacion horquilla que dacomo resultado dos PRWs estables relacionadas por simetrıa. Con unamayor disminucion de R las PRWs se inestabilizan, dando cuasiperiodicidada dos frecuencias, que tambien acaba por inestabilizarse a consecuenciade una bifurcacion de Hopf secundaria. Como resultado, aparecen dosatractores cuasiperiodicos a tres frecuencias. Finalmente esto dos 3-torosparecen fundirse, y como resultado surge la CRWs. Sin embargo, un examendetallado nos permite obtener una descripcion mucho mas detallada de laformacion del atractor tipo CRW. Con este proposito, usamos diferentestecnicas para arrojar alguna luz sobre este problema: exponentes deLyapunov, secciones de Poincare, estadısticas de los transitorios caoticos,una medida de la dimension de correlacion, un mapa de retorno, ... Laprimera conclusion es que el conjunto caotico de alta dimension –que semanifiesta como un transitorio caotico– se crea para un valor de R dondelos atractores del sistema son un par de toros. Esto ocurre a traves deuna doble conexion heteroclınica de las PRWs asimetricas con la simetrica.Ası, de forma analoga al sistema de Lorenz, llamamos a este mecanismo‘explosion heteroclınica’. En esta explosion se crea un numero infinito detoros inestables tridimensionales. El conjunto caotico se vuelve atractoren una crisis de borde (de cuenca de atraccion) que involucra los dos 3-toros inestables mas simples. Para un valor ligeramente inferior de R, losdos 3-toros estables mencionados mas arriba desaparecen al fusionarse condos inestables en sendas bifurcaciones silla-nodo gemelas. Por tanto, existeun pequeno intervalo de R donde el atractor caotico de alta dimensioncoexiste con la cuasiperiodicidad a tres frecuencias. La importancia deesta ruta al caos de alta dimension reside en la ausencia de un atractorcaotico de baja dimension intermedio. Una seccion del capıtulo se dedica aanalizar la robustez de la descripcion ofrecida aquı, y hasta que punto puedeconsiderarse como una aproximacion de la ruta real (que es probablementeimposible de describir debido a su complejidad inherente).

En el capıtulo 4 [PP01, PP] se trata el problema de la aparicion deun frente viajero en un sistema reaccion-difusion discreto con biestabilidadsimetrica. Nos restringimos a sistemas cuya dinamica local consiste de dospuntos fijos estables relacionados por simetrıa, y un punto de equilibrioinestable situado entre medias. Para sistemas continuos una bifurcacion porrotura de paridad del frente puede inestabilizar el frente estatico creandodos frentes viajeros que se propagan en sentidos opuestos. La situacion paraun sistema discreto es mas complicada. Concretamente, para un conjunto deunidades de Lorenz biestables acoplados en una dimension se encuentra que

Resumen xxiii

variando un parametro (por ejemplo el acoplamiento) la transicion ocurresiguiendo la siguiente ruta: frente estatico → oscilante → viajero. En elprimer paso el frente estatico se inestabiliza por una bifurcacion de Hopfsupercrıtica, dando lugar a un frente oscilante. El periodo de oscilacioncrece hasta que finalmente diverge a infinito al llegar a un punto crıtico.Mas alla de este punto aparecen dos soluciones contra-propagantes, conunas velocidades que crecen desde cero de una forma muy abrupta. Ladependencia funcional, del periodo de oscilacion y de la velocidad del frente,en el acoplamiento se explica en el contexto de una bifurcacion collageentre ciclos. Para ver esto se construye un nuevo espacio de fases cilındricodefiniendo dos variables reducidas. La transicion de frente oscilante aviajero presenta en este nuevo espacio de fases una gran analogıa con conla transicion de libracion a rotacion en el clasico problema del pendulo.Ası en el punto de transicion existe una doble conexion homoclınica dela dislocacion (estatica) inestable consigo misma. El acercamiento de unaorbita periodica a un punto de silla viene caracterizado por una divergencialogarıtmica de su periodo. Mostramos que el periodo de oscilacion, y lainversa de la velocidad del frente exhiben esta dependencia funcional. Noobstante, la situacion puede llegar a ser mucho mas complicada si la solucioninestable que media el proceso de “pegado” es un punto silla-foco en vezde una silla. En este caso, dependiendo del valor de una cantidad conocidacomo ındice de silla, uno puede observar dinamicas del frente exoticas. Haceanos, Shil’nikov probo que una conexion homoclınica a un punto silla-focopuede generar una herradura de Smale y, por tanto, caos. Esto explica porque se observan diversos regımenes oscilantes y viajeros nuevos, incluyendoun movimiento erratico (que refleja una dinamica caotica subyacente). Losresultados obtenidos para el conjunto de osciladores de Lorenz biestablesse reproducen para diferentes tipos de acoplamiento y tambien para unalınea de modelos de dinamo biestables.

Hemos estudiado tambien la version discreta y simetrica del bienconocido modelo de FitzHugh-Nagumo (FHN). Para este modelo latransicion es algo diferente. En vez de un regimen oscilante, hay una zonadel espacio de parametros donde los frentes estatico y viajero coexisten. Lasolucion viajera aparece en una bifurcacion silla-nodo de ciclos. Para unacoplamiento mayor, las dos soluciones viajeras inestables relacionadas porsimetrıa (creadas en bifurcaciones silla-nodo gemelas y que se mueven endirecciones contrarias) se fusionan y –como resultado– se crea una solucionoscilante inestable. Finalmente esta solucion inestable se encoge hasta quecolapsa a un punto. Este punto es la solucion frente estatico estable que

xxiv Resumen

pasa a ser inestable a partir de esta colision. Este ultimo paso no es mas queuna bifurcacion de Hopf subcrıtica, a partir de la cual la solucion viajeraes la unica estable. La dependencia de la velocidad para esta ruta presentarasgos especiales. Primeramente, debido a su aparicion en una bifurcacionsilla-nodo, la solucion viajera nace con una velocidad distinta de cero. Ladependencia funcional de la velocidad es –de acuerdo con esto– una raızcuadrada. No obstante, es subrayable que mas alla de un cierto puntopuede reconocerse una dependencia logarıtmica (que es consecuencia dela proximidad al punto de silla).

Tanto para el modelo consistente en sistemas de Lorenz acoplados comopara el modelo FHN, hemos investigado el lımite de muy alta difusion,teniendo en cuenta el hecho de que en el lımite de difusion infinita serecupera, para estos sistemas, el continuo. Las dislocaciones estable einestable se funden para formar la solucion continua. Por tanto, desde unpunto de vista formal existe una bifurcacion en ese lımite. A causa de lascondiciones de simetrıa, esta puede considerarse un bifurcacion horquilla.Las lıneas de bifurcacion observadas a difusion finita para ambos modelos(ristra de osciladores de Lorenz y FHN) convergen (para acoplamientoinfinito) a un punto, que en el caso del modelo de FitzHugh-Nagumo esel punto que fue previamente encontrado al estudiar la version continua.Desde el punto de vista de los modelos discretos esta singularidad situada enel infinito es un punto de codimension dos donde dos autovalores de las dossoluciones estaticas se anulan. Esta bifurcacion es conocida como Takens-Bogdanov (TB) y ha sido estudiada por diversos autores. Sin embargo,el caso con el que nos hemos topado aquı es tan particular que no hasido tratado previamente. Ası, hemos visto la forma normal contenida enel libro de Guckenheimer y Holmes que describe una TB con la simetrıaque nuestro problema, pero con el inconveniente de que no considera unespacio de fases cilındrico. Dependiendo de un parametro interno aparecendos posibles conjuntos de bifurcaciones cerca del punto de codimension dos.Cada uno de ellos muestra fuertes analogıas con las bifurcaciones observadasen nuestros experimentos numericos. Por tanto, conjeturamos que cualquierrotura de paridad de un frente que de lugar a frentes viajeros en un sistemacontinuo biestable simetrico se transforma –al discretizarlo– en una de lasdos rutas que se describen en esta tesis. A este respecto, es importanteresaltar que la implementacion numerica de un sistema continuo siemprelleva consigo algun tipo de discretizacion, y por tanto esperamos que losfenomenos estudiados aquı se manifiesten en un escala fina.

El capıtulo 5 [PMP01] estudia un sistema 1D de osciladores de Lorenz

Resumen xxv

acoplados que muestra frentes viajeros y que esta caracterizado por unamatriz de acoplamiento no diagonal. En la region en la que el osciladorde Lorenz es biestable, la transicion a la propagacion, demuestra serequivalente a la mostrada en el capıtulo 4 para osciladores acopladosdiagonalmente. De acuerdo con esto, se obtienen perfiles logarıtmicos parala velocidad del frente y para el periodo de oscilacion. Hemos ampliadonuestro ambito de estudio a la region de parametros donde los puntos fijosdel oscilador de Lorenz se vuelven inestables, de tal forma que este pasa a sercaotico. En esta region (caotica), el sistema exhibe dos tipos de caos espacio-temporal dependiendo de que exista o no propagacion de frentes. En laregion con propagacion encontramos dos procesos caracterısticos: creacionespontanea de frentes contra-propagantes, e inversion del movimiento de losfrentes. La lınea de bifurcacion que separaba regiones con y sin propagacionen el caso biestable marca ahora la frontera entre dos tipos de caos espacio-temporal. Ademas, hemos encontrado tambien que por encima de un ciertoacoplamiento, tanto en la region biestable como en la caotica, el sistemasufre una bifurcacion de onda corta. Este tipo de bifurcacion, y la aparicionde la propagacion de frentes por la ruta explicada arriba, se observansolamente en sistemas discretos. Tambien hemos comprobado que el patronde longitud de onda corta (que emerge de la bifurcacion homonima) inhibeel caos espacio-temporal, dando lugar a un patron ordenado.

Finalmente, las conclusiones principales de esta tesis, ası como algunasperspectivas, se reunen en el capıtulo 6.

Summary

Context of the work

The cooperation among mathematicians, physicists, engineers, biolo-gists and scientists from other areas of knowledge has given rise to what isnowadays known as cross-disciplinarity. In its realm, nonlinear science –orsimply nonlinearity– has become a meeting point that provides a commonlanguage thus facilitating the understanding among disciplines. Underthe light of common mathematical tools, unexpected relations betweenphenomena with different underlying physics are explained. Together withthe old statistical methods, bifurcation theory has become the keystonethat unifies into a common mathematical framework the results of avariety 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 ofchaos. It was first noticed by Poincare, who became astonished withits geometric complexity. In the first half of the 20th century, chaos inconservative systems was the main object of theoretical interest (Birkhoff,Kolmogorov, Arnol’d, Moser, ...). At the same time, there was a growinginterest in experiments with dissipative nonlinear oscillators (Van der Pol,Appleton...). The paper by Lorenz in 1963 revealed the pervasive nature ofchaos and the utility of computers, as numerical experiments were becomingpretty accessible.

In this way, since the 1970s chaotic dynamics has been the subject ofsignificant interest by a wide community of scientists. The ideas of Cantoron fractal sets were recovered by Mandelbrot, the connection between chaosand turbulence was pointed out by Ruelle and Takens, and Feigenbaumintroduced the concepts of universality and renormalization in the contextof dynamical systems.

Nowadays, the transitions that originate low-dimensional chaos from

xxviii Summary

regular motion (order) are well characterized; and the mainstream hasshifted to other subjects as high-dimensional chaos, chaotic synchroniza-tion, pattern formation, and applications of chaos to biology and othersciences.

Summary of the thesis

Chapter 1 intends to give a personal view of the world of chaos. Ihave chosen to write it in Spanish to best serve the purpose of being aprimer for (Spanish-speaker) newcomers to the world of chaos. It consistsin an introduction to the most fundamental concepts of chaos and relatedareas of mathematical analysis. Special emphasis is given to the simplestideas of bifurcation theory. This provides the basic tools to understand thedifferent scenarios leading from regular motion (steady state, periodicity orquasiperiodicity) to chaos. Also, a short description of the main types ofchaotic synchronization is included.

In Chapter 2, the problem of the transitions to phase and lag synchro-nization in coupled non-identical chaotic oscillators is addressed [PZK03].The Rossler oscillator is taken as a model of a phase-coherent chaoticoscillator. The explanation of the observed transitions is attempted, withthe help of the ‘unstable periodic orbits’ (UPOs) that are known to beembedded in every low-dimensional chaotic attractor. It is shown thatthe onset of phase synchronization corresponds to the appearance of aninfinity of UPOs (with 1:1 locking ratios) on the surface of the invarianttori existing in the limit of no coupling between both oscillators. Due to thenon-simultaneous emergence of these UPOs the system exhibits –in somerange of the coupling parameter– intermittent phase synchronization. Inthis state, long time intervals where the oscillators are phase synchronizedare interrupted by jumps of 2π in their relative phases. These jumpsoccur less and less frequently as the coupling parameter approaches thecritical value above which perfect phase synchronization settles down. Thejumps obey a scaling law what has been denominated ‘eyelet intermittency’.Accordingly, it seems that phase jumps occur when the trajectory passesnear those tori that remain unlocked (or are locked with a rotation numberdifferent of one). On the other hand, a description of the transition tolag synchronization by means of the UPOs, should clarify the apparentreduction in the complexity of the attractor. It is shown that most ofthe UPOs that made possible the onset of phase synchronization are notpresent when lag synchronization is reached. More precisely, the UPOs

Summary xxix

with a badly suited structure for lag synchronization disappear in severalbifurcations. Those out-of-phase UPOs that survive for larger values ofthe coupling strength give rise to a phenomenon known as ‘intermittentlag synchronization’, where the system exhibits exotic lag configurations atsome moments, whereas remains lag synchronized (with small lag) most ofthe time. Finally, for a large enough value of the coupling all the UPOs areassociated to (very approximately) the same lag, and the complexity of theattractor is the same as that of a single oscillator.

Chapter 3 is devoted to a novel transition to high-dimensionalchaos [PSM01, PM, SPM] in a system composed of three coupled Lorenzoscillators. The oscillators are unidirectionally coupled, and hence theyform a ring geometry with a cyclic symmetry. On a plane spanned bytwo parameters, regions with different behaviors are identified. We focuson a (monoparametric) line crossing along the different regions with thegoal of describing the transitions among them. For the smallest valuesof 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 motionknown as ‘periodic rotating wave’ (PRW) where each oscillator followsthe same periodic orbit but with the particularity that a 2π/3 phaseshift exists between them. The system is better understood when theordinary differential equations describing the dynamics of the ring are castin terms of two discrete Fourier modes k = 0, 1. Thus, there exists avery approximate analytical solution to the PRW, that consists in a steadystate of the uniform k = 0 mode, plus a sinusoidal solution for the spatialk = 1 mode. Concerning synchronous chaos, this state corresponds to achaotic motion of the k = 0 mode, and a vanishing k = 1 mode. Loss ofsynchronous chaos leads to a high-dimensional chaotic attractor known as‘chaotic rotating wave’ (CRW) where the k = 1 mode becomes differentfrom zero and oscillating. Then the dynamics is chaotic, at the same timethat a superimposed oscillating dynamics with 2π/3 phase shifts betweenadjacent units is observed.

As R is lowered, the PRW undergoes a pitchfork bifurcation thatyields two stable symmetry-related PRWs. With a further decrease of Rthe PRWs become unstable, giving rise to two-frequency quasiperiodicity,which also becomes unstable through a secondary Hopf bifurcation. Asa result, two symmetry-related three-frequency quasiperiodic attractorsappear. Finally, these two three-tori seem to merge, and as a consequencethe CRW emerges. However a detailed examination allows us a much more

xxx Summary

accurate description of the formation of the CRW attractor. With this aim,we used different techniques to shed some light into this problem: Lyapunovexponents, Poincare sections, statistics of the chaotic transients, a measureof the correlation dimension, a return map, ... The first conclusion isthat the high-dimensional chaotic set –that manifests itself as a chaotictransient– is created at a value of R where the attractors of the system area pair of two-tori. This occurs through a double heteroclinic connectionof the asymmetric PRWs with the symmetric one. Thus, analogously tothe Lorenz system, we call this mechanism ‘heteroclinic explosion’. Inthis explosion, an infinite number of unstable three-dimensional tori iscreated. The chaotic set becomes attracting at a boundary crisis thatinvolves the two simplest unstable three-tori. At a slightly smaller value ofR the stable three-tori mentioned above coalesce with the unstable ones intwin saddle-node bifurcations. Therefore, there exists a small interval of Rwhere the high-dimensional chaotic attractor coexists with three-frequencyquasiperiodicity. The importance of this route to high-dimensional chaoslies in the absence of an intermediate low-dimensional chaotic attractor.A section of the chapter is devoted to analyze the robustness of thedescription explained here, and to what extent can it be considered just anapproximation of the real route (that is probably impossible to be describeddue to its inherent complexity).

In Chapter 4 [PP01, PP] the problem of the onset of a travelingfront in a discrete reaction-diffusion system with symmetric bistability isaddressed. We restrict to systems whose local dynamics consists of twostable symmetry-related fixed points, and one unstable equilibrium locatedin between. For continuous systems, a parity-front bifurcation may renderunstable the static front, creating two counterpropagating traveling fronts.The situation for a discrete system is more complicated. Concretely, foran array of bistable Lorenz units it is found that varying a parameter(the coupling strength, for example) the transition occurs following thisroute: static → oscillating → traveling front. In the first step the staticfront becomes unstable through a supercritical Hopf bifurcation, givingrise to an oscillating front. The oscillation period grows until it diverges toinfinity at a critical point. Beyond this point two counter propagating frontsolutions appear, with their velocities growing from zero in a very abruptway. The functional dependence of the oscillation period and the velocity ofthe front on the coupling is explained in the context of a gluing bifurcationof cycles. To see this, a new cylindrical phase space is constructed definingtwo reduced variables. The transition from oscillating to traveling front

Summary xxxi

presents in this new phase space a strong analogy with the transitionfrom libration to rotation in the classical pendulum problem. Thus, in thetransition point there exists a double homoclinic connection of the unstable(static) dislocation of the front with itself. The approach of a periodic orbitto a saddle point is characterized by a logarithmic divergence of its period.We show that the oscillation period, and the inverse of the velocity ofthe front exhibit this functional dependence. Nonetheless, the situationmay become much more convoluted if the unstable solution mediating thegluing 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 thata homoclinic connection to a saddle-focus point may generate a Smalehorseshoe and, therefore, chaos. This explains why several new oscillatingtraveling regimes of the front are observed, including an erratic motion(reflecting an underlying chaotic dynamics). The results obtained for thearray of bistable Lorenz oscillators are reproduced for different couplingtypes and also for an array of bistable dynamo models.

We have also studied the discrete and symmetric version of the well-known FitzHugh-Nagumo (FHN) model. For this model, the transitionis somewhat different. Instead of an oscillating regime there is a zonein the parameter space where stable static and traveling fronts coexist.The traveling solution appears in a saddle-node bifurcation of cycles. Fora 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. Thispoint is the stable static front solution which becomes unstable from thiscoalescence. This last step is nothing but a subcritical Hopf bifurcation,above which the traveling solution is the only stable one. The features ofthe velocity dependence for this route are special. First of all, because ofits emergence in a saddle-node bifurcation, the traveling solution is bornwith a non-zero velocity. The functional dependence of the velocity is–accordingly– a square root. Nonetheless, it is remarkable that beyond acrossover, a logarithmic dependence (that is a consequence of the proximityto a saddle point) can be recognized.

For both the model consisting in coupled Lorenz systems and the FHNmodel, we have investigated the limit of very large diffusion, taking intoaccount the fact that in the limit of infinite diffusion, the continuumlimit is recovered for these systems. The stable and unstable dislocations

xxxii Summary

coalesce to form the continuous solution. Therefore, from a formal pointof view there exists a bifurcation at that limit. Because of the symmetryconditions, this can be considered a pitchfork bifurcation. The bifurcationlines observed at finite diffusion for both models (Lorenz array and FHN)converge (for infinite coupling) at a point, that in the case of the FitzHugh-Nagumo model is the point that was previously reported when studyingthe continuous version. From the point of view of the discrete modelsthis singularity located at infinity is a codimension-two point where twoeigenvalues of the static solutions become zero. This bifurcation is known asTakens-Bogdanov (TB) and has been studied by several authors. However,the case we have faced here is so particular that has not been addressedbefore. Thus, we have looked at the normal form provided in the book ofGuckenheimer and Holmes that describes a TB with the same symmetrythat our problem, but with the drawback that it does not considers acylindrical phase space. Depending on an internal parameter two possiblebifurcation sets close to the codimension-two point appear. Each of themshows strong analogies with the bifurcations observed in our numericalexperiments. Therefore, we speculate that any parity front bifurcationleading to traveling fronts in a symmetric bistable continuous system istransformed –when discretized– in one of the two routes reported in thisthesis. In this respect, it is important to emphasize that the numericalimplementation of a continuous system always involves some kind ofdiscretization, and therefore we expect the phenomena studied here tomanifest at a fine scale.

Chapter 5 [PMP01] studies an array of Lorenz oscillators, coupledthrough a non-diagonal matrix, where wavefront solutions arise. In theregion where the Lorenz oscillator is bistable, the transition to propagationis demonstrated to be equivalent to that shown in Chapter 4 for on-diagonalcoupled oscillators. Accordingly, logarithmic profiles for the velocity of thefront and the oscillation period are obtained. We have broadened our scopeto the parameter range where the fixed points of the Lorenz oscillatorbecome unstable, such that it is chaotic. In this (chaotic) region, thesystem exhibits two types of spatio-temporal chaos depending on whetherthere exists or not front propagation. In the propagating region we findtwo characteristic processes: spontaneous creation of counter-propagatingfronts, and front reversal. The bifurcation line that separated propagatingfrom non-propagating regions in the bistable case marks now the boundarybetween two kinds of spatio-temporal chaos. In addition, we have alsofound that above a certain coupling, in the bistable as well as in the chaotic

Summary xxxiii

regions, the system undergoes a short wavelength bifurcation. This kind ofbifurcation, and the onset of propagating fronts by the route explainedabove, are observed in discrete systems only. We have also noticed thatthe 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.

Capıtulo 1

El Caos: Fundamentos

1.1 Introduccion

A finales del siglo XIX, el gran matematico Jules Henri Poincare par-ticipo en un concurso organizado por el rey Oscar II de Suecia. Dichoconcurso consistıa en demostrar la estabilidad o inestabilidad del sistemasolar. Ninguno de los participantes fue capaz de resolver el problema, locual no es sorprendente puesto que ni hoy mismo se conoce la respuestacon total seguridad, pero fue Poincare quien gano el concurso. La victoriafue merecida puesto que, aunque se limito a estudiar un problema queconsistıa unicamente en tres cuerpos que se atraıan mutuamente por lagravedad, Poincare demostro que un caso tan aparentemente sencillo no eraintegrable y que, ademas, el comportamiento podıa resultar tan complicadoque resultase, en la practica, impredecible [Poi93].

Esta impredecibilidad de la que hablaba Poincare, en un sistemadeterminista, ha recibido diversos nombres como, por ejemplo, ruidodeterminista, pero se conoce actualmente como ‘caos’, y la primeraaparicion en la literatura con este nombre se remonta a 1975 [LY75]. Aunquela palabra ‘caos’ evoca entre el comun de la gente conceptos como desordeno cataclismo, como sucede con muchos otros conceptos usados en fısica,existe una definicion rigurosa del termino que no coincide con la usada enel lenguaje corriente. El caos se define como un movimiento deterministaque presenta sensibilidad exponencial a las perturbaciones. Es decir, doscondiciones iniciales ligeramente diferentes se alejaran, en promedio, launa de la otra exponencialmente, de forma que tras no mucho tiemposeran claramente distinguibles (y ademas estaran descorrelacionadas) lastrayectorias seguidas para cada condicion inicial. Por tanto, el caos

2 El Caos: Fundamentos

introduce un lımite en la predictibilidad de un sistema fısico aunque seconozcan la leyes que gobiernan el sistema, ya que cualquier mınimo erroren la determinacion de la condicion inicial se amplifica sin freno. Estainfluencia crucial de (casi) cualquier mınima perturbacion tiene importantesconnotaciones en lo referente al poder de prediccion que se le puedeatribuir a una teorıa fısica, ya que aunque una teorıa o modelo describaperfectamente un determinado proceso, la existencia de caos limita lacapacidad de prediccion en funcion del error en la determinacion de lacondicion inicial. Este concepto ha llegado al gran publico como ‘efectomariposa’ ya que se supone que la atmosfera tambien es un sistema quepresenta caos y, por tanto, una ınfima perturbacion, como el simple aleteode una mariposa puede provocar (o evitar) que haya, dıas o semanas mastarde, un tifon en la otra esquina del mundo. Diversos libros de divulgacionse han ocupado del caos, entre los que cabe destacar los de Ruelle [Rue93]y Lorenz [Lor95]. Tambien resulta interesante, en una primera lectura, unartıculo publicado en Investigacion y Ciencia en el ano 1987 [CFPS87].

Cabe preguntarse si el caos es un fenomeno frecuente o, si al contrario, seobserva solo en condiciones muy especiales. Pues bien, la dinamica caoticaha demostrado ser un comportamiento muy frecuente en la naturaleza, y,prueba de ello es que se ha encontrado en multitud de sistemas fısicos(ver por ejemplo [Cvi84, Hao90, BGK+02]) y quımicos [EKDO83]; eincluso, en la dinamica de poblaciones en ecologıa [ERG98]. Logicamente,los modelos matematicos que describen estos procesos muestran caos alrealizar una simulacion numerica, independientemente de que se trate desistemas dinamicos continuos (EDOs) o discretos (mapas). Por supuesto,tambien se encuentra caos cuando se tratan sistemas extendidos descritospor ecuaciones en derivadas parciales, pudiendose encontrar, ademas, caosespacio-temporal cuando existe dependencia sensible no solo en el tiemposino tambien en el espacio.

El caos se muestra al observador como un movimiento irregular, quepodrıa inducir a sospechar la existencia de una componente estocastica(ruido). Pero, como hemos dicho, el caos es un fenomeno puramentedeterminista, a pesar de que a primera vista haya una cierta aleatoriedadaparente. El que la transformada de Fourier de una senal caotica, compuestapor una variable del sistema frente al tiempo, sea de ancho espectro,a diferencia del movimiento periodico o cuasiperiodico, no tiene porque implicar la existencia de una componente estocastica.

Es evidente que existe una diferencia fundamental entre el sistemaplanetario estudiado por Poincare y la atmosfera, que no reside simplemente

1.1 Introduccion 3

en la complejidad de uno y otro problema. Mientras que Poincare estudio unsistema conservativo, la atmosfera es un sistema disipativo que recibeenergıa constantemente del Sol y que la disipa irradiandola hacia el exterioro por su friccion viscosa. En los sistemas conservativos, el volumen en elespacio de fases (definido, por ejemplo, por posiciones y velocidades) seconserva en el tiempo, como nos dice el teorema de Liouville. Por otrolado, en los sistemas disipativos un volumen en el espacio de fases, queevolucione con el flujo, se contrae a lo largo del tiempo y tiende a cero. Poresto, en los sistemas disipativos se puede hablar de atractores ya que lastrayectorias tienden a una region del espacio de fases de volumen cero, quepuede ser un punto de equilibrio, un ciclo, la superficie de un toroide, o unobjeto mas complicado como un ‘fractal’, en el caso de que el atractor seacaotico. No merece la pena extenderse mas, en la naturaleza (multi)fractalde los atractores caoticos (denominados por esto extranos, aunque ambosconceptos no son totalmente intercambiables), puesto que el tema rebasael espacio propio de una introduccion. Una referencia concreta sobre losfractales es el libro de Takayasu [Tak90], no obstante, la mayorıa de loslibros sobre caos tratan este tema con cierta profundidad.

La medida de cuan caotico es un determinado atractor, viene dada porlos exponentes de Lyapunov. Estos son un promedio adecuado de las tasasde divergencia y convergencia, en diferentes direcciones, de orbitas cercanasen el espacio de fases. En particular, si para un sistema k-dimensional deecuaciones diferenciales de primer orden

x ≡ dxdt

= F(x) (1.1)

consideramos una orbita desplazada infinitesimalmente x(t) + δx(t) y elvector y(t) = δx(t)/|δx(0)|, para el cual tenemos la ecuacion que nos da laevolucion del desplazamiento infinitesimal

y = DF(x(t)) · y (1.2)

donde DF(x(t)) denota las matriz Jacobiana k × k de derivadas parcialesde F(x(t)). Los exponentes de Lyapunov estaran dados entonces por

λ(x(0),y(0)) = lımt→∞

1t

ln |y(t)| (1.3)

La dependencia en y(0) es formal, puesto que el resultado es el mismopara (casi) cualquier perturbacion δx(t). Asimismo, el teorema ergodicomultiplicativo de Osedelec asegura que el resultado sera el mismo para


todo x(0) en la cuenca de atraccion del atractor, excepto para un posibleconjunto con medida de Lebesgue cero [ER85]. Un atractor caotico secaracteriza por tener un exponente de Lyapunov positivo, o mas de uno, encuyo caso se habla de atractor hipercaotico. Los exponentes positivos dancuenta de la dependencia sensible propia del caos. Ademas si el sistema esautonomo tendra un exponente nulo correspondiente a las perturbacionesen la direccion del flujo. Por otro lado, habra uno o mas exponentesnegativos, necesarios para que el sistema sea disipativo (y el flujo contraigael volumen). Por tanto, se necesitan k > 2 variables para poder encontrarcaos. Logicamente, existe una definicion analoga para los exponentes deLyapunov en mapas. En estos sistemas, una sola variable es suficiente paragenerar caos, siempre y cuando el mapa sea no invertible.

Otro aspecto importante es que el hecho de que la divergencia delas trayectorias en un atractor caotico sea justamente exponencial, y no,por ejemplo, simplemente polinomica, tiene importantes consecuenciasya que, en el primer caso, el error no puede ser acotado durante untiempo arbitrariamente grande, por mucha capacidad computacional quese utilice [For87]. ¿Quiere esto decir que cualquier simulacion numerica deuna trayectoria caotica carece entonces de sentido, debido a los errores deredondeo que introduce el ordenador al calcular cada paso de integracion?Afortunadamente, la mayorıa de los sistemas presentan lo que se conocecomo ‘shadowing’. Esto implica que, aunque nuestra solucion numerica nose corresponda con una determinada condicion inicial, sı que, tıpicamente,se aproxima a (o sombrea), durante un largo periodo de tiempo, unasolucion verdadera cuya condicion inicial esta cercana a la condicion inicialconsiderada primeramente (ver por ejemplo [HGY88, GHSY90, SY91]).De esta forma, aunque nuestras soluciones numericas carezcan de utilidadpredictiva, sı que representan soluciones reales del sistema, y por tanto, nosinforman sobre la naturaleza del atractor caotico.

En la seccion 1.2 se introducen los conceptos basicos de la teorıade bifurcaciones. A partir de ahı, se explican, en la seccion 1.3, lasdiversas ‘rutas’ por las que un atractor no caotico puede transformarseen un atractor caotico. En la seccion 1.4, se trata el tema de losinvariantes inmersos en un atractor caotico, especialmente las orbitasperiodicas inestables. Finalmente, la seccion 1.5 describe los diversos tiposde sincronizacion caotica, que se pueden producir por interaccion de dos omas sistemas caoticos.

1.2 Bifurcaciones locales 5

1.2 Bifurcaciones locales

En muchas ocasiones, no es necesario conocer la solucion exacta deuna determinada ecuacion diferencial (o un mapa), sino que es suficientecon saber que tipo de solucion presenta el sistema (punto fijo, ciclo lımite,etc). La rama de la matematica que se ocupa de las transiciones entreestados cualitativamente diferentes se conoce como teorıa de bifurcaciones.Una bifurcacion se puede producir en un sistema si se varıa alguno desus parametros; piensese, por ejemplo, en el numero de Reynolds de unfluido o en la resistencia de un resistor en un circuito electrico. En estaseccion se presentan las bifurcaciones llamadas ‘locales’ que se producen‘genericamente’ en sistemas diferenciables1. Estas bifurcaciones puedenestudiarse considerando unicamente el entorno de una solucion. Por otrolado, una bifurcacion es generica cuando se encuentra tıpicamente, sinnecesidad alguna de simetrıa o condicion adicional. Tambien se puede decirque es estructuralmente estable porque no es destruida por una pequenaperturbacion. El apelativo de genericas se puede sustituir por codimension-uno, que quiere decir que en un sistema n-parametrico estas bifurcionesse dan en variedades en el espacio de parametros de dimension n− 1. Porejemplo, si tenemos dos parametros, los estados cualitativamente diferentesse localizaran en regiones del plano, u otra variedad bidimensional, queambas definan. Por tanto, habra lıneas de bifurcacion que separen estadoscualitativamente distintos. Normalmente, uno cruzara, al variar alguno delos parametros, una de esas lıneas para pasar de un estado a otro. Tambienexiste la posibilidad, si uno tiene infinita precision, de que al variar losparametros uno cruce dos lıneas a la vez, es decir, pase por un puntodonde intersecten dos de ellas. En este caso, tan particular, tendremosuna bifurcacion de codimension dos (puede encontrarse un ejemplo en elApendice).

En sistemas continuos (EDOs, ecuaciones derivadas ordinarias) con unavariable existe unicamente un tipo de bifurcacion generica que se conocecomo bifurcacion tangente o silla-nodo (saddle-node). En esta bifurcacionaparecen una solucion estable y otra inestable. Esta asociada a un autovalornulo. Por ejemplo, la ecuacion

x = ax2 − ε (1.4)

no tiene soluciones de equilibrio para ε < 0 (suponiendo a > 0), mientras

1No repasamos, en cambio, el concepto de estabilidad asintotica y las condiciones deestabilidad lineal; si bien algunos apuntes aparecen en el Apendice.


−0.1 0 0.1−0.4

−0.2

0

0.2

0.4

ε

x

bif. tangente

a)

−0.1 0 0.1−0.4

−0.2

0

0.2

0.4

ε

bif. horquilla supercritica

b)

−0.1 0 0.1−0.4

−0.2

0

0.2

0.4

ε

bif. horquilla subrcritica

c)

Figura 1.1: Diagramas de bifurcacion de las Ecs. (1.4, 1.5). Las soluciones estables einestables aparecen con lıneas continua y discontinua, respectivamente.

que para ε > 0 exiten dos soluciones x = −√

ε/a (estable) y x =√

ε/a(inestable).

Aunque no es generica, porque precisa la existencia de una simetrıapor reflexion, merece la pena mencionar aquı la bifurcacion horquilla(pitchfork). En esta bifurcacion se pasa de tener una solucion a tener tres,dos de ellas relacionadas por simetrıa. Se tienen dos variantes, supercrıtica ysubcrıtica. En la primera una solucion estable se inestabiliza para dar lugara dos soluciones estables. En la variante subcrıtica una solucion estable sevuelve inestable cuando dos soluciones inestables colisionan con ella. Porejemplo, la ecuacion siguiente presenta una bifurcacion horquilla en ε = 0:

x = x(ax2 + ε). (1.5)

que es supercrıtica para a < 0 y subcrıtica para a > 0. Los diagramas debifurcacion de las Ecs. (1.4,1.5) se muestran en la Fig. 1.1.

La segunda bifurcacion generica es la bifurcacion de Andronov-Hopf, oen forma abreviada Hopf. Precisa de dos o mas dimensiones y ocurre cuandoun punto fijo cambia de estabilidad al cruzar dos autovalores complejosconjugados el eje imaginario. Entonces, en un entorno de la bifurcacion hayun ciclo lımite, con frecuencia angular aproximadamente igual al valor dela parte imaginaria de los autovalores. Al igual que la bifurcacion horquilla,tambien tiene variantes super y subcrıtica, dependiendo de que el ciclolımite sea estable o inestable. Por ejemplo, presenta una bif. de Hopf enε = 0 el siguiente sistema de ecuaciones:

x = −ωy + εx + (x2 + y2)(ax− by)

y = ωx + εy + (x2 + y2)(bx + ay). (1.6)

Pasando a coordenadas polares queda mas claro el significado de cada

1.2 Bifurcaciones locales 7

Figura 1.2: La bifurcacion de Hopf (Ec. (1.6)). El punto de equilibrio aparece en negroo gris y la solucion periodica con trazo continuo o discontinuo segun sean estables oinestables.

parametro:

r = εr + ar3

θ = ω + br2 (1.7)

En ε = 0 la solucion en el origen cambia su estabilidad. Para a < 0 (casosupercrıtico) un ciclo lımite estable aparece para ε > 0; en cambio, paraa > 0 (caso subcrıtico) un ciclo lımite 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 aquı como casosparticulares. Sin embargo, estas ecuaciones se conocen como formasnormales de dichas bifurcaciones. Cualquier sistema que sufre unabifurcacion puede ser reducido a la forma normal correspondiente,tras sucesivos cambios de coordenadas y despreciando los terminos deorden superior [GH83]. En un entorno de la bifurcacion el sistema secomportara de forma muy similar a como indica su forma normal, ya que


los termimos de orden superior son tan pequenos como arbitrariamente sequiera.

Las bifurcaciones que se dan en mapas son tambien importantes yaque, a partir de un flujo k-dimensional se puede construir un mapa conk−1 variables. Para esto, se tiene que realizar una seccion de Poincare queconsiste en tomar los puntos donde el flujo intersecta, en un sentido, una(hiper)superficie (k − 1)-dimensional. De esta forma, una orbita periodicase convierte en el mapa (de Poincare) asociado (F) en un punto fijo (x∗):

xn+1 = F(xn) / x∗ = F(x∗) (1.8)

La condicion de estabilidad de un punto fijo de un mapa es que todoslos autovalores de la matriz jacobiana en ese punto esten dentro de lacircunferencia de radio unidad en el plano complejo. Los mapas presentanbifurcaciones genericas que son analogas a las de los flujos: la bifurcacionsilla-nodo (asociada a un autovalor +1) y la bifurcacion de Hopf, tambienconocida como Neimark-Sacker (asociada a dos autovalores complejosconjugados que abandonan la mencionada circunferencia). Existe, ademas,una bifurcacion generica que es genuina de los sistemas discretos. Esta es labifurcacion de duplicacion de periodo o flip. Se produce cuando un autovalortoma el valor -1. Existen las variantes supercrıtica y subcrıtica dependiendode que la orbita de doble periodo (x1,x2,x1, ...) = (x∗+ δx1,x∗+ δx2,x∗+δx1, ...) existente en un entorno de la bifurcacion sea estable o inestable.Notese que la orbita de doble periodo da lugar a dos puntos fijos de la doblerecurrencia del mapa:

x1 = F(F(x1)) , x2 = F(F(x2)) (1.9)

Aunque no hemos mostrado aquı mas que los mınimos fundamentos dela teorıa de bifurcaciones, esto nos servira como base para describir, enla proxima seccion, las formas en las que un sistema pasa de tener uncomportamiento “ordenado” a tener un comportamiento caotico.

1.3 Rutas al caos

Como ya hemos mencionado, variando un parametro de control,un sistema transita entre estados cualitativamente diferentes a travesde bifurcaciones. Las bifurcaciones descritas en la seccion anterior sonuniversales, ya que se pueden dar genericamente en cualquier sistemadinamico, no importa que sistema fısico represente. Cabe preguntarse, por

1.3 Rutas al caos 9

tanto, que tipo de bifurcaciones deben ocurrir en un sistema para que ’estepase a exhibir un comportamiento caotico.

La transicion al caos de baja dimension se produce a traves de una seriede rutas (o escenarios) que han resultado ser universales. La descripcionteorica de estas rutas se ha realizado, normalmente a partir de mapas. Caberesaltar aquı que no deja de resultar sorprendente que mapas, en muchoscasos con una sola variable, sean capaces de modelizar las transiciones alcaos que se producen en sistemas tan sumamente complicados como unfluido. De hecho, gran parte de los primeros trabajos en este campo seconcibieron con la intencion de describir la transicion a la turbulencia. Erahabitual usar los terminos caos y turbulencia de forma casi equivalente.Hoy en dıa, sin embargo, se suele reservar el termino turbulencia parala dinamica observada a muy alto numero de Reynolds que no se puededescribir como un movimiento caotico de baja dimension. La turbulencia, otambien llamada turbulencia totalmente desarrollada, constituye aun unode los retos mas importantes de la fısica.

1.3.1 Cascada de duplicacion de periodo

Muchos mapas en una variable (ver Ec. (1.8)) muestran una sucesionde bifurcaciones de duplicacion de periodo que desemboca en un estadocaotico. Lo llamativo es que muchos sistemas fısicos complicados muestranuna ruta coincidente con la que se observa en un mapa de gran simplicidad.Si miramos el espectro de Fourier de una determinada variable de unsistema tenemos que el movimiento periodico se manifiesta con un pico enla frecuencia caracterıstica, digamos f1. Tras una duplicacion de periodo,aparece un subarmonico, que se ve como un nuevo pico en f1/2. La siguienteduplicacion nos da un nuevo pico en f1/4 (ası como en 3f1/4). Finalmente,la aparicion de infinitos subarmonicos nos lleva al caos caracterizado portener un espectro continuo. Es esta vision en terminos de espectro depotencias lo que hace que esta ruta tambien se conozca como cascadasubarmonica.

La ruta al caos por duplicacion de periodo esta indiscutiblemente unidaa Feigenbaum [Fei78]. El gran merito de Feigenbaum fue encontrar yexplicar la universalidad en una clase de mapas unidimesionales en unintervalo. En concreto, todos los mapas en un intervalo que satisfaganser unimodales. Unimodal quiere decir que tiene un solo maximo (que esaproximable por una cuadratica) y que la derivada Schwarziana es negativa


en el intervalo:

SF (x) ≡ F ′′′(x)F ′(x)

− 32

(F ′′(x)F ′(x)

)2

< 0 (1.10)

Esta ultima condicion implica que el sistema no tiene nunca mas deuna orbita periodica estable (aunque puede no tener ninguna comoocurre cuando hay caos). Feigenbaum demostro, mediante metodos derenormalizacion, que a medida que se producen mas duplicaciones deperiodo, todos los mapas unimodales tienden al mismo escenario universal.Ası, por ejemplo, si tenemos un parametro µ tal que las duplicaciones seproducen para los valores µ1, µ2, . . . , µn, . . ., el caos aparece en n = ∞.Entonces, se cumple que existe un numero universal δ:

δ = lımn→∞

µn − µn−1

µn+1 − µn= 4,6692016 . . . (1.11)

Consideremos, como ejemplo de la ruta al caos por duplicacion de periodo,el oscilador de Rossler [Ros76], definido por las ecuaciones:

x = y − z

y = x− ay (1.12)

z = f + z(x− c)

que contienen solamente un termino no lineal. Si se incrementa el valor delparametro c, mientras que se mantienen constantes a = 0,2 y f = 0,165, seobserva en la Fig. 1.3 que de una orbita periodica sencilla (que etiquetamoscomo de periodo-1 porque se cierra al cabo de un giro en torno al origen) sepasa mediante bifurcaciones de duplicacion de periodo a orbitas cada vezmas complicadas (periodo-2n) hasta llegar al caos. Sin embargo, el estadocaotico no es robusto, ya que aunque la medida (en el sentido de Lebesgue)en el parametro c de comportamiento caotico no es nula, existen “ventanas”periodicas que llenan densamente c. Es decir, tan cerca como queramos deun valor de c donde hay caos, existe otro valor de c donde el sistema presentaun comportamiento periodico estable. Esto puede evitarse con sistemas quesean suaves a trozos [BYG98].

Muchas ventanas periodicas no son fısicamente observables ya que parala mayorıa de las condiciones iniciales puede existir un transitorio muy largoantes de “caer” en la orbita periodica de la ventana. Sı que se suele observaruna ventana de periodo-3, de la que surge otra cascada de duplicaciones(periodo-3·2n) que desemboca en caos. La jerarquıa de las orbitas periodicas

1.3 Rutas al caos 11

−10 0 10−10

−5

0

5

10c=3

periodo−1

y

−10 0 10−10

−5

0

5

10c=4

periodo−2

−10 0 10−10

−5

0

5

10c=5

periodo−4

−10 0 10−10

−5

0

5

10c=5.22

periodo−8

y

x−10 0 10

−10

−5

0

5

10c=5.4

caos

x−10 0 10

−10

−5

0

5

10c=5.5

periodo−6

x

Figura 1.3: Proyeccion sobre el plano x − y de la trayectoria seguida por el oscilladorde Rossler para diferentes valores del parametro c de la Eq. (1.12).

en mapas unimodales fue desarrollada primeramente por Sharkovskii. Y fueposteriormente redescubierta por Li y Yorke [LY75], y puede resumirse enla siguiente frase: ‘periodo-tres implica caos’. En el capıtulo 2 de la presentetesis puede encontrarse como el mapa de Poincare del oscilador de Rosslerse corresponde con un mapa unimodal, y en el capıtulo 4 se muestra unacascada duplicacion de periodo en un sistema diferente.

1.3.2 Intermitencia

La transicion al caos por intermitencia fue descubierta por Pomeauy Maneville [PM80]. Variando un parametro de control el sistemapasa de exhibir un comportamiento periodico a mostrar una dinamicapracticamente periodica interrumpida de forma aperiodica por brevesepisodios de comportamiento complicado. En analogıa con los fluidos,los intervalos de tiempo con una dinamica aproximadamente periodica seconocen como fases laminares, mientras que las interrupciones caoticas dedichas fase se denominan bursts.

La desaparicion (o perdida de estabilidad) del estado regular (para uncierto valor crıtico del parametro de control µc) que da lugar al estadointermitente puede producirse de tres maneras, lo cual produce tres tiposde intermitencia. Cada uno de estos tipos presenta caracterısticas que leson propias, sobresaliendo el exponente crıtico (γ) de la ley de potencias


que satisface el promedio de la duracion de las fases laminares:

〈τ〉 ∼ (µ− µc)−γ (1.13)

En resumen, tenemos la siguiente clasificacion:

Intermitencia Bifurcacion asociada γ

Tipo I Silla-nodo 1/2Tipo II Hopf subcrıtica 1Tipo III Duplicacion de periodo sub. 1

Aunque las bifurcaciones Tipos II y III tienen el mismo exponente γ sediferencian, por ejemplo, en la distribicion de probabilidad de las faseslaminares [BPV88].

1.3.3 Cuasiperiodicidad

La cuasiperiodicidad se define como aquel movimiento en el quese superponen n frecuencias incomensurables. Esto correspone a unatrayectoria no cerrada en el espacio de fases que “llena” la superficie de untoroide n-dimensional (Tn). Ası, si tenemos un sistema con n frecuenciasΩ1,Ω2, . . . ,Ωn habra cuasiperiodicidad en un n-toro si dada la ecuacion:

m1Ω1 + m2Ω2 + · · ·+ mnΩn = 0, mi ∈ Z (1.14)

la unica solucion es mi = 0. Si existe otra solucion, entonces se diceque existe una resonancia. Por ejemplo, si tenemos dos frecuencias Ω1,2

tendremos cuasiperiodicidad si el cociente de las dos frecuencias esirracional, mientras que si el cociente es racional el movimiento sobre elT2 se cerrara sobre sı mismo, por lo que el movimiento sera periodico. Eneste ultimo caso se dice que las frecuencias estan enganchadas (locked).Desde un punto de vista topologico, el enganche de frecuencias hace quesobre la superficie del toro aparezcan, mediante una bif. silla-nodo, un cicloestable y otro inestable.

La cuasiperiodicidad se ha relacionado con el flujo turbulento desdeque Landau [Lan44] propuso su modelo de la turbulencia. Segun Landau,dado que la turbulencia muestra un espectro de potencias continuo, estadeberıa de aparecer a partir de un estado cuasiperiodico al que se leiban superponiendo cada vez mas frecuencias (incomensurables) hasta quellegaran a ser infinitas. Sin embargo fue Ruelle, quien se dio cuenta deque un espectro continuo podıa corresponder a un atractor caotico de baja


dimension, en contraste con el toroide infinito-dimensional de Landau. Apartir de esta idea, Ruelle y Takens se preocuparon por la estabilidaddel movimiento cuasiperiodico a varias frecuencias. Demostraron que paraun flujo cuasiperiodico en un T4, era posible realizar perturbacionesarbitrariamente pequenas tal que el flujo pasase a ser caotico dentro delT4 [RT71]. Ademas, este atractor caotico una vez creado no puede serdestruido por perturbaciones arbitrariamente pequenas. Posteriormente,en colaboracion con Newhouse [NRT78], demostraron que lo mismo, salvopor alguna restriccion para las perturbaciones, podıa decirse para unT3. Por lo tanto en un sistema con cuasiperiodicidad a dos frecuenciasla aparicion de una tercera deberıa desembocar directamente en caos.Diversos experimentos [GS75, LFL83] echaron por tierra la hipotesis deLandau al tiempo que dieron fuerza a la nueva ruta que se conocerıa comoRuelle-Takens-Newhouse. A la vista de esto, parecerıa imposible encontrarun sistema fısico que exhiba cuasiperiodicidad a tres o mas frecuencias.Sin embargo, se realizaron diversos estudios tanto numericos [GOY83a,GOY85] como experimentales [GB80, MLM84], que demostraron que lacuasiperiodicidad a tres frecuencias no es un fenomeno excepcional. Dehecho, como hizo notar Eckmann [Eck81], que el conjunto de los camposde vectores con atractor extrano sea un abierto (es decir, robusto bajopequenas perturbaciones) no significa que la medida de tal conjuntosea grande. En analogıa con la cuasiperiodicidad a dos frecuencias, elhecho de que la cuasiperiodicidad pueda ser destruida por pequenasperturbaciones no quiere decir que su medida sea pequena y que, portanto, la cuasiperiodicidad no sea una solucion fısica. En concreto, siacoplamos un conjunto de n osciladores con frecuencias incomensurables esde esperar que para una magnitud de acoplamiento suficientemente pequena(distinta de cero) el sistema tenga como atractor un Tn. En cambio, paraacoplamientos mayores, habra resonancias y, probablemente, caos. Comovarıa, esquematicamente, la medida de los distintos comportamientos paran = 3 puede encontrarse en la Refs. [Bat88, Ash98]. Asimismo, existeun artıculo [WKPS84] de convencion Rayleigh-Benard del ano 1984 en elque se encontro cuasiperiodicidad a cuatro y cinco frecuencias. Aunque,como explican los autores, los diversos modos estan localizados de formapredominante en diversas partes de la celula convectiva, lo cual hacesuponer que existe una interaccion debil entre ellos.

Pese a lo dicho arriba, si se quiere hablar de una ruta bien descrita alcaos por cuasiperiodicidad hay que referirse a la creacion de un atractorextrano a partir de un toro a dos frecuencias. Esta ruta, encontrada por


−20 −10 0

20

40

60

−20 −10 0

20

40

60

−20 −10 0

20

40

60

−20 −10 0

20

40

60

−20 −10 0

20

40

60

−20 −10 0

20

40

60

−20 −10 0

20

40

60

−20 −10 0

20

40

60

−20 −10 0

20

40

60

β=1.0 β=0.9 β=0.85

β=0.8

β=0.775

β=0.790

β=0.785 β=0.780

β=0.795

Figura 1.4: Seccion de Poincare para un sistema de ocho osciladores de Lorenzacoplados. Cuando el parametro de acoplamiento β toma los valores 1.0 y 0.9 la seccionde Poincare da una curva topologicamente equivalente a una circunferencia; lo cual indicaque tenemos cuasiperiodicidad a dos frecuencias (T2). En β = 0.85 tenemos un T3 puestoque la interseccion con el plano de Poincare da un objeto bidimensional. En β = 0.8, 0.795tenemos un locking del T3 lo cual nos da que el atractor es topologicamente equivalente aun T2. En β = 0.790 aparecen arrugas en la seccion del T2 lo cual, permite prever que, deacuerdo a la ruta Curry-Yorke nos estamos acercando a una zona de comportamientocaotico, que finalmente aparece en β = 0.785 y para valores inferiores del parametro β.


Curry y Yorke [CY78, ACHM82, DBC82] se caracteriza por un procesode rotura de un toro invariante (torus break-up), debido a la interaccionde las resonancias. El proceso de rotura del toro que origina un atractorextrano se inicia cuando la superficie del toro deja de ser suave (esto es,se arruga) para, finalmente, cambiar su topologıa (ver Fig. 1.4). Esta rutase enmarca dentro de las posibles maneras en las que se destruye un toroinvariante [AAIS89, ASC93].

1.3.4 Crisis

Una crisis se define como un cambio cualitativo de un atractor quemodifica su estructura bruscamente. Ası, la bifurcacion silla-nodo y lasbifurcaciones de duplicacion de periodo y de Hopf subcrıticas podrıanconsiderarse crisis. No obstante, hoy en dıa suele usarse el terminosolamente en el contexto de los atractores caoticos [GOY82, GOY83b].

Un atractor puede sufrir tres tipos de crisis. Primeramente puedesuceder que el atractor sea destruido. Tambien es posible que su tamanoaumente bruscamente. Finalmente, en los casos en los que existe simetrıa,puede ocurrir que dos (o mas) atractores se fundan.

Un atractor puede ser destruido, por ejemplo, siguiendo un caminoinverso al que se ha explicado en las secciones anteriores. Sin embargo,en el caso de la crisis no debe quedar un atractor en el espacio de fasesallı donde antes se encontraba el atractor caotico. Esto sucede cuandoel atractor colisiona con el borde de su cuenca de atraccion (boundarycrisis) [GOY82, GOY86, HB94], es decir, con el lımite de la region delespacio de fases que contiene todas las condiciones iniciales que tiendenasintoticamente al atractor. Una vez se ha producido la crisis, aquellastrayectorias que antes tenıan como lımite el atractor caotico ahora describenuna trayectoria aparentemente caotica (transitorio caotico) antes de serexpelidas a otra region del espacio de fases. La duracion promedio deltransitorio caotico sigue una ley de potencias con el parametro de control,como en la Ec. (1.13).

En lo que se refiere a los casos en los que un atractor aumenta sutamano o se funde con otro (interior crisis), esto da lugar a fenomenosde intermitencia (crisis-induced intermittency) [GORY87, DRC+89] conpromedios temporales de cada fase que divergen de acuerdo, nuevamente,a leyes de potencias.


Figura 1.5: Dos variantes de la bifurcacion saddle-loop. a) Un ciclo lımite establecolisiona con un punto de silla y desaparece. b) Una conexion homoclınica origina laaparicion de una orbita periodica inestable.

1.3.5 Bifurcaciones globales

Se entiende por bifurcacion global aquella que no puede ser tratadamediante un analisis de estabilidad local cerca de un punto fijo o una orbitaperodica (o un toro, aunque no consideramos aquı este caso). El libro deWiggins [Wig88] recoge de forma extensiva la teorıa de las bifurcacionesglobales y el caos. Las bifurcaciones globales se producen mediadas porconexiones homoclınicas y heteroclınicas. Las primeras consisten en orbitasque tienden a un punto fijo o un ciclo para t → ±∞. La segundas, sonorbitas que conectan dos puntos fijos, o un punto fijo y un ciclo, o dosciclos.

Un ejemplo tıpico de bifurcacion global es la bifurcacion silla-lazoo saddle-loop. En esta bifurcacion un ciclo lımite (estable o inestable)colisiona con un punto de silla, momento en el que pasa a ser la conexionhomoclınica del punto de silla. Se muestra un esquema en la Fig. 1.5.

En las secciones anteriores, se han descrito sucintamente las transicionesal caos en terminos de bifurcaciones locales o, simplemente, conexplicaciones intuitivas. Sin embargo, hay que advertir que todas ellas“esconden” conexiones globales. Y es el estudio de las bifurcaciones globalescorrespondientes lo que permite una comprension verdaderamente profundade ellas.

La cascada de doble periodo contiene una conexion heteroclınica entrecada nueva obita y la orbita de la que se ha bifurcado [CGS87]. La ruta


Curry-Yorke se basa en la conexion heteroclınica de las dos orbitas quesurgen en un ‘locking’, vease [Kir93] o la representacion grafica en lapag. 46 de la Ref. [AAIS89]. Las crisis se originan debido a conexioneshomo/heteroclınicas entre orbitas inestables (ver la Sec. 1.4) inmersas enel atractor caotico [GOY86].

Finalmente, la ruta por intermitencia (PM80) merece atencion aparte,ya que este mecanismo involucra la conexion homoclınica de una orbita quees no hiperbolica. En el punto crıtico (µ = µc), existe una orbita que no esasintoticamente estable o inestable. Las conexiones globales entre objetosno-hiperbolicos no han sido muy estudiadas. De todas formas, se sabe quetales conexiones pueden dar lugar mediante lo que se conoce como una‘Ω-explosion’ [IL99] a un atractor extrano.

El comportamiento caotico no surge solo a partir de movimientosperiodicos o cuasiperiodicos. Tambien es posible que un unico punto fijopueda dar lugar a un conjunto caotico (aunque para que este conjunto sevuelva atractor, en vez de ser solo un transitorio, puede ser necesaria laexistencia de una crisis). Para generar un conjunto caotico deben existiruna, o varias, conexiones homoclınicas a este punto de equilibrio [Hom93].

Como casos mas importantes, encontramos la doble conexion tipoLorenz, el mecanismo de Shil’nikov y la conexion bifocal. El sistema deLorenz presenta una doble conexion homoclınica a un punto de silla conforma de mariposa. El mecanismo de Shil’nikov consiste en una conexiona un punto de silla-foco. Finalmente, en la conexion bifocal participa unpunto foco-foco, es decir, un punto cuyas variedades (centrales) estable einestable son ambas bidimensionales. Este ultimo mecanismo no ha sidomuy estudiado (ver [FS91]) debido a su difıcil visualizacion, ya que senecesita que el espacio de fases sea, al menos, cuatridimensional.

Por supuesto, tambien son posibles conexiones entre varios puntos fijos(esto es, heteroclınicas) que den lugar a caos. No obstante esta posibilidadno ha sido objeto de un estudio tan pormenorizado en la literatura.

1.3.5.a Sistema de Lorenz

En 1963, Lorenz publico un artıculo de gran impacto [Lor63]. En el sedescribıa un sistema tridimensional de ecuaciones diferenciales ordinarias,que integrado numericamente presentaba soluciones extremadamentecomplicadas (que hoy llamamos caoticas). El origen de las ecuacionesde Lorenz es la aproximacion de Saltzmann de las ecuaciones de lahidrodinamica (Navier-Stokes, conduccion del calor y continuidad) alproblema de una celula convectiva. Sin embargo, estas ecuaciones han


sido estudiadas por muchos autores desde entonces, en tanto en cuanto,constituyen un ejemplo de sistema caotico bastante simple. El significadofısico de las ecuaciones no ha sido muy relevante, aunque sirvan paramodelizar tambien ciertas inestabilidades de un laser [Hak75], o algunsistema mecanico (ver p. ej. [Str94]). No obstante, si atendemos a los autoresde la Ref. [CCT99], las ecuaciones de Lorenz son importantes ya que sonla forma normal de una inestabilidad generica en un sistema macroscopicocuasirreversible.

En todo caso, si por algo son bien conocidas las ecuaciones de Lorenz espor haberse convertido en uno de los paradigmas del caos, ya que han sidoestudiadas de forma intensiva (ver p. ej. el libro de Sparrow [Spa82]). Dichasecuaciones contienen tres parametros positivos (σ, r, b) y dos terminos nolineales:

x = σ(y − x)

y = rx− y − xz (1.15)

z = xy − bz

Cabe resaltar que las ecuaciones poseen una simetrıa por reflexion:(x, y, z) → (−x,−y, z), y que el eje z es invariante.

Un primer analisis del sistema de Lorenz, debe centrarse en los puntosfijos del sistema. Es facil averiguar que para 0 < r < 1 el sistema posee ununico punto fijo, situado en el origen, que es estable. En r = 1, la solucionen el origen se inestabiliza por una bifurcacion horquilla que da lugar a laaparicion de dos puntos fijos estables C± = (±

√b(r − 1),±

√b(r − 1), r −

1). Estas dos soluciones sufren en r = rH una bifurcacion de Hopf(suponiendo que σ > b + 1):

rH =σ(σ + b + 3)

σ − b− 1. (1.16)

Para los parametros tomados en el artıculo de Lorenz σ = 10, b = 8/3,tenemos que rH ≈ 24.74. La bifurcacion de Hopf en rH es subcrıtica.Demostrarlo requiere un gran esfuerzo de calculo ya que tenemos tresdimensiones2.

2No es totalmente riguroso considerar la forma normal de la bifurcacion (Ec. (1.6))en el plano definido por los autovectores asociados a los autovalores imaginarios purosen rH . El calculo de los terminos no lineales al orden mas bajo, para una bifurcacionde Hopf en un espacio N -dimensional, puede encontrarse en [HW78]. Ref. [MM76] tratael caso concreto del Lorenz (ojo, tiene algun error ya que la bifurcacion de Hopf en elLorenz es siempre subcrıtica).


a) d)

e)

b)

f)

c)

g) h)

Figura 1.6: Esquema (correspondiente a una proyeccion sobre el plano x − z) de latransicion al caos en el sistema de Lorenz, ver detalles en el texto.

Para entender como aparecen las orbitas periodicas inestables queparticipan en la inestabilizacion de C± en rH , y principalmente, paraexplicar la aparicion del caos en el modelo de Lorenz hay que tener encuenta las bifurcaciones globales. En la Fig. 1.6 se presenta un esquemacorrespondiente a una proyeccion sobre el plano (x, z) de los puntos fijos,las variedades estable e inestables del origen y las orbitas inestables antesmencionadas. La Fig. 1.6(a) corresponde a r < 1, con un unico punto fijo enel origen. Entre (a) y (b) se produce la bifurcacion horquilla que da lugaral nacimiento de C±. Cuando r se incrementa aun mas, C± se vuelvenfocos (c). Cuando r alcanza un valor crıtico se produce una doble conexionhomoclınica (d). Este punto corresponde, para los parametros estandar deσ y b, a r1 ≈ 13,926. Con la doble conexion homoclınica aparecen lasdos orbitas inestables mostradas en (e) ası como un conjunto infinito3

de orbitas periodicas inestables (no mostradas) que puden clasificarse deacuerdo con la secuencia de giros en torno a C+ y a C−. Este conjuntoinfinito de ciclos inestables aparece gracias a que la conexion es doble, y sunacimiento coincide con la aparicion del transitorio caotico. Por lo general,las trayectorias en el espacio de fases acaban por caer a C+ o C− tras untiempo de comportamiento aparentemente caotico en el que la trayectoriavisita ambos lados del sistema. A medida que r crece las orbitas mostradasen la Fig. 1.6(e) se hacen cada vez mas pequenas, lo que produce que

3El nacimiento de este conjunto infinito de ciclos se conoce como ‘explosionhomoclınica’.


los transitorios caoticos sean mas prolongados . Llegado un punto crıticor2 ≈ 24,06, se establece una conexion heteroclınica (doble) entre la variedadinestable del origen y las orbitas inestables, (f). Esto corresponde a unacrisis [YY79, KY79b, GORY87], de tal forma que para r > r2 tenemosun atractor caotico (con forma de doble lobulo, como se puede ver en laportada de esta tesis), mientras que para r < r2 solo existe un transitoriocaotico. En el intervalo r2 < r < rH existe triestabilidad entre C± y elatractor extrano, (g). Por encima de rH , C± son inestables y el atractorcaotico resulta ser el unico del sistema (h).

Finalmente, cabe destacar que hasta muy recientemente no se ha podidodemostrar, de forma matematicamente rigurosa, que el atractor de Lorenzexiste como tal [Tuc99, Ste00]. La simple evidencia numerica no satisfacea los matematicos.

1.3.5.b Caos de Shil’nikov

El caos de Shil’nikov tambien se conoce como caos homoclınico, aunqueesta es una denominacion un tanto imprecisa ya que hay varios tipos de caoshomoclınico. Se observo por primera vez en un sistema quımico [AAR87]y en un laser [AMG87]. No obstante, el estudio teorico habıa sido iniciadoanos atras por Shil’nikov.

El problema de la dinamica en las cercanıas de una conexion homoclınicaa un punto silla con autovalores complejos (ver Fig. 1.7) atrajo la atencionde Shil’nikov, quien primeramente lo estudio en 1965 [Shi65]. Shil’nikovdemostro que, para el caso que se muestra en la figura, se forma unasituacion propicia para la existencia de caos (concretamente, una herradurade Smale). Para esto debe cumplirse que el ındice de silla

δ = − ρ

λu(1.17)

sea menor que uno. Tambien es posible la situacion inversa en el tiempo:variedad estable unidimensional y variedad inestable bidimensional. En elartıculo de Glendinning y Sparrow [GS84], se estudia la jerarquıa en laque infinitas orbitas inestables aparecen en un entorno del punto crıtico delparametro de control donde se encuentra la conexion homoclınica.

1.4 Orbitas periodicas inestables (UPOs)

Los atractores caoticos se caracterizan por tener un conjunto infinito deobjetos inestables en su interior. Estos objetos suelen ser orbitas periodicas

1.4 Orbitas periodicas inestables (UPOs) 21

Figura 1.7: Orbita homoclınica (Γ) a un punto silla-foco. La orbita sale del punto fijo en la direcciontangente al autovector asociado al autovalor λu, yentra tangente al plano definido por los autovectoresasociados al autovalor imaginario λs = ρ±iω (ρ < 0).

inestables (unstable periodic orbits, UPOs). Ya mencionamos este hechoal estudiar las rutas al caos en el sistema de Lorenz y en el escenario deShil’nikov. Es facil percatarse de la existencia de infinitas UPOs en el casode la ruta al caos por duplicacion de periodo, ya que cada duplicacion deperiodo deja como residuo una orbita inestable. Si hablamos de sistemasque son disipativos en todo el espacio de fases, es claro que las UPOs seranobjetos tipo silla; ya que si fueran totalmente inestables el volumen delespacio de fases se expandirıa en un entorno. Ası pues, una trayectoriacaotica puede entenderse como aquella que va “saltando” de una UPO aotra. La variedad estable de una UPO le permite acercarse a esta hasta quees expulsada por su variedad inestable, para ser captada por otra UPO.

Como las UPOs constituyen el esqueleto del atractor extrano, elcompleto conocimiento de estas proporciona toda la informacion delatractor [ACE+87, Cvi88, GOY87]. Como veremos en la Sec. 1.5 y en elcapıtulo 2 las UPOs juegan un papel muy importante en la sincronizacioncaotica. El inconveniente de las UPOs es que son infinitas y hallarlas paraun determinado sistema no es una tarea facil. En la siguiente seccion sedescribe el metodo mas sencillo de estabilizacion de UPOs.

1.4.1 Metodo de Newton-Raphson

El metodo de Newton-Raphson se caracteriza por su rapida convergen-cia, pero tiene el inconveniente de que solamente converge en un entornopequeno de la solucion. Esto hace que sea necesario tener una buenaestimacion inicial de la localizacion de la UPO.

Para hallar las orbitas inestables de un sistema continuo k-dimensional,primeramente hay que realizar una seccion de Poincare del sistema. Comola integracion numerica de la trayectoria, por un metodo Runge-Kuttau otro, va “a saltos”, se recomienda utilizar algun ardid como el deHenon [Hen82] con objeto de obtener un resultado preciso. El mapa (enk − 1 variables) de Poincare asociado nos dice cual es la posicion de cadainterseccion en funcion de la anterior. Por supuesto, no tenemos la formuladel mapa, ası que solo podemos integrar numericamente el sistema. Si


estamos interesados en una UPO que se cierra tras n cortes con la seccionde Poincare, tendremos que encontrar alguno de los n puntos fijos de lan-esima iteracion del mapa de Poincare, correspondientes a esa orbita.

Si el atractor caotico tiene una dimension fractal cercana a dos, puedeobtenerse una estimacion bastante buena de la posicion de las orbitasinestables, ya que la seccion de Poincare del atractor es practicamenteunidimensional. De esta forma puede obtenerse un mapa de retorno en unavariable a partir del cual interpolar la posicion aproximada de las UPOs.

Formalmente tenemos un (n-esimo) mapa de Poincare F, tal que la UPOen la que estamos interesados intersecta en x∗ = (x∗1, x

∗2, . . . , x

∗k−1)

T =F(x∗). Supongamos que tenemos una estimacion razonable de x∗, quellamaremos x0, entonces se cumplira que:

x0 = F(x0) ∼= x∗ + A(x0 − x∗) (1.18)

donde A es la matriz jacobiana del mapa F. Ası pues, el problema resideen encontrar la matriz A ya que:

x∗ ∼= (I−A)−1(x0 −Ax0) = x1 (1.19)

lo que nos da una mejor estimacion x1. Un proceso iterativo x0,x1,x2, . . .nos permite aproximarnos a la solucion tanto como queramos: x∞ = x∗.

Para encontar los valores de los elementos de matriz de A = (aij), hayque realizar una iteracion del mapa sobre una condicion inicial ligeramenteperturbada x0 + ~ε :

x0 = F(x0 + ~ε) ∼= x∗ + A(x0 + ~ε− x∗) (1.20)

Sustrayendo la ec. (1.18) llegamos a que:

x0 − x0 ∼= A~ε. (1.21)

Si la perturbacion, tiene solo el r-esimo elemento distinto de cero: ~ε =(0, . . . , ε, . . . , 0)T podremos estimar los elementos de matriz de la r-esimacolumna de A, a partir de la ec. (1.21):

a0ir∼=

x0i − x0

i

ε. (1.22)

Por tanto, para cada estimacion de la UPO tenemos que tomar k − 1perturbaciones para estimar todos los elementos de matriz de A que nospermitan conseguir una mejor estimacion con la Ec. (1.19). Ademas, dadoque la matriz A es la jacobiana del mapa de Poincare asociado a esa orbita,el calculo de sus autovalores nos informa de sus propiedades de estabilidad.Al tratarse de una orbita inestable, al menos uno de sus autovaloresestara fuera de la circunferencia unidad en el plano complejo.

1.5 Sincronizacion caotica 23

1.5 Sincronizacion caotica

Sincronizacion y caos parecen dos terminos contradictorios. Debido a ladependencia sensible del caos, parecerıa imposible que dos o mas sistemascaoticos pudieran sincronizarse. Sin embargo, la interaccion de dos sistemascaoticos puede, efectivamente, provocar una sincroniazcion entre ellos.

Los primeros artıculos sobre sincronizacion caotica [FY83, Pik84] noprovocaron demasiado interes en la comunidad cientıfica. Es a partir delfamoso artıculo de Pecora y Carroll [PC90], cuando comienza un estudiointensivo en la sincronizacion. Parte de este interes se debio a la posibilidadde usar el fenomeno para comunicaciones encriptadas [CO93, Lor00] omultiplexacion [Mar99]. Por otro lado, el fenomeno de la sincronizacion deosciladores acoplados ha sido de interes desde tiempos de Huygens [Huy86],tanto para la mecanica [Ble88] como para la biologıa [SS94, Gla01]. Por lotanto, es natural extender estas teorıas al ambito de los osciladores caoticos.Fruto de este interes han aparecido multitud de publicaciones, entre las quepodemos destacar el libro de Pikovsky, Rosenblum y Kurths [PRK01] y elartıculo de revision de Boccaletti y otros [BKO+02].

Hasta aquı no hemos hecho uso de la definicion del termino sin-cronizacion. Ası pues, primeramente tendremos que introducir el significadode lo que entendemos por sincronizacion. Lo cierto es que, actualmente,se distinguen varios tipos de sincronizacion. Los fundamentales son lossiguientes:

• Sincronizacion completa: Dos sistemas caoticos identicos describen lamisma trayectoria x1(t) = x2(t) debido a su interaccion [PC90]. Elsistema evoluciona caoticamente en el subespacio x1 = x2, conocidocomo variedad de sincronizacion. El atractor en estado sincronizadoes igual al de cada uno de los sistemas por separado y, por tanto,contiene el mismo espectro de orbitas inestables. Estas orbitas soninestables solo en una direccion que esta contenida en la variedadde sincronizacion. Ahora bien, si al variar un parametro de control,alguna de estas UPOs se inestabiliza en una direccion transversa a lavariedad de sincronizacion se producira el fenomeno conocido como‘attractor bubbling’ [ABS94]. Aunque en promedio el atractor seatransversalmente estable, si la trayectoria se acerca mucho a algunade estas orbitas sera expelida lejos de la variedad de sincronizaciona lo largo de la direccion inestable tranversalmente. Esto produceque la sincronizacion sea interrumpida por episodios de perdida demomentanea de esta. En resumen, entre la sincronizacion y la no


sincronizacion suele encontrarse una region (mas o menos pequena)que muestra sincronizacion de “baja calidad”.

• Sincronizacion generalizada: Dos sistemas caoticos no identicosajustan sus dinamicas de tal forma que ambas quedan relacionadaspor una funcion χ, tal que: x1(t) = χ(x2(t)) [RSTA95].

• Sincronizacion de fase: Solo se puede definir para aquellos sistemascaoticos que tienen una fase bien definida [PROK97]. Existira sin-cronizacion n : m si sus fases (definidas en la recta real, es decircada giro suma 2π) satifacen |nφ1 −mφ2| < cte. [RPK96], al mismotiempo que sus amplitudes permanecen altamente descorrelacionadas.Notese, que aunque los dos osciladores (caoticos) sean identicos estacondicion no se satisfara al no ser que exista un acoplamiento superiora un umbral, debido a que la dinamica caotica hece que el periodo decada giro fluctue en cada oscilador.

• Sincronizacion 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 capıtulo 2 se estudiaran en profundidad la sincronizacion de fasey de retardo. Por otro lado, en el capıtulo 3 aparecera la sincronizacioncompleta entre osciladores de Lorenz.

1.5 Sincronizacion caotica 25

La mayor parte de los contenidos que se tratan en este capıtulo sepueden encontrar en muchos libros. En la siguiente lista se incluyen aquellosque son adecuados para una primera lectura:

• Schuster [Sch88]: antiguo pero bastante completo.

• Berge, Pomeau & Vidal [BPV88]: orientado hacia una vision fısica,especialmente experimental (ediciones en frances e ingles).

• Alligood, Sauer & Yorke [ASY97]: muy pedagogico y con grancantidad de problemas. Se estudian los mapas mas que en el resto.

• Glendinning [Gle94]: matematicamente riguroso sin resultar pesado.No trata todos los “tipos de caos”.

• Strogatz [Str94]: pedagogico a la vez que amplio en el tratamiento delos diversos problemas de la fısica no lineal.

• Sole & Manrubia [SM96]: trata muchos temas relativos a la fısicano lineal, incluido un capıtulo bastante extenso al caos determinista.Tiene la ventaja de que es el unico de la lista que esta escrito encastellano.

• Ott [Ott93]: probablemente el mas completo en lo que al temaconcreto del caos se refiere. La nueva edicion (2002) trata lasincronizacion caotica.

Chapter 2

Role of Unstable PeriodicOrbits in Phase and LagSynchronization betweenCoupled Chaotic Oscillators

Abstract. Interaction between chaotic oscillators leads to adjustment oftheir characteristics. Depending on the strength of the coupling, interactingsubsystems can share different dynamical features. Under relatively weakcoupling, only the timescales of chaotic motions get adjusted; this is knownas “phase synchronization”. A stronger coupling can enforce a convergencebetween phase portraits: a subsystem imitates the sequence of states of theother one, either immediately (“complete synchronization”), or after a timeshift (“lag synchronization”). With the help of unstable periodic orbits embeddedinto the chaotic attractor, we investigate the transition from nonsynchronizedbehavior to phase synchronizaton, and further to lag synchronization. Wedemonstrate that onset of phase synchronization requires locking on the surfacesof unstable tori, and relate intermittent phase jumps to local violations ofthis requirement. Further, we argue that onset of lag synchronization ispreceded by the disappearance of many unstable periodic orbits whose geometry isincompatible with the lag configuration. We identify orbits which are responsiblefor intermittent deviations from the state of lag synchronization.

2.1 Introduction

Synchronization is a universal phenomenon that often occurs whentwo or more nonlinear oscillators are coupled. Its discovery dates backto Huygens[Huy86], who observed and explained the effect of mutualadjustment 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 wehave witnessed the successful extension of the basic ideas of synchronizationto the realm of chaotic dynamics [PRK01]. Since chaotic oscillations aremore complicated than periodic ones, such extension is neither obvious norstraightforward. The instantaneous state of a periodic process is adequatelycharacterized by the current value of its phase; on the contrary, completeinformation about the state of a chaotic variable includes, in general,more characteristics. Different degrees of adjustment between thesecharacteristics correspond to different kinds of synchrony: from completesynchronization where the difference between two chaotic signals virtuallydisappears [FY83, Pik84, PC90], through the “generalized” synchronizationwhere the instantaneous states of subsystems are interrelated by afunctional dependence [RSTA95, KP96], to phase synchronization. In thelatter case coupled chaotic oscillators remain largely uncorrelated, but themean timescales of their oscillations coincide or become commensurate.

Phase synchronization appears to be the weakest form of synchronybetween chaotic systems; it does not require the coupling to be strong. Incertain situations, the increase of coupling leads through the further, moreordered stage of synchronized motion: the lag synchronization [RPK97]. Inthis state –which precedes the complete synchronization– phase portraitsof 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 counterpartX2 from the second subsystem by a mere time shift: X1(t) = X2(t + τ).

Different aspects of phase and lag synchronization have been inves-tigated mostly from the point of view of global characteristics (Lyapunovexponents, distributions of phase jumps, statistics of intermittent violationsof lag configuration, etc.) Below, we intend to have a closer look at thelocal changes which occur in the phase space of mathematical models. Weconcentrate on invariant sets and their restructurings, which simplify thedynamics by gradually transforming the non-synchronized chaotic attractorinto 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 thecoupling, we trace the fate of unstable periodic orbits (UPOs) embeddedinto the attractor. A universal and powerful tool for the exploration ofchaotic dynamics [Cvi91], unstable periodic orbits proved to be especiallyefficient in the context of synchronization [Rul96]. Interpretation interms of UPOs helped to understand the onset of phase synchronizationin 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 coupledperiodic and chaotic oscillators. Below, we extend this approach to asystem of two bidirectionally coupled non-identical chaotic oscillators.This situation is more complicated, since now each of the participatingsubsystems possesses an infinite set of UPOs. In the following section webriefly characterize the properties of the unstable periodic orbits which existin both subsystems in the absence of coupling. In section 2.3 we describethe changes in the structure of the attractor of the coupled system whichoccur a the coupling strength is increased. We interpret the onset of phasesynchronization in terms of phase locking on unstable tori, and argue thattransition to lag synchronization should be preceded by extinction of mostof the unstable periodic orbits. In sections 2.4 and 2.5 respectively, thesequalitative arguments are supported by numerical results which illustratethe role of UPOs in the intermittent bursts close to thresholds of both phaseand lag synchronization.

2.2 Periodic orbits in the absence of coupling

As an example, we consider the system of two coupled Rossler oscillatorsunder the same set of parameter values for which lag synchronization wasreported for the first time [RPK97] (note that two terms ω1,2 have beenintroduced with respect to Eq. (1.12)) in order to get two oscillators withdifferent frequencies):

x1,2 = −ω1,2y1,2 − z1,2 + ε(x2,1 − x1,2)

y1,2 = ω1,2x1,2 − ay1,2 (2.1)

z1,2 = 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 theseparameter values have been discussed in subsequent publications [SBV+99,BV00, ZWL02].

In each of the subsystems, taken alone, this combination of parametersensures chaotic oscillations (Fig. 2.1(a)). Projected onto the xy plane ofthe corresponding subsystem, these oscillations look like rotations around


-20

-10

0

10

-10 0 10 20

yx

(a)

0.01

0.015

-15 -10 -5

z

x

(b)

5

10

15

5 10 15

-xn+

1

-xn

(c)

Figure 2.1: Rossler oscillator at ω = ω1 = 0.99: (a) projection of phase portrait; solidline: location of the Poincare plane, (b) trace of the attractor on the Poincare plane, (c)one-dimensional return mapping.

the origin; this allows us to introduce phase geometrically, as a lift of theangular coordinate in this plane:

φ1,2 = arctany1,2

x1,2+

π

2sign(x1,2) (2.2)

The mean frequency of the chaotic oscillations is then defined as themean angular velocity: Ω(1,2) = 〈dφ1,2/dt〉. Difference in the values of theparameters ω1,2 makes the mean frequencies of uncoupled oscillators slightlydifferent: 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 orderto enforce phase synchronization, the coupling should be able to suppressthis drift by adjusting the rotation rates.

In order to understand the role of unstable phase orbits in the phasespace of the coupled system, it is helpful to start with the classification ofsuch orbits in the absence of coupling. In its partial subspace, each of the3-dimensional flows induces the return map on an appropriate Poincaresurface (it is convenient to use for this purpose the trajectories whichin the i-th system intersect the surface yi = 0 “from above”). Thistwo-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 Poincaresurface is graphically almost indistinguishable from a one-dimensionalcurve (Fig. 2.1(b)). Parameterizing this curve (e.g. by the value of thecoordinate x), we arrive at the non-invertible one-dimensional map shownin Fig. 2.1(c). Since the latter turns out to be unimodal, its dynamics iscompletely determined by the symbolic “itinerary” [CE80]: the sequenceRLL... in which the j-th symbol is R if the j-th iteration of the extremumlies to the right from this extremum, and L otherwise. According tonumerical estimates, for ω = ω1 = 0.99 the itinerary is RLLLLRLLL...,and for ω = ω2 = 0.95 it becomes RLLLLLRLL... The starting segmentsof the two symbolic strings coincide, the first discrepancy occurs in the 6-th symbol; therefore, the number of unstable periodic orbits which make lturns around the origin is the same in both subsystems, if l does not exceed5. 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 thelogistic mapping xi+1 = A xi(1− xi) shows that flows with ω = ω1 = 0.99and ω = ω2 = 0.95 correspond to maps with A = 3.9977031 . . . andA = 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 ofeach single revolution (turn) in the phase space appears to be of littleimportance: what matters for phase dynamics is the mean duration of theturn, i.e. the overall period of the orbit divided by the number of turnsin 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 returnsonto the Poincare plane depends on the position on this plane, the periodsof all periodic solutions are –in general– different. It is convenient tocharacterize periodic orbits in terms of “individual frequencies” Ωi; theseare not the usual inverse values of the corresponding overall periods, butmean frequencies per one turn in the phase space: for an orbit with periodT consisting of l turns, Ωi ≡ 2πl/T . Fig. 2.2 presents the distributionsof individual frequencies for periodic solutions for both subsystems in theabsence of coupling. Since usually the orbits with relatively short periodsare sufficient for an adequate description of the whole picture [HO96], werestrict ourselves to orbits with length l ≤ 10; this yields 164 UPOs atω = 0.99 and 196 UPOs at ω = 0.95.

As shown in Fig. 2.2, two frequency bands are separated by a gap. Forω = 0.99 the individual frequencies belong to the interval between Ω(1)

max =1.035519 . . . (orbit with length 1) and Ω(1)

min = 1.014042 . . . (one of the orbits


0.96

0.98

1

1.02

1.04

2 4 6 8 10

Ωi

symbolic length

Figure 2.2: Frequencies of unstable periodic orbits embedded into the attractors of theRossler equations. Circles: ω = 0.99; crosses: ω = 0.95.

with length 5). For ω = 0.95 the values are distributed between Ω(2)max =

0.9927899 . . . (the same orbit with length 1) and Ω(2)min = 0.9790416 . . . (one

of the orbits with length 6).Besides periodic orbits, the Rossler equations possess a saddle-focus

fixed point located close to the origin. Although this point does not belongto the chaotic attractor, it is not irrelevant: under coupling, it interactswith periodic orbits of the complementary subsystem and contributes tothe general scenario. To unify notation, we refer to this fixed point as the“length-0 orbit”

2.3 Attractors of a coupled system: role ofunstable tori in synchronization transitions

Formally, at ε = 0 (absence of coupling) the attractor in the joint phasespace of the two systems contains a countable set of degenerate invariant 2-tori: direct products of each periodic orbit from the 1st subsystem with eachperiodic orbit from the 2nd one. Again, for the purpose of comparison ofphase 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 onerevolution around the torus for the projections onto two subsystems be,respectively, τ1 and τ2. Then the rotation number is introduced as theratio ρ = τ1/τ2. If the equality ρ = 1 holds, within a sufficiently long timethe projections of trajectories make an equal number of rotations in thesubspaces of subsystems: the torus is “phase locked”. Generalization of thisinterpretation for other rational values of ρ is straightforward. Obviously,at ε = 0, the rotation number is ω1/ω2 where ω1 and ω2 are the individualfrequencies (per one rotation, as discussed above) of the two periodic orbitswhich form the torus.

As soon as the infinitesimal coupling between the subsystems isintroduced, the degeneracy of tori is removed. The UPOs shown in Fig. 2.2produce 164 × 196 = 32144 tori whose rotation numbers (in the abovesense) lie between Ω(2)

min/Ω(1)max = 0.92737 . . . and Ω(2)

max/Ω(1)min = 0.97904 . . ..

In general, each torus persists in a certain range of ε, and its rotationnumber ρ is a devil’s staircase-like function of ε: intervals of values of εcorrespond to rational values of ρ.

Since the periodic orbits in the subsystems are unstable, the tori arealso unstable: for small values of ε, a trajectory on the toroidal surface hasat least two positive Lyapunov exponents.

The boundaries of “locking intervals” of ε for each torus are marked bytangent bifurcations of periodic orbits. Such a bifurcation creates/destroyson the surface of the torus two closed trajectories, one stable (with respectto disturbances within the surface), the other one unstable. Since themotion along the torus is parameterized by the phases of subsystems, belowwe refer to these orbits as “phase stable” and “phase unstable” [PZR+97],respectively.

The following argument demonstrates that on each torus the phase-stable and phase-unstable orbits are not necessarily unique. Let the torusoriginate from the direct product of two periodic orbits: an UPO from thefirst subsystem with length l and an UPO from the second subsystem withlength m. Then the main locking (1 : 1 in our notation) assumes that thephase curve is closed after n turns, n being the least common multiple of land m. Let us take the projection of the periodic orbit onto the subspaceof the first subsystem, and select some particular point on it (e.g., thehighest of the n main maxima for one of the variables). By translatingforwards and backwards the partial projection onto the other subsystem,we get n configurations in which one of the n maxima of the second variableis close to the selected point. Fig. 2.3 shows such “appropriate for the


Figure 2.3: Configurations favor-able for the locking on the torusoriginated from the direct productof UPOs of length 2: (a) “in-phase”; (b) “out-of-phase”. Solidcurves: x1(t); dashed curves: x2(t).

-10

0

10

0 5 10

x 1,2

t

(a)

-10

0

10

0 5 10

x 1,2

t

(b)

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 shouldexpect to observe on the surface of a single torus up to n coexisting pairs ofphase stable and phase unstable periodic orbits. [Naturally, the argumentis not rigorous; in principle, not all of n possible configurations shouldnecessarily be exhausted; on the other hand, the existence of additionallockings cannot be totally excluded as well]. A locking interval in theparameter space ranges from the birth of the first couple of curves with theprescribed locking ratio, to the death of the last such couple. Uniqueness ofthe rotation number forbids the coexistence on the same torus of periodicorbits with different locking ratios.

In the course of time a chaotic trajectory repeatedly visits theneighborhoods of unstable tori; in each of them it spends some time windingalong the surface, until being repelled towards some other unstable torus.During the time T spent in the vicinity of the torus with rotation numberρ, 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 passageclose to a torus results in a phase drift. On the other hand, if the torus islocked in the ratio 1 : 1, a passage of a chaotic trajectory along one of thephase-stable UPOs on the toroidal surface leads neither to a phase gain norto a phase loss. Therefore we can expect that in the phase synchronizedstate all of the tori embedded into the chaotic attractor are locked andhave the same frequency ratio. From this point of view, in the course ofthe transition to phase synchronization, each of the tori present at ε = 0should either reach the main locking state or disappear, from the attractoror from the whole phase space. Note that, even if a single torus within theattractor remains not locked, the ergodic nature of chaotic dynamics willensure that from time to time the trajectory will approach this torus closeenough 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 theUPOs in uncoupled subsystems into two sequences U (k)

i , k = 1, 2; i =1, 2, . . .. The ordering can be done by means of criteria which take intoaccount the symbolic length and topology (expressed e.g. by symbolicitinerary) of the orbits. This induces labeling among the tori of the coupledsystem: the torus Tij originates from the interaction of the orbit U

(1)i from

the first subsystem and the orbit U(2)j from the second one.

As discussed above, for the values of ε beyond the threshold of phasesynchronization all the tori inside the attractor should have the samerotation number 1, hence they should possess periodic orbits. In fact,at finite values of ε neither smoothness nor even the very existence of a2-dimensional toroidal surface can be guaranteed, but this circumstanceappears to be of little importance: in the synchronized state the decisiverole is played not by the entire torus or its global remnants, but by relativelysmall segments near closed phase-stable and phase-unstable orbits. Thetorus may break up, but periodic orbits persist. Therefore, in our discussionbelow the symbol Tij denotes not so much the actual two-dimensional torus,but rather the set of (possibly several) periodic orbits corresponding to thelocking 1:1 on this torus. If U

(1)i and U

(2)j 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 land m. Symbolic labels B(1) and B(2) for projections of Tij respectivelyonto the first and second subsystem consist of n symbols: B(1) is n/l timesrepeated 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. Thesymbolic 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 tobring one of these projections very close to the other by time shift: for allvalues of such shift the time-averaged difference between these projectionswill neither vanish nor become very small. According to this argument,only those Tij for which two generating UPOs have the same length andtopology can persist in the attractor of lag-synchronized state. Presence ofthe “non-diagonal” Tij is incompatible with lag synchronization. Thereforeall such tori and associated closed orbits should, in the course of increaseof ε, either disappear, or leave the attractor.

Further, Tij with identical symbolic labels may contain several phase-stable periodic orbits. However, only the passage close to the “in-phase”


orbit would allow for lag synchronization with small (compared to the meanduration of one turn) value of lag. For “out-of-phase” configurations, whichare obtained from the “in-phase” ones by cyclic permutation of maxima,the appropriate time shift would be close to several durations (lengths ofthe shift) of the turn. Apparently, only the “in-phase” orbits contributeto the motion in the lag-synchronized state. For example, the UPO inFig. 2.3(a) can participate in the lag-synchronized dynamics, whereas theUPO 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 bepreceded by extinction of most unstable periodic orbits which populatethe attractor at the onset of phase synchronization. In fact, a set of twooscillators in the state of lag synchronization behaves almost the sameway as one of them taken separately; in this sense, the complexity of lagsynchronization is relatively low.

If the above interpretation is correct, intermittency of respectivecharacteristics, observed below the threshold values of the coupling strengthboth for phase [LKL98] and lag synchronization [BV00, ZWL02] should becaused by the passage of chaotic trajectories close to the last obstructinginvariant sets. In the first case these sets are the last non-locked tori, andin the second case they are either the last remaining UPOs from “non-diagonal” Tij or the “out-of-phase” UPOs.

For completeness, it should be mentioned that there are certain UPOswhich 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) onone side, and an UPO on the other side. Obviously, such orbits are alsoincompatible with lag synchronization, and should disappear in the courseof increase of ε.

In the following sections we test these qualitative conjectures about theonset mechanisms of phase and lag synchronization against the numericaldata obtained by integration of Eq.(2.1); UPOs have been computedthrough combination of the Schmelcher-Diakonos[SD98] and Newton-Raphson methods.

2.4 Phase synchronization

Phase synchronization in Eq. (2.1) is observed beyond the thresholdvalue ε = εps. For ε > εps, the difference of phases between two oscillatorsremains confined within a narrow interval for t →∞; below this threshold

2.4 Phase synchronization 37

0.037 0.038 0.039 0.04 0.041

104

106

ε

<

Tpj

>

20 22 24 26 2810

−7

10−6

10−5

10−4

(εps

−ε)−1/2

freq

uenc

yFigure 2.4: Mean time 〈Tpj〉 between phase slips and frequency f of slips vs. couplingstrength ε.

it grows unboundedly. According to our computations, εps ≈ 0.0416(this is somewhat higher than the value 0.036 reported in [RPK97]). Infact, already at ε ≥ 0.036 the phases of two oscillators stay synchronizedfor most of the time: the plot of phase difference as a function of timereminds a staircase in which long nearly horizontal segments are interruptedby relatively short transitions. Such transitions (phase slips) are notinstantaneous; usually, it takes several dozens of turns in the phase space,to increase the phase difference by 2π. However, compared to the averageduration of the synchronized segment, phase slips are fast: as seen inFig. 2.4, when ε approaches εps, such duration grows from hundreds ofturns through tens of thousands to millions and further on. The value0.0416 is the highest value of ε at which we were able to observe a phasejump (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 spaceas a kind of repeller-attractor collision [POR+97, PZR+97, ROH98]: thelocal bifurcation (tangent bifurcation in which a phase-stable and a phase-unstable UPOs are born), is simultaneously the global event: disappearanceof the last channel for phase diffusion. Rare violations of synchronizationbelow the threshold were named “eyelet intermittency”, since escapes fromthe phase-locked state were due to the very accurate hitting of a vicinityof the last non-locked torus.

The same mechanism is at work in our case just below εps: of infinitelymany tori Tij embedded into the chaotic attractor, almost all are lockedin 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


difference. Since the tori are unstable, the chaotic trajectories are mostlykicked out from their neighborhoods before producing a noticeable phasedifference. Only the trajectories which come very close to the non-lockedtori, stay long enough in their vicinities in order to gain a phase slip. Thefrequency f of such events depends on the distribution of the invariantmeasure on the attractor. Assuming, for simplicity, that this measureis 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 asa function of the coupling strength ε. The vertical “amplitude” coordinateon this plot is fictitious: it plays the role of appropriately re-scaled andshifted coordinate values (if actual values of coordinates were used, mostof the branches would overlap, strongly hampering understanding of thebifurcation sequences).

Tangent bifurcations are marked with asterisks (*), and period-doublings are denoted by small circles (). The notation m − m′ standsfor the locking on the torus which is the direct product of the length-mand the length-m′ orbits of the first and second oscillators, respectively.Thus, the 2-2 torus undergoes two lockings: at the moment of birth ofcorresponding UPOs, phase lags between both oscillators are respectively∼ π/2 and ∼ 2π + π/2; as ε grows, the values of these lags decrease. Inaccordance with the above classification, we call these orbits in- and out-of-phase lockings. It may be seen in Fig. 2.5 that the tangent bifurcationswhich create orbits of length 1 and 2, occur in a small interval aroundε = 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 otherbranches (see below) indicates that the smoothness of the correspondingtoroidal surfaces is already lost.

We remind that label 0 denotes orbits which are born from the directproducts of the steady solution with periodic solutions. The plot showsthat, as expected, such orbits disappear relatively early: the branch 2-0joins 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 thebranches born on the torus 1-1.

As a further illustration, in Fig. 2.6 we show solution curves andbifurcation points for orbits of length 3. This case is richer, insofar aseach isolated oscillator contains two UPOs of this length (cf. Fig. 2.2);


0 0.05 0.1 0.15 0.2 0.25

2−2 in

2−1

2−2 out

1−1 1−2

1−0

2−0

2

1

stab

le w

indo

w o

f len

gth

5

ε

Figure 2.5: Bifurcation diagram showing UPOs of length 1 and 2. Notation: ∗ -saddle-node bifurcations, - period-doubling bifurcations. Solid, dashed, and dottedlines: orbits unstable in 1, 2 and 3 directions, respectively.


they are labeled 3a and 3b. Since every orbit possesses three maxima ofx1,2, 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, withphase lags ∼ 2π and ∼ 4π, respectively. Now, besides tangent and period-doubling 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 createUPOs, occur at ε < 0.04.

in−phase (a)

3a−3b

3b

3a

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

3b−3b

ε0.033 0.034 0.035 0.036 0.037 0.038 0.039 0.04 0.041 0.042 0.043

3b−3b

3b−3a

ε

2π−out−of−phase (b)

3a−3b

3a−3a

0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07

3b−3b

ε

4π−out−of−phase (c)

3a−3b

3a−3a

Figure 4c, D. Pazo et al. (2002)

Figure 2.6: Bifurcation diagramfor UPOs of length 3. Notation:© - Hopf bifurcations; others as inFig. 2.5.

A remarkable feature here are the isolas in Fig. 2.6(b),(c): each familyof out-of-phase lockings is not connected to families of periodic solutionsand exists only in the relatively small interval of values of ε. As seen inFig. 2.6(a), for sufficiently high values of ε of all the UPOs of length 3, onlytwo in-phase orbits survive.

Several further families of UPOs are not shown on these plots. Whenε is increased, the orbits of the type 0 − 2n disappear one-by-one in theinverse period-doubling cascade, and finally the last of them, the UPO 0-1,


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 ina saddle-node bifurcation, as well as the orbits 0-3a and 0-3b. The tori3a-1 and 3b-1 annihilate each other in the same way as the 2π-out-of-phaselockings, 3a-3b and 3a-3a in Fig. 2.6(b). On the other hand, we failed tolocate numerically the 1-3a and 1-3b lockings; it seems that both tori alsocollide and disappear in a saddle-node bifurcation (or they get locked buttheir UPOs survive in a very narrow range of ε).

Calculations for UPOs of other lengths have shown qualitatively similarpictures, with tangent bifurcations around ε ≈ 0.04 and short-lived out-of-phase lockings.

We have also performed numerical experiments in order to verify theconjecture that phase jumps occur when the trajectory approaches a non-locked torus. Since we are presently unable to locate numerically in thephase space the two-dimensional unstable tori, sometimes it is difficult toassign the jump to the passage near a particular torus. Nevertheless, incertain cases it was possible to identify a configuration which provoked aphase jump. In such situations, at the beginning of the jump the segmentsof 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 closeto a 1-3 torus can be recognized. In general, the farther from 1 is therotation number ρ on the degenerate torus at ε = 0, the higher shouldbe the magnitude of coupling required for the locking. In the frequencydistribution of Fig. 2.2, the highest individual frequency belongs to theorbit of length 1; the tori, built with the participation of this UPO, requirerelatively strong coupling in order to get locked. In accordance with this,many of the phase jumps close to the threshold of phase synchronizationare 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 valueε = 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 systemfor times higher than 109 mean rotation periods (our longest runs) in orderto experience such jumps. We cannot point out which torus is the last oneto be locked. Among the relatively short orbits, the closest to εps lockingappears to be the tangent bifurcation, which creates orbits of length 4 atεps = 0.0414302.


4.22 4.225 4.23 4.235 4.24

x 105

0

2

4

6

8

10

Time (t.u.)

∆φ, ∆

φ−2π

−200

20

−20

0

200

5

10

x1

y1

z 1

−200

20

−20

0

200

10

20

x2

y2

z 2

4.23 4.2305 4.231

x 105

−20

0

20

x 1

4.23 4.2305 4.231

x 105

−20

0

20

Time (t.u.)

x 2

Figure 2.7: Occurrence of a phase jump for ε = 0.0403. The upper left figure showsthe phase difference (and the phase difference less 2π as a reference) between the twosubsystems. The time lap corresponding to the passing through an eyelet is indicated bya segment and an arrow. The dynamics of each subsystem during the crossing is plottedin two time series and two snapshots.

2.5 Lag synchronization

The lag synchronized state in Eq. (2.1) was found to exist above thecritical value of the coupling strength εls ≈ 0.14[RPK97]. In this state, thedynamics of both oscillators is very similar to the one that they exhibitbeing isolated, but now they are related by a time lag: x1(t) ≈ x2(t + τ0).

The transition from phase synchronization to lag synchronization wasshown to be preceded by a intermittent region where lag synchronizationwas interrupted by bursts [RPK97]. Since the Rossler oscillator isapproximately isochronous, the time lag is practically equivalent to thephase lag. In Fig. 2.8(a) the value of the mean phase difference 〈∆φ〉between both oscillators is shown, as well as the corridor formed by thisdifference ± its standard deviation σ. For ε > 0.14 this corridor is rathernarrow (albeit non-zero); when ε is decreased below 0.14, the deviationrapidly grows. However, the minimal and maximal values for deviations of

2.5 Lag synchronization 43

0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.260.1

0.2

0.3

0.4

0.5

< ∆

φ>, <

∆φ>

+σ,

<∆φ

>−

σ

0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26−0.2

0

0.2

0.4

0.6

0.8

ε

(∆φ)

max

,min

10 turns4

10 turns5

10 turns6

10 turns7

(b)

(a)

Figure 2.8: (a) Mean phase difference between subsystems, and the bounds setby standard deviation. (b) Maximum and minimum phase difference computed fortrajectories 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 increasingcomputing time, we were able to detect larger deviations from 〈∆φ〉 athigher values of ε; the plot shows dependencies estimated from chaoticorbits of different length.

What is the role played by UPOs in this intermittent transition to lagsynchronization? We begin the discussion with the observation that growthof the coupling strength reduces the volume of phase space occupied by theattractor. Evolution of the system towards this state is illustrated by returnmaps for one coordinate, recovered from the intersection of the attractorwith the Poincare 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 remainsa small proportion of points which lie at a distance from the parabola-like


5

10

15

5 10 15

-x1(n

+1)

-x1(n)

(a)

5

10

15

5 10 15

-x1(n

+1)

-x1(n)

(b)

5

10

15

5 10 15

-x1(n

+1)

-x1(n)

(c)

5

10

15

5 10 15-x

1(n+

1)

-x1(n)

(d)

Figure 2.9: Return maps for the variable x1 on the Poincare 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 UPOssimilar to that of an isolated Rossler oscillator; according to Fig. 2.2, forε > εls there should be one UPO of length 1, one UPO of length 2, and twoUPOs of length 3. Characteristics of unstable periodic orbits for ε slightlybeyond εls are shown in Fig.2.10. According to Fig.2.10(a), correspondencewith an isolated oscillator is not yet reached: the full system possesses twoUPOs of length 2, as well as four orbits of length 3 and four orbits of length4, whereas the description based on the unimodal mapping prescribes oneorbit 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 eventuallydisappear: two orbits of length 3 annihilate each other throught the tangentbifurcation at ε = 0.154856; then at ε = 0.15694, a period-doublingbifurcation unifies the orbit of length 4 with the “superfluous” orbit oflength 2, and finally the branch of the latter UPO (which will be separatelydiscussed 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 allthese frequencies coincide). To characterize the time shift between thesubsystems, we use the value of the phase lag between them at the moment

2.5 Lag synchronization 45

0.99

0.995

1

1.005

1.01

1 2 3 4 5

Ωi

Orbit length

(a)

0

0.1

0.2

0.3

0.4

1 2 3 4 5

τ

Orbit length

(b)

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 Poincare plane y2 = 0. Since we are interestedin instantaneous values, each length-l UPO delivers l values of ∆φ. Asseen in Fig.2.10(b), most of the values of the phase lag belong to thenarrow range between 0.27 and 0.3; however, large deviations from thisrange are also present. Notably, most of these deviations belong to the“superfluous” orbits. As understood from Fig.2.8(b), noticeable outburstsof phase difference are very rare; this means that a chaotic trajectoryonly seldom visits the neighborhoods of these UPOs; accordingly, theircontribution into the dynamics is relatively small.

Growth of ε beyond the values shown in Fig. 2.9 leads to furthercondensation of the points of the return map onto the one-dimensionalbackbone; the proportion of deviations becomes smaller. It appears that inthe space phase there exists a pattern (at the moment we know too littleabout its properties to call it an “invariant manifold”), which is responsiblefor the lag structure and on which dynamics is adequately represented by aunimodal map. This pattern is locally attracting almost everywhere, exceptfor certain “spots”; a chaotic trajectory which hits such a spot, makes ashort departure from the pattern and disturbs the lag synchronism.

Note that at large values of ε the UPOs have only one unstable direction


Figure 2.11: Poincare section ofthe attractor for ε = 0.2; fiveregions with the largest densityof points, corresponding to the“ghost” of the stable window oflength-5, can be distinguished. Thelocation of the “transversally unsta-ble” length-1 orbit is indicated witha circle.

−14 −12 −10 −8 −6 −4 −2−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

1

23

4

5

ε = 0.2

0.5 (x1+ x

2)

0.5(

x 1− x

2)(one characteristic multiplier outside the unit circle); this corresponds tothe instability of all periodic points of the unimodal mapping. Whenε is gradually decreased, the first orbit to become unstable in a seconddirection is the one of length 1 (ε = 0.23892). This bifurcation was reportedin [SBV+99] where synchronization transitions for different mismatchesbetween ω1 and ω2 were studied. At this critical point, the length-1 periodicorbit embedded into the “lag attractor” undergoes the period-doublingbifurcation. As a result, an orbit of length 2 is created. Tracing thisnew orbit down to small values of ε, we observe that it ends up as a phase-stable orbit on the torus formed by the length-1 and length-2 UPOs of thedecoupled subsystems; the corresponding bifurcations are shown in Fig. 2.5.The configuration of this orbit (two approximately equal maxima in theprojection onto one subsystem versus two unequal maxima in the secondsubsystem) is obviously incompatible with the requirements of the lag-synchronized state. Thereby, the loss of perfect lag synchronization occursbecause 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 periodicorbit is born at ε(5) in the saddle-node bifurcation. Below this value,a typical type-I intermittency behavior is observed, and the distributionof invariant measure on the attractor is very non-uniform: the periodicorbit leaves a “ghost”, the density of imaging points is rather high in fivecorresponding regions of the Poincare section, and the length-1 UPO, whichlies aside from these regions, is very seldom visited. Probably, this is thereason why the intermittent behavior above ε = 0.145 was not observedearlier.

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”S2(τ) = 〈(x2(t)−x1(t−τ))2〉/(〈x2

1(t)〉〈x22(t)〉)1/2 to vanish (or nearly vanish)

for some τ0; naturally, τ0 is the lag duration. In [BV00] it was noticedthat besides the main minimum at τ0/T 1, the similarity function hassecondary minima at τ = τ0 + mT , where m = 1, 2, ..., and T is closeto the mean duration of one turn in the phase space. When perfect lagsynchronization is lost, the magnitudes of the secondary minima of S2

decrease. It turns out that intermittent violations of lag synchronizationconsist of jumps from the main lag configuration [x1(t) = x2(t + τ0)] toconfigurations 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 intermittencywhich precedes the onset of lag synchronization, is caused by passagesnear the out-of-phase UPOs. Our numerical data shed more light on thenature of these jumps and allow us to identify the orbits responsible for theintermittency. According to Fig. 2.5, among the orbits belonging to theout-of-phase locking of two UPOs of length 2, the least unstable one (theorbit 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) showsx1,2- and y1,2-projections of this UPO embedded into the attractor. Weobserve that part of this orbit is “transversal” with respect to the bulk ofthe attractor. In the course of the intermittent bursts, chaotic trajectorieswhich leave the bulk region, move along this UPO. During this motionthe dynamics of both oscillators gets approximately correlated, and the lagbetween 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 andonset of lag synchronization between two coupled chaotic oscillators areaccompanied by profound changes in the structure of the attracting set.Unstable periodic orbits serve as mediators in these processes: when thecoupling strength is increased, they should first appear in the phase space inorder to enforce the entrainment of phases, and then, most of them shouldagain disappear, in order to leave in the attractor only suitable patternsfor lag synchronization. Absence of necessary UPOs in the first case, andpresence of “non-suitable” orbits in the second case are the reasons for


0.05 0.1 0.15−4

−3

−2

−1

0

1

ε

Re(

λ i)

0 5 10

−15

−10

−5

0

5

10

15

Time (t.u.)

x

x2

x1

ε=0.14

−20 −10 0 10 20−15

−10

−5

0

5

10

15

20

x1

x2

−20 −10 0 10 20−20

−15

−10

−5

0

5

10

15

y1

y2

(a)

(b)

(c) (d)

Figure 2.12: Role of the out-of-phase UPO of length 2 in intermittent lagsynchronization. (a) Eigenvalues of the second iteration of the Poincare map at theUPO as functions of ε. Because of the strong transversal contraction of the Rossleroscillator, two of the eigenvalues are very close to zero and are not depicted. (b) Timeseries of x1 (black line) and x2 (gray line) over one period; (c,d) Dots: chaotic orbit onthe “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 intermittencyis caused by passages near the 2-tori which are not yet locked, or lockedin ratios different from 1:1; in the latter case such intermittency is, in fact,a certain form of “other” short-lived synchronization. The intermittencypreceding the onset of lag synchronization is due to the passages near theperiodic orbits, in which two oscillators are locked “out-of-phase”; suchpassages make the system exhibit momentary exotic lag configurations. Wefeel that unstable periodic orbits are an appropriate tool for the analysisof intricate details of these transitions; further numerical advances wouldprobably require the technique for the calculation of unstable 2-tori.

The studied system of two coupled oscillators is nonhyperbolic, at leastin the parameter region around the onset of phase synchronization. Asseen in Fig. 2.5, increase of ε leads to the decrease in the dimension of theunstable manifolds of UPOs. As a result, over large intervals of ε we observecoexistence of UPOs with 2-, 3- and 4-dimensional unstable manifolds. Ingeneral, this phenomenon, known under the name of ‘unstable dimensionvariability’, has important implications for dynamics itself as well as for thevalidity and applicability of numerical algorithms [BS00]; its significance inthe context of synchronization is yet to be analyzed.

Chapter 3

Transition toHigh-Dimensional Chaosthrough a Global Bifurcation

Abstract. We study a novel transition to high-dimensional (d ≈ 4)chaotic behavior in a system composed of three unidirectionally coupled Lorenzoscillators. The transition involves a global explosion that creates a high-dimensional chaotic set, formed by an infinite number of unstable tori. Thechaotic set becomes attracting after a boundary crisis, and exhibits multistabilityin a range of parameters, coexisting with two (stable) symmetry-related three-dimensional tori (the only attractors before the crisis). These two toridisappear after a saddle-node bifurcation in which they annihilate with two three-dimensional unstable tori.

3.1 Introduction

One of the most fundamental problems in the study of nonlineardynamical systems is the characterization of the routes through whichthese systems undergo a transition to chaotic behavior. In low dimensionaldissipative dynamical systems, chaos quite often appears through a fewwell characterized routes (or scenarios) [Eck81]: (a) the period-doublingcascade [Fei78]; (b) the intermittency route [PM80]; (c) the route involvingquasiperiodic tori [NRT78, CY78]; (d) the crisis route [GOY83a]; (see also[BPV86, Ott93] for a survey). Another possibility is that chaos sets inthrough a global connection to a fixed point, as is the case of the Lorenzsystem [Spa82], and of Shil’nikov chaos [AAIS89, Hom93].

Low-dimensional chaos, occurring in at least three-dimensionaldissipative 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 dissipativenature of the flow implies the existence of a third negative exponent suchas the sum of all the exponents is negative. We will represent the Lyapunovspectrum 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 theLyapunov 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 high-dimensional 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 increasethe dimensionality of phase space, at least by one.

An obvious possibility to transit to high-dimensional chaos is to take asstarting point a low-dimensional chaotic attractor. This possibility has beenusually considered in the context of desynchronization between coupledchaotic oscillators. A possible scenario, and probably the most studiedone [HL99, KMP00] involves generating a hyperchaotic attractor, i.e., achaotic attractor with two (or more) unstable directions (associated topositive 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 tohave a chaotic attractor with two null Lyapunov exponents. Although thissituation is not generic, it has been shown [MGP+97, SMP00, Yan00] tooccur in unidirectionally coupled chaotic systems through a symmetric Hopfbifurcation2 [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 fourthexponent is smaller in absolute value than the positive one.

Less obvious is the possibility of a direct transition to high-dimensional chaos without an intermediate low-dimensional chaoticattractor. Restricting to autonomous ordinary differential equations, the

1According to this formula, actually a conjecture, D1 = K +∑K

j=1(λj/|λK+1|), being

K the largest integer such that∑K

j=1 λj ≥ 0, where the λj are the Lyapunov exponents

ordered from larger to smaller. With, e.g., 2+ we shall indicate the order of the integercontribution to the dimension.

2One can claim that, close enough to the onset of the bifurcation, two Lyapunovexponents are zero, or, at least, very close to zero. In practical terms the situation isanalogous to what happens when studying chaotic phase synchronization in two coupledRossler oscillators, where two Lyapunov exponents remain quite close to zero for a finiteparameter range [RPK96].

3.1 Introduction 53

transition to high-dimensional chaos has been found to be associatedto quasiperiodicity. Thus, the works by Feudel et al. [FJK93], and byYang [Yan00], observe a three-frequency quasiperiodic window before thesystem passes from two-frequency quasiperiodicity to high-dimensionalchaos. A more geometrical view is contained in the route (to high-dimensional chaos) from a two-dimensional torus reported by Moon[Moo97]. It comprises a global bifurcation whose structure amounts toadding one dimension to each building block of the Lorenz attractor.

In the present thesis we shall describe a route to chaos whose probablymost novel aspect is that it implies the sudden creation (i.e. withoutmediating low-dimensional chaos) of a high-dimensional chaotic attractorwith D1 > 4. Our approach combines the computation of the Lyapunovexponents (as Refs. [FJK93, Yan00]) with an investigation on the globalbifurcations (as Ref. [Moo97]) giving rise to the chaotic attractor. Thechaotic set is first created through a double heteroclinic connection betweencycles. This set, that is visible by the existence of transient chaos,becomes attracting through a boundary crisis. At this crisis the chaoticattractor touches its basin boundary that is constituted by two symmetryrelated unstable three-dimensional tori T3 (and their stable manifolds).The resulting chaotic attractor has a single unstable (chaotic) directionand two neutral directions, as indicated by its Lyapunov spectrum:(+, 0, 0,−,−, · · · ). This high-dimensional attractor is known as chaoticrotating wave (CRW) [MGP+97, SMP00]. The CRW may be created in aring of three coupled chaotic oscillators in which the synchronized chaoticstate (low-dimensional chaos) is destabilized by a k = 1 Fourier spatialmode3.

Our system of coupled Lorenz oscillators also exhibits a periodicrotating wave (PRW) [MPL+97, SM98, MPM98] (this route was foundexperimentally in Ref. [SM99, San99]). The PRW is found for strongercoupling and it is the starting point of our study. It consists roughly in adynamics where the synchronous k = 0 mode is in a steady state.

In addition to its fundamental interest, the synchronization and theselection of particular phase relations is a fundamental topic in a biologicalcontext, ranging from neurology and brain function [RGL+99] to animallocomotion [GSBC99]. In this latter case, circular geometries of coupledcells are used for modeling central pattern generators [CS93, CS94] becausethe system’s symmetries provide the different phase patterns observed fordifferent gaits.

3If the ring were larger, other modes could become unstable [PMPP00].


The plan of this chapter is as follows. In Sec. 3.2 the system is describedand an overall picture of the route to chaos is presented. Section 3.3provides an analytical approximation to the state known as periodicrotating wave. Section 3.4 discusses at some detail the creation and thecharacterization of the two- and three-frequency quasiperiodic attractorsfound in the system. Section 3.5 presents the numerical evidences that havebeen used to understand the complex route to chaotic behavior presentedby the system. Section 3.6 presents a characterization of the route tohigh-dimensional chaos through a return map similar to that of the Lorenzsystem. The goal of Sec. 3.7 is precisely to present and discuss at somelength the route to chaos exhibited by the system. Finally, Secs. 3.8 and3.9 present some further remarks on this work and a summary of the mainresults, respectively.

3.2 System and overall picture

As already advanced in the Introduction, the goal of the present chapteris to study the transition to high-dimensional chaos in the 9-dimensionalsystem formed by three Lorenz [Lor63] systems coupled according to thesynchronization method introduced in [GM95]. The evolution equationsfor the ring can be written in the form [MPL+97],

xj = σ(yj − xj)yj = R xj − yj − xj zj

zj = xj yj − b zj

j = 1, . . . , N = 3 , (3.1)

where xj = xj−1 for j 6= 1, introduces the coupling. The periodic boundaryconditions used in this work imply that x1 = x3, while the followingparameters 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 suggestedby the results of the experimental study of three coupled Lorenz oscillatorscorresponding to these parameters [SM99].

This system exhibits synchronous chaos for R < Rsc ≈ 32.82. AtRsc the system exhibits a Hopf instability but from a chaotic state,yielding a behavior that was first found in rings of unidirectionally coupledChua’s oscillators, and called a Chaotic Rotating Wave (CRW) [MGP+97].The Hopf bifurcation exhibited by the system is called symmetric, asit is originated from the cyclic (Z3) symmetry of the ring. The CRWbehavior is characterized by the fact that it is formed by the combinationof two different dynamical behaviors, at least close to the onset of

3.2 System and overall picture 55

the symmetric bifurcation: the Lorenz waveform, characteristic of thesynchronization manifold existing for R < Rsc (when the system exhibitschaotic synchronization) and the oscillation created by the symmetricHopf bifurcation (that occurs in the subspace which is transverse to thesynchronization manifold). The oscillation created in the symmetric Hopfbifurcation is characterized by a behavior in which neighboring oscillatorsdiffer by a phase of 2π/3 (2π/N in the general N -oscillator case), andbecause of this picture this behavior is the analog of a traveling rotatingwave 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 stillcharacteristic of the subspace transverse to the synchronization manifold,k = 1 Fourier mode. Two of such periodic behaviors, called PeriodicRotating Wave (PRW) in Ref. [MPL+97], are found depending on theinitial conditions in the range R ∈ [35.26, 39.25] (approximately). AtR = Rpitch ≈ 39.25 both solutions merge giving rise, through a pitchforkbifurcation, to a centered stable symmetric periodic behavior.

Now, let us consider a useful representation in the study andcharacterization of these discrete rotating waves: the use of thecorresponding (discrete) Fourier spatial modes [OS89, HCP94, MPL+97].These modes are defined as follows,

Xk =1N

N∑j=1

xj exp[2πi(j − 1)k

N

], (3.2)

where N = 3, as already indicated, and i is the imaginary unit. Theevolution equations in terms of these modes are as follows,

X0 = σ(Y0 −X0)Y0 = R X0 − Y0 −X0 Z0 −X1 Z∗1 −X∗

1 Z1

Z0 = X0 Y0 − b Z0 + X1 Y ∗1 + X∗

1 Y1

X1 = σ(Y1 −X1)Y1 = R X1 − Y1 −X0 Z1 −X1 Z0 −X∗

1 Z∗1Z1 = X0 Y1 + X1 Y0 − b Z1 + X∗

1 Y ∗1

(3.3)

with R = R exp(2πi/3).

3.2.1 Lyapunov exponents and attractors

In previous studies [MGP+97], it was demonstrated that thecomputation of the transverse Lyapunov exponents to the synchronization


Lyap. spectr. Attractor(−,−,−,−, · · · ) Fixed Point(+, 0,−,−, · · · ) Synchronous Chaos(+, 0, 0,−, · · · ) Chaotic Rotating Wave I & III(+,+, 0,−, · · · ) Chaotic Rotating Wave II(0, 0, 0,−, · · · ) 3-Torus(0, 0,−,−, · · · ) 2-Torus(0,−,−,−, · · · ) Periodic Rotating Wave

Table 3.1: Correspondence between Lyapunov spectra and attractors. Fourth to ninthLyapunov exponents are negative in all cases.

manifold allowed to know (approximately) the region of the parameterspace where synchronized chaos was stable. Nonetheless, if one wishesto know more about the spatio-temporal structures emerging fromdesynchronization, it is worthwhile to compute the Lyapunov exponents(LEs) for the whole set of variables (nine in our case). Since Lyapunovexponents are related to the exponentially fast divergence, or convergenceof nearby orbits, they can be used to identify attractor types. With thisaim we computed the LEs .

The method used for the calculation has been the one developed byBenettin et al. [BGGS80] and Shimada and Nagashima [SN79], anddescribed in [WSSV85]. For the orthonormalization process we have useda modified Gram-Schmidt method. The integration of the system ofdifferential equations and the copies of the linear system have been done bymeans of an adaptive stepsize algorithm based in a fifth order Runge-Kuttamethod [PTVF92]. The results presented here in Fig. 3.1 were obtainedcomputing 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 R-σ (b fixed to b = 3) with different Lyapunov spectra, and consequentlywith different attractors can be observed (see Table 3.1). Thus, we finda region (denoted FP in Fig. 3.1) where all the LEs are negative, whichindicates that the three oscillators “collapse” to the same fixed pointC± = [±

√b(R− 1),±

√b(R− 1), R − 1]. Also, there exists a region

with synchronous chaos (SC) where the Lyapunov spectrum contains onepositive and one vanishing exponents.

For larger values of the parameter R, the synchronized state becomesunstable through a supercritical blowout bifurcation. This means that asR is increased, more and more unstable periodic orbits embedded into the


29 30 31 32 33 34 35 36 37 3818

18.5

19

19.5

20

20.5

21

21.5

22

R

σ

SC

PRW

FP

T2CRW

I II III

Figure 3.1: Regions of the (R, σ) plane where the marked states are achieved by asystem composed of three Lorenz oscillators. Notation: FP, fixed point; SC, synchronouschaos; 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-frequencyquasiperiodicity; PRW, periodic rotating wave. Symbols “x” and “+” indicate the lociof RBC and RH (see Eq. (1.16)), respectively.

synchronous attractor become transversely unstable, in such a way thatabove 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, sincethe instabilities of the UPOs occur through supercritical Hopf bifurcations.This oscillatory instability gives rise to the previously mentioned structurecalled Chaotic Rotating Wave (CRW). This state was first found for aring of Chua’s circuits [MPL+97, MGP+97, SMP00] and later for a ring ofLorenz oscillators [SM99]. Also, for coupled maps in a ring, an analogousbehavior was found in [YP99]. The CRW is characterized by a fastoscillation, associated with a 2π/N phase difference between neighboringoscillators (where N is the number of units in the ring), superimposedto the chaotic motion. We distinguish three regions within this statedepending on the LEs. In regions I and III, there exist one positive and


two vanishing Lyapunov exponents (since the degeneracy of this exponentis not theoretically proved, it is plausible that one exponent has very smallmagnitude and is numerically indistinguishable from zero). In region IIthere are two (clearly) positive and one vanishing LEs.

At larger values of R we observe a small region that exhibits two-frequency quasiperiodicity (T2). The Lyapunov spectrum does not containany positive exponent, hence chaos has disappeared; instead, there are twonull Lyapunov exponents which correspond to the two incommensuratefrequencies of the quasiperiodic regime. Finally, for large R we find periodicdynamics (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 isdue 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 thechaotic attractor. The fixed points become unstable through respectivesubcritical Hopf bifurcations at RH ; whereas, as explained by Yorke andYorke [YY79], the chaotic attractor is born (or dies) at a boundary crisis atsome 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 twoheteroclinic connections between the fixed point at the origin and bothunstable cycles surrounding C+ and C−.

The loci of RBC and RH on the (R, σ) plane are depicted in Fig. 3.1with “×” and “+” symbols, respectively. Between both lines the Lorenzoscillator is tri-stable. The basin of attraction of the (synchronized) chaosis 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 shouldlead 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 explainsthe existence of the mentioned wedge-like (see Fig. 3.1) region where thesystem 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 toCRW; instead, we are more interested on the transitions from PRW toCRW, i.e. we go ‘from order to chaos’. The reader should notice thatby CRW we refer to a high-dimensional chaotic attractor characterized


10−4

10−3

10−2

10−1

100

101

−20

−15

−10

−5

0

5

10

15

20

R−35.093

X0

Rpitch

Rh1

Rh2

Rbc

Figure 3.2: Bifurcation diagram. Points represent intersections with the Poinacaresection Im(X1) = 0, Im(X1) > 0. The logarithmic scale in the x-axis has been adoptedto better resolve dynamics existing in quite different interval ranges of R.

by an oscillation with a 2π/3 phase shift between neighboring oscillatorssuperimposed to an underlying chaotic behavior. Taking a Poincare surfaceof section (Im(X1) = 0, Im(X1) > 0), and plotting the coordinates X0

of the intersections we obtain Fig. 3.2. Going from right to left, we firstdistinguish a pitchfork bifurcation (R = Rpitch) that mediates the transitionfrom a centered PRW to a pair of symmetry related PRWs. At R = Rh1

the PRWs undergo a Hopf bifurcation giving rise to two symmetry relatedtwo-frequency quasiperiodic attractors (the flow fills densely the surface of atorus: T2). At a lower value of R = Rh2, two three-frequency quasiperiodicattractors (T3) are born at a secondary Hopf bifurcation. Finally, Rbc

marks the point where the CRW appears and the trajectory visits positiveas well as negatives values of X0. The time series for the different behaviorsstudied 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 thesize of the former T2 (the time scale has been broadened in Fig. 3.3(c) toappreciate this).


0 0.5 1 1.5 2

−15

−10

−5x 1, x

2, x3

Time (t.u.)

a)0 0.5 1 1.5 2

−15

−10

−5b)

x 1, x2, x

3

Time (t.u.)

0 100 200 300−20

−10

0 c)

x 1, x2, x

3

Time (t.u.)0 100 200 300 400 500

−20

0

20

x 1, x2, x

3

Time (t.u.)

d)

Figure 3.3: Time series of the system Eq. (3.1) with σ = 20 and b = 3: (a) R = 35.5Periodic Rotating Wave (b) R = 35.2 T2 (c) R = 35.095 T3 (d) R = 35.093 ChaoticRotating Wave. Note the different time scale for each panel.

More precisely, a view of the parameter region that will be consideredin this work can be found in Fig. 3.4, in which the Lyapunov spectrumcorresponding to the four largest Lyapunov exponents is presented. InFig. 3.4(b), seen from right to left, the two symmetry-related waves (limitcycle attractors) exhibit a (supercritical) Hopf bifurcation when R isdecreased, namely at R = Rh1 ≈ 35.26, yielding two symmetry-related two-frequency quasiperiodic attractors (two vanishing Lyapunov exponents).Diminishing R further, the system exhibits another (supercritical) Hopfbifurcation at R = Rh2 ≈ 35.0955, that yields two symmetry-related three-frequency quasiperiodic attractors (three vanishing LEs). Decreasing Rfurthermore the system exhibits a boundary crisis, at R = Rbc ≈ 35.09384,in which the chaotic attractor is born (or destroyed when seen from theopposite side). The chaotic attractor is characterized by two vanishing(at least, very approximately) Lyapunov exponents, and a single positiveLE 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 intoaccount that the K-Y conjecture cannot be stated oversimply; for example,when synchronous chaos exists, a small Lyapunov exponent may indicatea transverse contraction to the synchronization manifold, and as long asthis direction does not participate in the stretching-and-folding mechanism(associated to chaos) such a Lyapunov exponent should not be consideredwhen calculating the dimension.

The presence of 2-D and 3-D quasiperiodic attractors may lead to

3.3 The centered periodic rotating wave: analytical solution 61

30 32 34 36 38 40

−0.5

0

0.5

1

R

λi

(a)

R1

FP SC

CRW

PRW "Pitch."

λ1

λ2

λ3

λ4

33 34 35

−0.03

0

35 35.1 35.2 35.3 35.4−0.1

−0.05

0

0.05

0.1

0.15

λi

R

(b)

PRW

T2CRW

35.094 35.096 35.098

−3

0

x 10−4

T2T3

Figure 3.4: (a) The four largest Lyapunov exponents of a ring of three unidirectionallycoupled Lorenz systems. The inset shows from second to fourth LEs in the intervalwith CRW dynamics. It may be seen the the second LE becomes slightly positive whichindicates the existence of hyperchaos. (b) Detailed figure of the transition from PRW toCRW; the existence of three-frequency quasiperiodicity is confirmed in the inset wherethree vanishing LEs exist.

think that chaos appears through a quasiperiodicity transition to chaos (seeSection 3.4). However, it will be shown below that the chaotic attractorappears at a boundary crisis, and coexists with the two T3 attractors untilthe 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 creationof an infinite number of unstable 3-D tori. Thus, chaos is created througha global connection in which the reflection symmetry of the system seemsto play a fundamental role, analogously to what happens for the Lorenzsystem [Spa82]. Although symmetry plays an analogous role as in Lorenzmodel, the higher number of dimensions available in phase space implies amore complex route to (high-dimensional) chaos in this system.

A schematic diagram of the whole set of bifurcations linking the PRWand synchronous chaos is shown in Fig. 3.5. As mentioned above, in theinterval of R where the CRW is found, the shape of the attractor changes.Anyway, in this work we are not interested in the transitions betweendifferent types of CRWs.

3.3 The centered periodic rotating wave: analyt-ical solution

Inspection of the dynamics observed under simulation of theEqs. (3.1,3.3) suggests to try the following ansatz for the centered periodic


H.-D. CHAOTIC ATTRACTOR (CRW)

35.09367 35.09384 35.0955 35.11 35.26

CHAOTIC TRANSIENT

(2)T3

(2)T2

(2)T1

boundary crisis ‘heteroclinic explosion’

“saddle-node bif.” Hopf bif. Hopf bif. pitchfork bif.

T1

(PRW)

R =

CO

EXIS

TENC

E

39.2532.82

“Hopf blowout”

SYNCHRONOUS CHAOS

Rsc Rsn Rbc Rh2Rexpl Rh1

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 = AXeiωt (3.6)

Y1 = AY ei(ωt+φY ) (3.7)

Z1 = AZei(−2ωt+φZ) (3.8)

where time invariance lets us set φX = 0. We first examine the effectof X1, Y1, Z1 on the dynamics of the zero mode, to check whether it iscompatible with a steady state for the mode k = 0. We observe that theequations for the mode k = 0 in Eq. (3.3) are identical to the equations fora Lorenz oscillator with two forcing terms K and L,

K = X1Z∗1 + X∗

1Z1 (3.9)

L = X1Y∗1 + X∗

1Y1. (3.10)

Substituting our ansatz (Eqs. (3.6-3.8)) we obtain:

K = 2AXAZ cos(3ωt− φZ) (3.11)

L = 2AXAY cos(φY ). (3.12)

The term L is constant in time, whilst the contribution of K consists ina oscillation with frequency 3ω and zero mean value. The value of ω isexperimentally found to be appreciably larger than the natural frequencyof the Lorenz oscillator. Therefore, we take K = 0 when working withthe equations for the mode k = 0. In this way, to take X0 = Y0 = 0 andZ0 = const = L/b is a good approximation. Note also that the oscillating

3.3 The centered periodic rotating wave: analytical solution 63

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 becomputed taking the ansatz above as starting point. We substitute ouransatz in the equations (recall that X0 and Y0 vanish) obtaining:

iωAX = σ(AY eiφY −AX) (3.13)

iωAY eiφY = (R− Z0 −AZe−iφZ )AX −AY eiφY (3.14)

−2iωAZeiφZ = AXAY e−iφY − bAZeiφZ (3.15)

Z0 =L

b=

2AXAY 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 obtainthe following relations:

AY eiφY = (1 +iω

σ)AX (3.17)

AZeiφZ =1− iω

σ

b− 2iωA2

X , (3.18)

which allow to express Z0 (see Eq. (3.16)) as a function of AX :

Z0 =2A2

X

b. (3.19)

Finally, substituting Eqs. (3.17, 3.18, 3.19) into Eq. (3.14) we get, discardingthe trivial solution AX = 0, a complex equation with two unknowns (AX

and ω): (2b

+1 + iω/σ

b + 2iω

)A2

X = R−(

1 +iω

σ

)(1 + iω). (3.20)

We get one equation for the real part,(2b

+b + 2ω2/σ

b2 + 4ω2

)A2

X =ω2

σ− R

2− 1, (3.21)

and another for the imaginary part:

(b/σ − 2) ω

b2 + 4ω2A2

X =√

3R

2− ω

(1 +

1σ

). (3.22)

Assuming ω2 > σ2 b2 = 9, Eqs. (3.21,3.22) become highly simplified:


(2σ

+8b

)A2

X∼= 4

(ω2

σ− R

2− 1)

(3.23)

(b

σ− 2)

A2X

∼= 4ω

[√3R

2− ω

(1 +

1σ

)]. (3.24)

Casting the value of A2X as a function of ω2 thanks to Eq. (3.23) and

introducing the result in Eq. (3.24) we obtain a quadratic equation for ω:

αω2 + βω + γ = 0 → ω =−β +

√β2 − 4αγ

2α(3.25)

where the + sign is taken because we are assuming a high frequency (recallthat we are considering ω2 b2), and:

α = 1 +1σ−

1− b2σ

1 + 4σb

= 1.017 . . . (3.26)

β = −√

3R

2(3.27)

γ =(R + 2)(2− b/σ)

4 (1/σ + 4/b). (3.28)

If we consider that 4αγ/β2 ∝ 1/R 1, then, we may approximate ω fromEq. (3.25) by:

ω ≈ −β

(1α− γ

β2

). (3.29)

In Fig. 3.6, we compare the numerically observed frequencies with theones obtained from Eq. (3.25) as a function of R. The theoretical values ofω provide, using Eq. (3.24), the values of AX , so we show in Fig. 3.6 themaximum and minimum values of AX as a function of R. Also it must beremarked 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 numericalresults agree with the analytical ones for R > Rpitch ≈ 39.25. Belowthis point the centered solution renders unstable and two symmetry-relatedcyclic solutions appear. For these solutions X0, Y0 are no longer close tozero, and the analysis of the new solutions becomes much more convolutedbecause cross-terms arise, yielding other components of the Fourier seriesof the mode k = 1.

3.4 Transition to quasiperiodic behavior 65

40 45 50 55 6030

35

40

45

50

55

ω

R

39 39.5 40 40.532.5

33

33.5

34

34.5

R

ω

40 45 50 55 606

7

8

9

10

11

12

R

AX

39 39.5 40 40.56.5

7

7.5

R

AX A

Xmax

AXmin

Figure 3.6: Numerical results for the frequency ω (left panels) and the max. and themin. values of the amplitude of the coordinate X1 (right panels) as function of R. Thelower panels show zooms at the pitchfork bifurcation Rpitch. Solid lines correspond totheoretical predictions (see text).

3.4 Transition to quasiperiodic behavior

As described above, the (symmetric) periodic rotating wave, that is thestarting point of our analysis, exhibits a transition to quasiperiodic behaviorT3, mediated by a pitchfork bifurcation, in which two (asymmetric) periodicrotating waves are born; the latter, in turn, exhibit two consecutive Hopfbifurcations. A small note is in place about the stability of this type ofunusual attractors, as in many places in the literature it is stated thatthese attractors are intrinsically unstable.

The question is: which is the behavior that one expects to find typicallyin a system where n frequencies have been generated (n ≥ 2): chaoticor (quasi)-periodic? According to the Newhouse-Ruelle-Takens (NRT)Theorem [NRT78], if one has a flow on a torus Tn = Rn/Zn (n ≥ 3) everyneighborhood 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


case, not covered by the NRT Theorem, a torus can be destroyed followingone of the routes proposed by Afraimovich and Shil’nikov [AAIS89]. Thus,the destruction of the torus due to the interaction of resonances can lead tochaotic behavior, as in the route first studied by Curry and Yorke [CY78].

Quasiperiodic motion, living on a torus, is quite fragile, becauseresonances constitute a fat fractal that densely fills parameter space(although their joint Lebesgue measure may be small) [Ott93]. In thesimplest setup, corresponding to the analysis of the circle map [Arn65,Sch88, Ott93] and valid for the analysis of nonautonomous systems, rationalvalues for the winding number, that have zero measure when the nonlinearparameter is zero, are the seeds for finite measure regions (so-called Arnoldtongues) 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 bornin a Neimark-Sacker bifurcation [Kuz95], is a bit more involved. First ofall 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 numberchanges along this line as a function of the system parameters. Again,points in this line for which the winding number is rational, p/q, exhibita resonance, and as one goes into the T2 region one gets a resonance hornemanating from each rational; in general this is a multidimensional objectin parameter space. For q ≤ 4, the so called ‘hard resonances’, the structureis 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 arecharacterized by the creation of a pair of orbits (one stable and the otherunstable) in a saddle-node bifurcation on the surface of the torus as oneenters the resonance horn. In our case we have not detected resonances inthe T2 regime. This may occur due to the fact that along R, the windingnumber does not cross hard resonances, the largest denominator is q = 11(corresponding to the winding number 4/11), so the resonance horns maybe quite small. Perhaps the different nature of both frequencies (one isspatial with the phase shifts along the ring whereas the other is related toa uniform oscillation) is responsible of the weak interaction between them.

Concerning the existence of a 3D-torus –which in principle wouldbe forbidden by the NRT theorem– it must be stressed that numericalstudies [GOY83a, Bat88], and also a number of experiments [GB80, LC89,AR00], give support to the existence of three-frequency (and even four-frequency, and, perhaps, higher-frequency) quasiperiodic attractors. Thekey in understanding the prevalence of these high-frequency quasiperiodic

3.4 Transition to quasiperiodic behavior 67

Figure 3.7: Plot of a Poincare 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 Poincare section); (right) R = 35.095, for which the systemexhibits three–frequency quasiperiodic behavior (two–frequency quasiperiodic in thePoincare section). The representation has been carried out by using the (complex) moderepresentation of the system (3.3), and the three axes correspond to the x coordinate andare the uniform (k = 0) mode, and the real and imaginary parts of the k = 1 mode. ThePoincare cross section has been defined by the condition: Z0 = 34 ≈ R− 1 with Z0 > 0.

attractors is to understand that the existence of robust strange attractorsin its neighborhood does not mean that the measure of quasiperiodicity ina 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 regardingthe robustness of 3D-tori in systems with conserved quantities. In theirstudy of a model of a solar dynamo [FJK93] these authors showed that ina range of parameters one of the variables in their analysis was cyclic andthus, could be decoupled from the equations. Thus, such a system could beconsidered as a three-frequency torus consisting of a generic two-frequencytorus plus an extra frequency arising from this conservation law. It is inthis sense that the conclusions of the NRT Theorem [NRT78] should notapply to this case. In our case it can be numerically checked that thesystem exhibits three-frequency quasiperiodic behavior by looking at thePoincare cross section (see Fig. 3.7) of the torus in phase space (and also


by a careful computation of the Lyapunov spectrum). Also, Ref. [Yan00]reports the existence of three-frequency quasiperiodicity in a ring of coupledLorenz oscillators. We believe that the key point is the very small frequencyassociated to the secondary Hopf bifurcation, that is about two orders ofmagnitude smaller than the other two. It appears as a very slow modulationof the 2D-torus (see Fig. 3.3(c)) so the relative winding numbers have largedenominators. A more physical point of view is the possibility of applyingsome kind of adiabatic principle to understand the very weak interaction ofthis very low frequency with the other two. In fact, a case of a thirdvery low frequency associated to a 3D-torus has also been observed inRefs. [WN87, LM00].

Let us finish this section by pointing out that when quasiperiodic motionis present in a dissipative dynamical system one is, at first sight, inclinedto think (with good sense) that the system will exhibit chaos due to globalconnections originating in the Arnold tongues (because of to its abovementioned fragility) that typically lead to wrinkling and corrugation ofthe torus. Instead, as it will be shown in subsequent sections, this systemis an example in which chaos does not arise in this way, but rather in aglobal connection that is quite linked to the reflection symmetry exhibitedby the system. See Ref. [Moo97] for a further example of a route to chaosin a system exhibiting quasiperiodic behavior through a global connection,and not due to resonant interactions of the torus (that imply themselvesglobal connections, but of a different kind).

3.5 Numerical evidences of the route to chaosexhibited by the system

In this section we are going to analyze the route through which a chaoticattractor is born in this system. As this has some similarities to the classicalroute to chaos for an isolated Lorenz system, we are going to draw someuseful analogies with this system, although it is warned that the analogyis not complete (otherwise, we would not have a new route to chaos). Asit can be found in textbooks [Ott93], the Lorenz system exhibits a routeto chaos through a double homoclinic connection of the fixed point at theorigin, 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 ofthe butterfly (taking into account how the unstable directions reenter the

4For 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 (ahomoclinic explosion [YY79, Spa82]). The closure of this set is formedby the infinite number of unstable periodic orbits that can be classifiedaccording to their symbolic sequences of turns around the right (R) andthe left (L) fixed points (C+ and C−) [Spa82]. The existence of these UPOsreflects the dramatic change undergone by the stable manifold of the fixedpoint that allows initial conditions at one side of the phase space jump tothe other side (before falling to one of the two symmetry related stable fixedpoints C+ and C−), being these jumps impossible for R < RHOM . For alarger value of R, R = RBC , the chaotic set becomes stable in a boundarycrisis, that occurs precisely when the two shortest symmetry related length-1 unstable periodic orbits (R and L) located at each lobe, and that atR = RHOM coincide with the two homoclinic orbits, shrink, such that thechaotic set has a tangency with these two orbits [Spa82]. At this value ofR there exists a double heteroclinic connection between the equilibrium atthe origin and the mentioned length-1 UPOs. For RBC < R < RH thesetwo orbits (and their respective tubular stable manifolds) form the basinboundaries between the chaotic attractor and the two stable asymmetricfixed points (C±), and, consequently, the system exhibits multistability.These two fixed points loose their stability in a collision (a subcritical Hopfbifurcation) with the two mentioned length-1 orbits, that occurs when theseshrink to a point coinciding with (C+, C−), that become unstable from thispoint.

First of all, and in analogy to our analysis in the Section devoted to theanalysis of the quasiperiodic motions, we shall reduce the dimensionality ofthe problem by eliminating the fast frequency, that involves a phase shift by2π/3, as it leads to a conserved quantity (this time lag) and consequentlya null Lyapunov exponent. In the mode representation of Eq. (3.3) thisamounts to perform a cut through the Poincare section Im(X1) = 0. So, invisualizing objects cycles will become fixed points, T2-tori cycles and T3-toriwill become T2-tori, although we shall refer to these objects referring to thecomplete phase space, and not to this cut. Now, we shall shed light on thenature of the transition by performing a number of numerical experimentsand 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 Lorenzoscillators is that the two T3 attractors are not directly involved in the birth


67 70 73 76 79 82−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

r

R = 35.093 + r × 10−5 λ

i (×1

0−4 )

slave locking

λ1

λ2

λ3

λ4

Figure 3.8: Blowout of the four largest Lyapunov exponents for the T3 attractors in theregion in which they coexist with the chaotic attractor. The fourth Lyapunov exponentapproaches zero in a square-root fashion.

of the high-dimensional chaotic attractor. Indeed, one can easily performthe following numerical experiment for values of R close to the critical valueat which the (high-dimensional) chaotic attractor is born (more preciselyfor Rsn < R < Rbc).

If one chooses an initial condition that corresponds to a T3 attractorobtained in the nearby range Rbc < R < Rh2, a stable T3 is obtained,although not so large perturbations in this initial condition yield chaoticbehavior. This evidence implies that the system exhibits multistabilitybetween the high-dimensional chaotic and the T3 attractors in the rangeRsn < R < Rbc, with the latter attractor having a relatively small basinof attraction. This remark is important, because it implies that the high-dimensional chaotic attractor is not created through some route involvingthe 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


Figure 3.9: Schematic of a ‘slavelocking’. A portion of a section ofthe tori (stable and unstable) areshown with bold lines. The tori arenot differentiable at the foci.

is represented in Fig. 3.8. It can be seen that close to R = Rsn = 35.09367the fourth Lyapunov exponent exhibits a square-root profile as expected fora saddle-node bifurcation. This suggests that the stable T3 is approachingan unstable T3. Also one can appreciate quite clear lockings5 where thethird and fourth Lyapunov exponents become equal.

As R approaches Rsn the existence of a smooth invariant 3D-toruscannot be guaranteed, because if one locking occurs the strength of thenormal contraction can be of the same order than the rate of attractionof 2D-tori on the 3D-torus. Thus, in Fig. 3.9 a possible scenario for theinteraction of the tori, that would explain the lockings observed in Fig. 3.8is 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, weobserve two identical non-vanishing exponents that indicate the existenceof a stable focus-type T2 on the surface of the T3. The T3 continues to existbut is non-differentiable at the stable T2 located on its (hyper)surface.

The participation of the unstable 3D-torus in the final annihilation ofthe stable one is not trivial (the existence of the unstable T3 is corroboratedby Sec. 3.5.4). As we said before, the fast spatial frequency can beconsidered as a kind of forcing and interacts very weakly with the othertwo. Then we can argue that we may take use of the routes leading tothe 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 boththat leads to a saddle-node bifurcation between twin cycles on both tori.

5Recall that the classical theory for two-frequency quasiperiodicity tell us that when atwo-torus locks a pair of stable-unstable orbits are born on its surface through a saddle-node bifurcation. According to this, one of the Lyapunov exponents becomes slightlynegative, 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 visitssome (formally infinity) Arnold tongues where its rotation number is rational, as expectedfrom the existence of a stable periodic orbit on its surface. Analogous resonances appearfor 3D-tori.


If the first saddle-node bifurcation involves the stable cycle no attractorremains 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 (whichprobably undergoes the saddle-node bifurcation a bit later). We speculatethat one of these routes, with the proper substitution of cycles and two-toriby two- and three-tori respectively, occurs in our system. Unfortunately,because of the precision of our computations we are not able to resolve thisfine 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 transientsapproach 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 theneighborhood of the other torus.

Following the analogy with the Lorenz system, we conjecture thatprecisely at R = Rexpl a global bifurcation occurs, and past this value(R < Rexpl) an infinite number of unstable objects are created. To checkthis we stabilized the symmetric PRW (an unstable fixed point in thePoincare map) by a Newton-Raphson method and observed the fate ofthe trajectories starting from (approximately) that solution. In Fig. 3.10the evolution of the trajectories for two values of R above and below Rexpl

is shown. After approaching one the asymmetric PRWs (its location ismarked 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 oneof 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 areidentical). As one can see, fifth to ninth LEs are quite similar, indicatingthat 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 isabout four, the dynamics can be simplified to the study of a four-dimensional branched manifold. A theoretical justification of this statement


0 50 100 150 200 250 300 350 4000

5

10

15

20

X0

0 50 100 150 200 250 300 350 400−20

−10

0

10

20

Time(t.u.)

X0

R=35.12

R=35.10

Figure 3.10: Numerical experiment showing trajectories starting in an initial conditionin 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-dimensionalgeneralization of) the Birman-Williams Theorem [Gil98]. The theoremstates that –for a hyperbolic low-dimensional chaotic attractors– underidentification of points in phase space with the same future, the strangeattractor projects down to a two-dimensional branched manifold. Undersuch projection, no orbit cross through each other and their topologicalorganization is invariant. The B-W theorem requires the strange attractorto be hyperbolic, and this is clearly not fulfilled even for the Lorenz systemalone (mostly due to the fact that the fixed point at the origin is part ofthe chaotic set), nonetheless this theorem has demonstrated to be a usefultool to manage usual (i.e. non-hyperbolic) chaotic attractors.

In our case we can argue that our high-dimensional chaotic attractorhas, according to Kaplan-Yorke conjecture [KY79a], a value of theinformation dimension D1 that is slightly larger than 4. And the mentionedprojection (or “deflation” of the attractor) occurs when the dimensionapproaches 4, considering that the negative Lyapunov exponent that does


λi CA T3

λ5 −5.2547 −5.2027λ6 −5.2548 −5.2027λ7 −18.6124 −18.6519λ8 −18.6124 −18.6519λ9 −24.2734 −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, seeTable 3.2, λ5 → −∞ (see [Gil98] for a presentation of this argument),

D1 = 4 +λ1 + λ4

|λ5|→ 4 (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 thealgorithm of Grassberger and Procaccia (see Fig. 3.11), obtaining a valueclose to 4 (recall that D1 ≥ D2).

A more correct picture of the attractor amounts to consider that this4-dimensional manifold is a leave, and that many of these thin leaves makeup the attractor (4+). Of course that this reduced 4-dimensional picturecan be only considered to be a more or less faithful representation of thesystem when it is orbiting in one of the subspaces created by the symmetryhyperplane. In the epochs in which a trajectory jumps to the other side(subspace), which means reinjections through an extra dimension, it iswhen the existence of branching is needed in order to the trajectory not tointersect itself. In our case the ‘tear point’ is the symmetric PRW. This isin complete analogy with what happens with the Lorenz system, where theattractor can be understood as a template composed of a branched two-dimensional manifold with a tear point at the origin. Rotations around oneof 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 sameas a cycle divides R2). Then in the regime with coexistence, between chaosand three-frequency quasiperiodicity, it is the pair of unstable 3D-tori whatdefines the basin boundary of the chaotic attractor. In the Lorenz system


−2 −1 0 1

−4

−2

0

2

4

6

log10

(ε)

log 10

(C(ε

))

Figure 3.11: Determination of the correlation dimension for R = 35.05. We used thealgorithm of Grassberger and Procaccia [GP83] with the attractor sampled with 4× 106

points. We measured the average of points C(ε) inside of a ball of radius ε centered atone point of the attractor. The average was made over 104 points, except for ε < 10−1

where 105 points were used. The fitting slope is D2 = 3.96 ± 0.05. To be comparedwith the information dimension obtained from the Lyapunov exponents according to theKaplan-Yorke conjecture: D1 ≈ 4.005.

the length-1 unstable periodic orbits divide the approximately 2D-manifoldin two regions, where the chaotic and the fixed point attractors live; and alsoacts as the basin boundary. In the multistability region, trajectories spiralin one lobe away from the fixed point (say C+) because of the repulsive effectof the UPO surrounding that fixed point. After some turns the trajectoryjumps to the other lobe by using the third dimension (i.e. thanks to thebranching). If the trajectory approaches close enough to C− surpassing the‘barrier’ constituted by the length-1 UPO, it is captured by this fixed pointand no more jumps occur (i.e. the chaotic transient finishes). Analogously,the unstable T3-torus acts as a dividing hypersurface for trajectories in the4-dimensional branched manifold.


−5 −4.5 −4 −3.5 −3 −2.52

3

4

5

6

7

log10

(R−Rbc

)

log 10

(<τ>

)

Figure 3.12: Log-log representation of the average chaotic transient as a function of thedistance to the critical point Rbc. Each point is an average over 100 realizations. Datafit 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 havemeasured the average time of the chaotic transients (〈τ〉) when approachingR = Rbc. We observe that these transients diverge satisfying a power law(〈τ〉 ∼ (R − Rbc)γ) for an asymptotic value R = Rbc = 35.093838, asexpected 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. WhenR 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 phasespace and no chaotic transient is observed. However, for R just aboveRbc the result of using a point in this orbit as initial condition may beseen in Figure 3.13 (the PRW is a fixed point in the figure, as it has been

3.6 Description in terms of a return map 77

0 500 1000 1500−20

−10

0

10

20

X0

Time (t.u.)

(a)

1.4 1.45

x 104

−20

−10

0

10

20

(b)

Figure 3.13: Numerical experiment showing a trajectory (only points at the Poincaresection are shown) starting in an initial condition at one of the asymmetric unstablePRWs 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 minimumwidth along time is larger for the stable 3D-torus as expected for a torus located insideanother (see the discussion about the branched manifold).

stroboscopically cut through the Poincare hyperplane Im(X1) = 0). Thetrajectory seems to approach a T3 before ‘falling’ to the T3 attractor. Thus,we state that is the unstable T3 which constitutes the basin boundary ofthe chaotic attractor and the object involved in a global connection thatmarks the birth (or the death, depending on the viewpoint) of the chaoticattractor. Notice also, that the stable and the unstable T3-tori have quitesimilar sizes. Ultimately, at Rsn, the multistability region has an endwhen the two T3-tori disappear in the saddle-node bifurcation (previouslydiscussed in Sec. 3.5.1).

3.6 Description in terms of a return map

Lorenz described a nice technique for reducing the complexity of thesolutions of the Lorenz equations. By recording the successive peaks ofthe variable z(t), he reduced the dynamics of the Lorenz systems to aone dimensional map. Denoting the nth maximum of z(t) by Mn, heplotted successive pairs (Mn,Mn+1) of maxima, finding that points lay(very approximately) along a Λ-shaped curve. In this way, the dynamics isreduced to the “Lorenz map”: Mn+1 = Λ(Mn).


43 44 45 46 47 48

43

44

45

46

47

48

Nj

Nj+

1

R = 35.094, TRANSIENT CHAOS

← T3

unstable T3 →

chaotic transient →

43 44 45 46 47 48

43

44

45

46

47

48 R = 35.0937, COEXISTENCE

Nj

Nj+

1

← T3

chaotic attractor →

Figure 3.14: Return map of the maxima of the variable Z0 satisfying to be larger thantheir 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). AsR 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 becomean attractor. The lower plot shows the attractors occurring for R into the coexistenceinterval.

In our case we may try to reduce the dynamics by means of a returnmap. The fast dynamics concerning the k = 1 mode is approximatelyfiltered when considering the k = 0 mode. Then, the T3 attractor is seenin the k = 0 framework as a T2 plus some small oscillating component. Areturn map of the variable Z0 would reduce the dimension of the attractorby 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 ofthe chaotic attractor to a one dimensional curve (notice that the chaoticattractor has got a dimension larger in approximately one unit to the three-torus). 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 tookthe subset of maxima whose neighboring maxima were smaller: Nj =Mn,Mn > Mn±1. A plot of successive pairs (Nj , Nj+1) gives a fixed pointfor the T3, approximately because of the residual fast dynamics of the k = 1mode. 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 withthe Lorenz map. Probably, the existence of the residual fast componentmakes the attractor to deviate significantly from one-dimensionality. In thelight of the paper by Yorke and Yorke [YY79] who studied the transition tosustained chaotic behavior in the Lorenz model with the Lorenz map, it isfound that our results are consistent with a chaotic attractor of dimensionaround four, and a boundary crisis mediated by an unstable three-torus.

3.7 Route to chaos: theoretical analysis 79

a) b) c) d)

e)e) f) g) h)

Figure 3.15: Three-dimensional representation of the proposed heteroclinic route tocreate the high-dimensional chaotic attractor. Black and gray points correspond to stableand unstable fixed points (cycles in the global phase space), respectively.

3.7 Route to chaos: theoretical analysis

Condensing all the information obtained from numerical experimentsin the previous sections, we suggest the route to high-dimensional chaosrepresented in Fig. 3.15 (where the usual cross section through the fastrotating wave is understood). The high dimension of our attractor makessomewhat convoluted a geometric visualization. As we explained above, wepostulate a chaotic attractor whose structure may be simplified in termsof a 4-dimensional branched manifold, and therefore the Poincare sectionreduces the attractor to a 3-dimensional branched manifold. Figure 3.15represents a projection onto R3, hence some (apparently) forbiddenintersections between trajectories appear because of the branching. Asoccurs with the Lorenz attractor when it is projected on R2 (say x − z),the intersections between trajectories coming from different lobes, and alsoof 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 extradimension is needed, and this is provided by the definition of a branchedmanifold.

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 symmetryrelated PRWs appear (b). At a supercritical Hopf bifurcation (b→c) the


T2 appears. When R is slightly decreased the 2D-torus becomes focus-type (the leading Lyapunov exponent becomes degenerate as may be seenin Fig. 3.4). Hence the unstable manifold of the asymmetric PRW formsa whirlpool when approaching the T2 (d). At Rhet a double heteroclinicconnection between the asymmetric PRWs and the symmetric one occurs(e). At this point the chaotic set is created, that includes a dense setof unstable 3D-tori. In (f) the two simplest 3D-tori are represented withdotted lines, because of the heteroclinic birth one of the frequencies ofthese tori is very small. Note that the plot shows that the unstablemanifold of one of the asymmetric PRWs intersects the stable manifoldof the symmetric PRW which in principle violates the theorem of existenceand uniqueness. This occurs, as we said above, because we are projectingthe Poincare section onto R3 (our particular ‘flatland’). Twin secondaryHopf bifurcations (f→g) render unstable the 2D-tori and give rise to twostable T3 (g). When R is further decreased the unstable manifold of theasymmetric PRWs do not asymptotically tend to the stable T3 (h), and thechaotic set becomes attracting.

A two-dimensional cut of the schematic shown in Fig. 3.15 is depicted inFig. 3.16. A one-to-one relation between subplots of both figures does notexist. Thus, Fig. 3.16(h) corresponds to R = Rbc, whereas (i) correspondsto R = Rsn.

It is to be stressed that the existence of a stable T3 is not a fundamentalpart of the transition to high-dimensional chaos. Just a focus-type T2 isneeded such that the unstable manifold of the asymmetric PRWs forms awhirlpool. In fact, Ref. [SNN95] shows that such a whirlpool structure isrelated to the existence of a torus that becomes a heteroclinic connectionbetween saddle-foci, in the vicinity of a codimension-two point. In thisway, regarding the chaotic attractor, no fundamental change occurredif the unstable T3 shrank to collide with the stable T2 in a subcriticalHopf bifurcation. This picture would be more similar to the transition inthe Lorenz system where C± become unstable through a subcritical Hopfbifurcation.

3.8 Further remarks

In this section we want to address some subtle aspects concerning thetransition to chaos shown in this chapter.

Analyzing bifurcations occurring in high-dimensional systems maybecome a quite convoluted task. Thus, when describing the disappearance

3.8 Further remarks 81

a) b) c) d) e)

f) g) h) i)

Figure 3.16: Two-dimensional representation of the proposed heteroclinic route tocreate 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 bydots surrounded by small circumferences.

of the T3 attractors in Sec. 3.4 it was noted that the existence of anattractor-saddle collision between 3D-tori is expected to be nongeneric.Therefore, making an intense zoom at R ∼ Rsn would reveal the exactmechanism leading to the disappearance of the 3D-tori. A possible routewas pointed out there. Nonetheless, from a practical point of view, it canbe assumed that the T3 attractor disappears in a saddle-node bifurcation aslong as the final step occurs in a extremely small interval of the parameterR.

A similar situation occurs when studying codimension-m (m > 1)bifurcations6. When increasing the highest order n, of the termsconstituting the normal form (or n-jet) or adding a non symmetric term,substructures in the bifurcation set emanating from the codimension-mpoint can be resolved. For example, the effect of adding non-axisymmetricterms 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 representinga map (instead of a continuous dynamical system) because a Poincaresection of the fast dynamics involving the spatial mode is assumed. Globalconnections between hyperbolic cycles have been studied mostly in amathematical framework (excluding period-doubling and quasiperiodicity

6The codimension is the number of constraints imposed on the control parameters toreach the critical point.


routes to chaos that have been also studied intensively from experiments) bymeans of diffeomorphims in the plane (see [GH83] and references therein).But an important aspect of the route shown here is that the fast spatialrotating wave is conserved along the transition. The weak interaction of thisfrequency is manifest by the absence of visible lockings of the T2 attractors(R ∈ [Rh2, Rh1]) and for the presence of an additional vanishing Lyapunovexponent in the high-dimensional chaotic region. Thus, the fast spatialfrequency is somehow ‘orthogonal’ along the transition and therefore phasespace may be understood as a direct product of this frequency with thetransition shown in Fig. 3.15 that can be assumed not to differ too muchfrom a continuous system.

Let us consider further which is the meaning of the Lyapunov spectrumof the chaotic region. According to the route described by Figs. 3.15, 3.16an infinity of unstable T3 is created at the heteroclinic explosion (recallthat only the two simplest 3D-tori are shown). Hence, we could expect tohave a chaotic attractor with three vanishing Lyapunov exponents. Whyonly two are observed?

One could be tempted to think that due to the non-hyperbolicity ofquasiperiodicity it happens that a finite ratio of the 3D-tori set correspondsto locked tori (probably with large denominators), contributing to shift thevanishing Lyapunov exponents from zero. Nonetheless, it may be seen inthe paper by Rosenblum et al. [RPK96] that two coupled Rossler oscillatorsexhibit (very approximately) a pair of null exponents in a appreciable rangeof the coupling strength (below phase synchronization). This occurs inspite of the finite ratio of tori that (according to theoretical arguments) arelocked. Of course, the large denominators of the lockings cannot induce asignificant shift on the exponents, but it is important to regard this, mainlyfrom a fundamental perspective.

However we believe that the absence of a third vanishing Lyapunovexponent is due to the following. It must be noted that theheteroclinic connection between the symmetric and the asymmetricPRWs is structurally unstable, so for R < Rh1 it should disappear.Nonetheless, we have numerically observed (see Fig. 3.10) that theconnection (approximately) persists. Also, the axial symmetry of Fig. 3.15has not a theoretical justification.

Hence we expect that, a perturbation on the mechanism shown inFig. 3.15 will destroy its nice simplicity. Inspired on previous works [Gas93,Kir91] dealing with the effect of non-symmetric terms on codimension-two bifurcations, we believe that homoclinic connections will replace

3.9 Discussion 83

heteroclinic connections. A double (‘figure-eight’) homoclinic of thesymmetric PRW as well as homoclinic connections of the asymmetricPRWs will occur. Symmetric and asymmetric PRWs are both saddle-foci so it is important to know their saddle indexes7 in order to find outwhether (Shil’nikov) chaos will arise in a neighborhood of homoclinicity.Thus, computing the eigenvalues from the (8-dimensional) return map (thesame mapping that allowed us to stabilize these orbits), and finding thelogarithms of their modules, we get the saddle indexes for the symmetricand the asymmetric PRWs. Thus, for the asymmetric PRW we get undertime-reversal a saddle-index larger than one, which means that when thereal time is recovered there exist a single orbit of repeller type. For thesymmetric PRW the saddle index is smaller than 1/2, which means thathomoclinic chaos will occur.

The existence of homoclinic chaos seems to be nothing new, becausethe creation of infinite 3D-tori is substituted by an infinity of 2D-tori,which will produce homoclinic chaos of the type first reported years agoin Ref. [ACT81] plus an extra superimposed fast spatial wave. But it isimportant to emphasize that as long as the exact mechanism is very relatedto that shown in Fig. 3.15 the first negative Lyapunov exponent is very closeto zero which makes the information dimension to be larger than four (orlarger than three if the spatial oscillation is considered extra and/or trivial)

The next question is how the two simplest unstable 3D-tori appear ifthe route is not exactly as is shown in Fig. 3.15. A possibility is that theyappear though a saddle-node bifurcations between two unstable 2-tori, butwe have no way to find out this.

The reader may become disappointed with the observations of thissection, because the exact route of the system is not clarified. However, thisis typical of the kind of attractors that are called quasi-chaotic attractorsdue to the presence of structurally unstable homoclinic orbits. As observedin [SNN95] a complete description of a quasi-attractor in unattainable dueto infinity many non-controlled bifurcations of various types.

3.9 Discussion

In this chapter we have studied by numerical and theoretical argumentsthe transition to high-dimensional chaos, in a model of three coupled

7For a saddle-focus with three eigenvalues λs = ρ ± iω (ρ < 0), λu > 0 (as thesymmetric PRW), the saddle index is defined to be δ = −ρ/λu, and chaos will occur forδ < 1. Time reversal must be applied to study the asymmetric PRW.


Lorenz oscillators. The transition from a periodic rotating wave to achaotic rotating wave has been investigated. The structure of the globalbifurcations between cycles, underlying the creation of the chaotic set, issuch that a chaotic attractor with dimension d ≈ 4 emerges. The transitionis not mediated by low-dimensional chaos. Also it must be noted that evenif 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. Asoccurs with the Lorenz system, the existence of a reflection (Z2) symmetryseems to play a fundamental role.

The high-dimensionality of the chaotic attractor is not reflectinghyperchaos. Far from it, there is only one positive Lyapunov exponentsbut high-dimensionality is possible due to the existence of two vanishingand one slightly negative LEs. Hence, according to the Kaplan-Yorkeconjecture the information dimension is above four. We have also measured,with the algorithm of Grassberger and Procaccia, the correlation dimensionobtaining a value very close to four. The degeneracy of the null LE makeus think that a set of unstable tori is embedded in the attractor.

We have focused on giving a geometric view of the bifurcations occurringin the 9-dimensional phase space of the system. Although the precisesequence of bifurcations is probably resistant to analysis, we have been ableto give a geometric view of the transitions that explains the emergence ofthe chaotic set, through a ‘heteroclinic explosion’, and its conversion intoan attractor. This step occurs through a boundary crisis when the chaoticattractor collides with its basin boundary formed by two unstable 3D-tori.In consequence a power law for the mean length of the chaotic transientsis observed.

Chapter 4

Onset of Traveling Fronts inan Array of CoupledSymmetric Bistable Units

Abstract. A symmetry breaking mechanism is shown to occur in an arraycomposed of symmetric bistable Lorenz units coupled through a nearest neighborscheme. When the coupling is increased, we observe the route: standing →oscillating → traveling front. In some circumstances, this route can be describedin terms of a gluing of two cycles on the plane. In this case, the asymptoticbehavior of the velocity of the front is found straightforward. However, itmay also happen that the gluing bifurcation involves a saddle-focus fixed point.If so, front dynamics may become quite complex displaying several oscillatingand propagating regimes, including chaotic (Shil’nikov-type) front propagation.These phenomenology occurs for different couplings as well as for other discretebistable media. The case of the discrete FitzHugh-Nagumo model shows somepeculiarities that are analyzed. Thus, we suggest that there exist two universalforms of symmetry breaking leading to traveling fronts. These two routes wouldcorrespond to the two cases of a Takens-Bogdanov codimension-two bifurcationoccurring at the continuum limit.

4.1 Introduction

Several areas of science use model equations of the reaction-diffusion type. Moreover, besides its special interest in some fields aschemistry [Tur52, Fif79] and biology [Mur89], reaction-diffusion equationsare considered as abstract models for pattern formation [CH93].

Real systems are frequently composed of discrete elements, wherematerial models and biological cells are just two examples. Therefore, it isnatural to deal with the discrete space version of the reaction-diffusionequation. 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 simpleanalogies of the continuous ones.

Here, we focus on a discrete reaction-diffusion model such thatits local dynamics (reaction term) is bistable, with two stable fixedpoints. Particularly important examples of bistable dynamics are foundin optics [BCR81, Fir88], chemical systems [LE92], and biology [Mur89,McK70, Fit61]. The main interest of these systems lies in the behaviorof the fronts that constitute the boundaries of the domains of both(stable) states. Intuitively, the front moves from the most to the lessstable state, enlarging a domain and shrinking the other. In fact, greatattention has been devoted to the phenomenon of ‘propagation failure’ or‘pinning’ [Kee87, EN93, PPC92, MPG+95, MS95, KMB00, CB01], becauseit is usual that in discrete systems some threshold must be surpassed toachieve propagation.

In principle, one could expect that propagation does not succeed insystems with symmetric bistability. Nonetheless, it was demonstrated timeago [IMN89] that such possibility exists. Hagberg and Meron [HM94]studied a continuous bistable reaction-diffusion system, the FitzHugh-Nagumo model, showing that, in the symmetric case, propagation occurredafter a symmetry breaking front bifurcation. In this scenario, the frontbreaks its symmetry through a pitchfork bifurcation, at the same timethat propagation is initiated. Also, an analogous bifurcation was foundfor the complex Ginzburg-Landau equation [CLHL90] with the name ofnonequilibrium Ising-Bloch bifurcation.

In this chapter, we report the existence of front propagation in discretesymmetric bistable media. The transition is exclusive for discrete systemsand 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 thefunction that describes the local dynamics is not continuous [Fat98]. Itis shown that the mechanism leading to propagation is not a pitchforkbifurcation. Contrastingly, it consists in a Hopf bifurcation followed by aglobal bifurcation, that is equivalent to a gluing bifurcation of cycles. Incorrespondence, the velocity of the front shows a logarithmic dependencewith the coupling strength close to the onset. We also explain underwhich circumstances chaotic motion of the front is to be expected, whichmakes 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.

4.2 The system 87

4.2 The system

We consider a discrete reaction-diffusion equation in one dimension:

rj = f(rj) +D

2Γ(rj+1 + rj−1 − 2rj), (4.1)

where D accounts for the coupling strength between neighbors, andthe coupling matrix Γ says which variables get coupled. In whatconcerns the local bistable dynamics, we deal with the well-known Lorenzoscillator [Lor63] as has been made in previous works [JW00, BPP00,Par01]. It exhibits several characteristic behaviors depending on its internalparameters [Spa82]: monostability, bistability, limit cycle, ‘butterfly chaos’,‘noisy periodicity’, etc. We consider the range where the system containstwo stable symmetry related fixed points (bistability). In what concernsthe 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 (thatwe take equal to one). The case Γ = I has been studied very recently[BPP00, Par01]. Three situations are of special interest as long as theypresent front propagation Γ = γkl = δk2δl2, γkl = δk2δl1 and γkl = δk1δl2.The last possibility is studied in the next chapter, and we focus here in thefirst case. Thus, the differential equations describing the temporal evolutionof the array read:

xj = σ(yj − xj)

yj = rxj − xjzj − yj +D

2(yj+1 + yj−1 − 2yj) (4.2)

zj = xjyj − bzj j = 1, . . . , N

The standard values are chosen for the σ and b parameters: σ = 10and b = 8/3. 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 Hopfbifurcation. Also, it is important to note that there exists a saddle fixedpoint 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 tothe system. Results on front propagation do not depend on the boundaryconditions provided that the array is large enough. Thus, if one wishesthe system evolve for long time in the regime of traveling front, one should


move the boundary whilst the front propagates. Also, one may imposedto the system periodic boundary conditions, with two fronts diametricallyopposed 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 samedirection, 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 amongstanding (static), oscillating and traveling fronts. In Fig. 4.1, a centeredstep-like initial condition is imposed to the system with free ends and r =14; two different non-static regimes are achieved depending on the valueof the coupling strength D. Note that the Lorenz model is symmetric andthe coupling that appears in Eq. (4.2) destroys this local symmetry butpreserves 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 equallylikely. A diagram of the different regions on the r − D plane is shown inFig. 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 significativethat oscillating fronts always exist between the standing and the travelingfronts regions. The line (Dos), that defines the boundary between thestanding and the oscillating regions, approaches D = 0 as r → rH , sincethe nature of the fixed points (C±) deeply influences the dynamics of thefront. This is not very surprising, but indicates that oscillating fronts willbe usually found in bistable systems with stable foci, rather than nodes.Somehow, the stability properties of the fixed points are transmitted to thefront. 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 isthe 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 consideringconsists of rj = C+, j = 1, . . . , 25 and rj = C−, j = 26, . . . , 50 (N = 50). Asthe coupling increases this solution can be smoothly continued. But thereexist two constrains for the stationary solutions, that come from Eq. (4.2):

4.3 Standing, oscillating and traveling fronts 89

010

2030

4050

−505

time

D = 39

i

xi

010

2030

4050

−505

time

D = 40

i x

i

Figure 1 D. Pazo and V. Perez−Muñuzuri (2002)

Figure 4.1: Spatio-temporal evolution of the front for two different values of the couplingstrength D and r = 14. A step-like initial condition is imposed for an open array ofN = 50 oscillators. When the coupling surpasses the critical value D = Dth ≈ 39.63the front propagates through the array shifting the oscillators from C+ to C−. The timeinterval shown is 100 time units.

12 14 16 18 20 22 240

10

20

30

40

50

60

Dth

DosI

II

III

r

D

I − static front

II − oscillating front

III − travelling front

Figure 2D. Pazo and V. Perez Muñuzuri (Chaos, 2002)

Figure 4.2: Three main types of dynamics are distinguished as a function of r and D.The boundaries among them are the critical lines Dos and Dth.


xj = yj , zj =xjyj

b∀j ∈ 1, 2, ..., N (4.4)

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. Theprojection of the parabola onto the plane (x, y) is a straight line, thebisectrix of the first and third quadrants. However, to better visualizeoscillations out of the parabola, in Fig. 4.3 we performed a 45 rotationof the reference system. Obviously those units closer to the front oscillatewith larger amplitude that those located far from the front that are almostinsensitive to the bifurcation. When Dth is reached a multiple collisionoccurs; the orbits of neighboring units collide creating two “channels” goingfrom C± to C∓. Like in the symmetric FitzHugh-Nagumo model studied byHagberg and Meron [HM94], the traveling front is not symmetric. However,in our case the mechanism leading to propagation is not a pitchforkbifurcation, instead, a Hopf bifurcation that creates the oscillating frontis the precursor of the traveling front.

The description provided by Fig. 4.3 is illustrative, but it is incompleteif one does not realize that besides the static solution that loses itsstability 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, calledstable 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 referto them as A- and B-state, respectively, and Fig. 4.4 shows a sketch ofthem. It is the B-state which mediates the transition at Dth, and it willbe shown below that the stability properties of this solution will determineimportant features of the onset of the wavefronts.

4.4 Gluing of cycles

In this section we demonstrate how the transitions presented in theprevious section may be described in the context of a gluing bifurcation,in which two limit cycles become a two-lobed cycle by involving anintermediate saddle point. The gluing bifurcation is usual in systems withZ2 symmetry [LM00, RFH+95, HFP+98], because provided the existenceof this symmetry it becomes a codimension-one bifurcation.

4.4 Gluing of cycles 91

−8 −6 −4 −2 0 2 4 6 8−0.04

−0.02

0

0.02

0.04

C+

C−

(−x i+

y i)/21/

2

i=25i=24 i=26 i=27

D = 8

−8 −6 −4 −2 0 2 4 6 8−0.04

−0.02

0

0.02

0.04

(−x i+

y i)/21/

2

D = 39i=25

i=24

i=26

i=27

C+

C−

−8 −6 −4 −2 0 2 4 6 8−0.04

−0.02

0

0.02

0.04

(xi+y

i)/21/2

(−x i+

y i)/21/

2

D = 40

C+

C−

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 thetrajectory 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 frontoscillates. Then, the whole set of variables (xj , yj , zj) is discarded, andinstead, we chose to build a reduced phase space to study the dynamics.Notice that if the medium is infinite, there exists a perfect symmetry undertranslation along the array. With this in mind, we define two auxiliary


variables ξ and η, that have sense for large enough N and when the frontis far enough from the boundaries:

ξ =1√2

N∑j=1

xj + yj mod(2√

2b(r − 1))

(4.5)

η =1√2

N∑j=1

−xj + yj . (4.6)

Variable ξ accounts for the propagation of the front and is not boundedby default, then it must be defined within the range ∆ξ = [2

√b(r − 1) +

2√

b(r − 1)]/√

2 = 2√

2b(r − 1), that is the magnitude that ξ increases ordecreases when the front moves one position along the array. On the otherhand, η is defined so that it is bounded under front propagation, noticethat it is defined from Eq. (4.4). If the medium consists of a number Neven of units, the static solutions are located in this cyclic phase space in:

A : ξ = η = 0 (4.7)

B : ξ = ±√

2b(r − 1), η = 0 (4.8)

If N is odd both solutions exchange their coordinates. In what follows, weonly take N even, unless it is specified.

The transition shown in Fig. 4.3 is now shown in the cylindrical phasespace in Fig. 4.5. Notice that when the front is static, we have anequilibrium point at ξ = η = 0 (A-state). When the front oscillatesa limit cycle exists, and finally when the coupling goes beyond Dth wefind two symmetry related limit cycles, that turn around the cylindricalphase space in opposite directions. The situation is quite similar to apendulum where libration corresponds now to oscillation, and rotationof the pendulum corresponds to propagation. Thus, at D = Dth thereexist two homoclinic loops that connect the B-state with itself (calledseparatrices for the pendulum). This bifurcation is called gluing bifurcationbecause it involves the collision of two cycles to create another; althoughin our case it could be more appropriate to talk of a inverse gluing or asplitting bifurcation.

Nonetheless, it is better to visualize our gluing bifurcation in R2. Apossible transformation consists, topologically, in what follows: First of all,cut the cylinder with two planes perpendicular to its axis in such a way


−5 0 5−0.1

−0.05

0

0.05

0.1

ξ

η

D = 8

−5 0 5−0.1

−0.05

0

0.05

0.1

ξ

D = 39

−5 0 5−0.1

−0.05

0

0.05

0.1

ξ

D = 40

Figure 5D. Pazo and V. Perez−Muñuzuri (2002)

Figure 4.5: Evolution on the reduced cylindrical phase space spanned by ξ and η forthree different values of D and r = 14; from left to right: standing, oscillating, andtraveling front. At D = Dth ≈ 39.63 a double homoclinic connection to the B-statearises.

that the saddle point and the heteroclinic orbits stand between both planes.Then compress both circumferences that limit the “cylinder” inbetween toa point. Thus, we have got an object topologically equal to a sphere. Thesetwo steps could be substituted by making the section of the cylinder equalto zero at η = ±∞ and then making a transformation that makes ourobject finite. The last step is to make a hole at ξ = η = 0 and deform whatremains into a plane.

In Fig. 4.6 the sketch of a gluing bifurcation equivalent to the one shownin 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 saddlepoint (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 asaddle, called saddle-loop bifurcation, is characterized by a logarithmiclengthening of the period of the cycle [Gle94, Str94]. For the scenariodepicted in Fig. 4.6, we expect to find the following dependencies for theperiods (T1,2) in a neighborhood of Dth:

T1 = a1 −2λu

ln(Dth −D) (4.9)

1c

= T2 = a2 −1λu

ln(D −Dth), (4.10)


D < Dth D = Dth D > Dth

λu

λs

Figure 4.6: Schematic of a (inverse) gluing bifurcation on R2. For D < Dth thetrajectory approaches twice per cycle the saddle point and the front oscillates. Beyondthe critical value D = Dth two symmetry related orbits, corresponding to both senses ofpropagation of the front, appear. λu and λs stand for the eigenvalues of the saddle fixedpoint at homoclinicity.

where λu is the unstable eigenvalue of the saddle fixed point. Taking intoaccount that one turn around the cylinder is equivalent to the movement ofthe front in one cell, it is clear that T2 is the inverse of the velocity of thefront (c). In the equation for the period T1 a factor 2 appears because theorbit approaches twice per cycle to the neighborhood of the saddle point.Moreover, it is expected that the ‘fast dynamics’ (the motion far from thesaddle) contained in variables a1,2 will be approximately the same at bothsides of the transition, and then, a1 ≈ 2a2. We show here the results forr = 16 in Fig. 4.7.

Notice that the velocity of the front grows quite abruptly from zero atD = Dth. The derivative of c = c(D), near Dth is, according to Eq. (4.10):

dc

dD=

1λu(D −Dth)

(1

a2 − 1λu

ln(D −Dth)

)2D→Dth−→ ∞. (4.11)

4.4.3 Quantitative analysis

In order to verify, not only qualitatively, but quantitatively the tendencyof T1 and c when D → Dth we computed numerically the value of theeigenvalues of the B-state at D = Dth for finite N . To preserve the


18.5 19 19.5 20 20.5 21 21.52

3

4

5

6

7a)

D

1/c T

1/2

−8 −6 −4 −2 02

3

4

5

6

7b)

~ −λu−1

ln | D−Dth

|

1/c T

1/2


Figure 4.7: 1/c (solid line) and T1/2 (squares) as a function of D − Dth (a) andln(|D −Dth|) (b) for r = 16. The behavior agrees with that predicted by Eqs. (4.9) and(4.10). The critical coupling is found to be Dr=16

th = 20.13952. In (b) a straight linewith slope −λ−1

u is depicted, being λu = 1.7959 the unstable eigenvalue of the B-statecomputed numerically for an array of N = 21 units.

symmetry, when computing the eigenvalues, the number of units N waschosen to be odd, with the central unit located at the origin. The jacobianmatrix (J) for a discrete reaction-diffusion model (see Eq. (4.1)) exhibitsthe following tridiagonal structure:

J =

H1 D 0 · · · 0 0D H2 D · · · 0 00 D H3 · · · 0 0...

......

. . ....

...0 0 0 · · · HN−1 D0 0 0 · · · D HN

, (4.12)

which, according to Eq. (4.2) has got the components:

Hj =

−σ σ 0r − zj −1−D −xj

yj xj −b

,D =

0 0 00 D/2 00 0 0

, (4.13)

recall that x(N+1)/2 = y(N+1)/2 = 0. Therefore, −b is always an eigenvaluewith trivial eigenvector (0,0, ...,0, (0, 0, 1),0, ...,0)T . Using the programmatlab we obtained the eigenvalues for an array of N = 21 units, that weexpect to be large enough to inform us about the properties of a front inan 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 andD. Fig. 4.8 (a) shows the eigenvalues (λi) for D = 0 and r = 14, which


−20 −10 0

−5

0

5

r = 14, D = 0

−b

Re(λi)

Im(λ

i)

−10 −5 0

−5

0

5

r = 14, D = Dthr=14

λu

λs

Re(λi)

−10 −5 0

−5

0

5

r = 16, D = Dthr=16

λu

λs

λs*

Re(λi)

a) b) c)


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 simpleeigenvalues corresponding to the oscillator located in the center of thearray whose coordinates are x(N+1)/2 = y(N+1)/2 = z(N+1)/2 = 0, recallthat we are considering a B-state. Moreover, there are three (N − 1)-degenerate eigenvalues, corresponding to the oscillators with coordinatesC±. Eigenvalues are depicted with circles of different sizes to better observedegeneracy. When D grows above zero degeneracy is lost. In Fig. 4.8 (b),we graph the eigenvalues for D = Dth ≈ 39.63 (only those eigenvaluessatisfying Re(λi) > −10 are shown). The unstable eigenvalue correspondsto the fixed point at the origin of a Lorenz oscillator. There is also aslightly negative eigenvalue (λs) that is the leading eigenvalue among thewhole spectrum of negative eigenvalues.

A different scenario is found for r = 16, as shown in Fig. 4.8 (c). Inthis case, it is not so clear which stable eigenvalue must be consideredas the leading one in the gluing process. There are several eigenvaluesmaking up two lines at both sides of the real axis in the complex plain, butthese are not relevant, as long as they are the eigenvalues associated withthe stability properties of each domain at both sides of the front wherequasi-homogeneous domains around C± exist. As well, if the eigenvaluelocated at −b is considered, its eigenvector becomes trivial and it cannotbe expected to participate in the gluing bifurcation. Then, the solitaryeigenvalue denoted by λs(= ρ + iω) is the one (with its complex conjugateλ?

s) that defines the leading (two-dimensional) stable manifold. In fact,by increasing continuously r along Dth, the eigenvalue λs could be followeddirectly from that obtained for r = 14 (see Figs. 4.8 (b,c)). According to thestatements exposed above, the graph of η vs. ξ, for D close to Dth(r) and

4.5 Exotic front dynamics 97

Figure 4.9: Projection onto (ξ,η)at r = 16, D = 20.136 < Dth ≈20.1395. In the inset, the spiralingapproach of the trajectory to the B-state (filled circle) may be observed.

−10 −8 −6 −4 −2 0 2 4 6 8 10−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

ξ

η

8.8 8.9 9

−0.01

0

0.01

0.02

ξ

η

B

−10 −8 −6 −4 −2 0 2 4 6 8 10−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

ξ

η

8.8 8.9 9

−0.01

0

0.01

0.02

ξ

η

B

−10 −8 −6 −4 −2 0 2 4 6 8 10−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

ξ

η

8.8 8.9 9

−0.01

0

0.01

0.02

ξ

η

B

−10 −8 −6 −4 −2 0 2 4 6 8 10−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

ξ

η

8.8 8.9 9

−0.01

0

0.01

0.02

ξ

η

B


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 mediatedby a saddle-focus. The cyclic definition of ξ has been relaxed in order to geta better observation of the spiral trajectory near the B-state. Nonetheless,if one looks at the asymptotic properties of the oscillation period of thefront and its velocity, no difference with the case of the planar connectionis found. So, Fig. 4.7 already confirmed the Eqs. (4.9) and (4.10), and theslopes agree with the numerical value: λu ≈ 1.7959.

A sketch of the gluing bifurcation in the saddle-focus case is shown inFig. 4.10. Homoclinic orbits, at D = Dth are denoted by Γ0 and Γ1 . Theprocess of gluing is drawn in analogy to Fig. 4.6. Although, from bothfigures, one could think that the complex and the real cases are almostequivalent, there exist very fundamental differences between both cases.Thus, the transition from oscillating to traveling fronts may become quiteconvoluted in the complex case. This is observed for larger values of r andis the subject of the next section.

4.5 Exotic front dynamics

In Figure 4.11 the value of the imaginary part of the leading stableeigenvalues (ω), along the line D = Dth, is shown as a function of r witha dashed line. It is found that above r = rsf ≈ 15.45 the leading stableeigenvalues are complex conjugates, and therefore, the gluing bifurcationoccurs mediated by a saddle-focus point.

Many works have been devoted to the problem of saddle-loopbifurcations involving a saddle-focus fixed point (see e.g. [Str94, Gle94,

1One should define a new extra variable ζ to work in a hyper-cylindrical phase space.One could define, for instance, ζ =

∑xjyj − bzj . Nonetheless, we shall continue to work

with ξ and η only, although keeping in mind that we are projecting the third variable.


D < Dth D = Dth D > Dth

Γ0

Γ1

Figure 4.10: Schematic of a gluing bifurcation mediated by a saddle-focus. Twohomoclinic trajectories Γ0 ,1 exist at D = Dth.

13 14 15 16 17 18 19 20 21 22 23 240

1

2

3

4

5

6

r

ω, δ

rsf

r1


Figure 4.11: Dashed line: Imaginary part of the stable leading eigenvalue ω of theB-state. Solid line: saddle index δ. For r > rsf , ω > 0; and for r > r1, δ < 1. Bothmagnitudes are computed as a function of r, along the curve D = Dth(r).


GH83, Wig88] and references therein). A deep presentation of this problemis out of the scope of this thesis, so we just recall some of the most relevantresults.

For the case of a single homoclinic orbit, occurring at a critical value ofthe control parameter (say µ = 0), it was found by Shil’nikov [Shi65] thatwhen the ratio

δ = −Re(λs)λu

= − ρ

λu, (4.14)

called saddle index, is less than one, there are a countable infinity ofperiodic orbits in a neighborhood of the homoclinic orbit, all of whichare saddles (see. Sec. 1.3.5.b). These saddle orbits are created in asequence of tangent bifurcations at both sides of µ = 0, such that the stablebranches (provided δ > 1/2) become unstable through period-doublingbifurcations. This scenario, where infinite orbits, with periods rangingfrom some finite value to an infinite one, arrange in a wiggly curve aroundthe critical value, is known as Shil’nikov wiggle. And it is, with the Lorenzand the bifocal mechanisms, one of the universal routes to chaos throughhomoclinicity [GH83, Wig88].

However, the results by Shil’nikov apply in a very small neighborhoodof the critical parameter. Therefore, the relationship between the local(theoretical) results and the global (observed) behaviors cannot be statedoversimply. It was shown by Glendinning and Sparrow [GS84] that thedifference between the δ < 1 and the δ > 1 approaches to homoclinicitymay not be easily observed numerically and may not be relevant from aglobal point of view. In fact, for the lowest branch (the orbit with thelowest period) that already exist far from the critical parameter, deviationsfrom 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 aremultiple-pulse homoclinic orbits, i.e. orbits that pass several times near thestationary point without achieving homoclinicity.

In case the homoclinic connection is double (two loops, Γ0 and Γ1 ), asoccurs in the gluing bifurcation, some differences are appreciated with thecase described above. The most important point is that, although for δ > 1no Smale horseshoe exist as in the single case, the approach of orbits toΓ0 ∪ Γ1 (Fig. 4.10) is chaotic [Hol80, Wig88].

Figure 4.11 shows the value of δ (solid line) as a function of r alongthe 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


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 solutionsfound, but just to show that the results have got the “flavor” expectedaccording to previous theoretical studies on bi-homoclinicity to a saddle-focus [Hol80, Gle84, ACT81].

4.5.1 r = 20 (δ > 1)

In the following, we analyze the transition from an oscillating front toa traveling front for r = 20. Fig. 4.12 (a) shows the dependence of theperiod of oscillation of the front (T1) on D. It is found that the perioddoes not grow continuously, and instead, there are some jumps, which giverise to hysteretic cycles (Fig. 4.12(b) shows an inset of Fig. 4.12(a)). Thesewiggles (note that unstable branches join consecutive stable branches) arenot surprising according to [GS84], and are expected to appear whenapproaching δ = 1. Unfortunately, we have not been able to find oscillationswith semi-periods larger than 3.2 t.u. This occurs because as the periodgrows, and therefore (bi-)homoclinicity is reached, the chaotic transientsbecome larger and larger.

The different regimes may be labeled by their representative symbolicsequences of 0’s and 1’s, depending on the number of turns at each side ofthe saddle point. For example, the uniform propagating regimes are labeledby 0 or 1, but if a period-doubling occurs the new (stable) regimes arelabeled by 02 or 12, respectively. On the other hand, the oscillatingregime, described by a two-lobed cycle, is represented by the code 01 (or10).

For the traveling front region, the dependence of the velocity with Dshows some features that were also reported in [GS84], where a systemof three ordinary differential equations, that exhibits a single homoclinicconnection to a saddle-focus, was studied. In Fig. 4.12 (c,d) the inverse ofthe velocity is plotted as a function of D. In those intervals of D where noline appear, the front continues to exhibit chaotic motion, characterized by“spontaneous” front reversals, after, at least, 30000 t.u. of transient; butsome small ‘windows’ with regular motion can be found.

Figure 4.13 shows the dynamics of the front for D = 17.85 during 200t.u., after some transient. In Fig. 4.13 (a), the variable x of the oscillatorsof the array is represented in gray scale, whereas Fig. 4.13 (b) shows thephase portrait (ξ, η). The front reverses its propagation several times anddisplays a quasi-erratic motion. Although the behavior is likely to be achaotic transient, from a practical point of view, it is indistinguishable of


15 15.5 16 16.5 17 17.5 180.5

1

1.5

2

2.5

3

3.5

D

T1 /

2

17.68 17.73 17.781.6

1.8

2

2.2

2.4

2.6

D

T1 /

2

17.5 18 18.5 19 19.5 201.5

2

2.5

3

D

1 /

c

17.8 17.85 17.92.4

2.6

2.8

3

3.2

D

1 /

c

17.78 17.785 17.79

3

3.1

3.2

0

02

0

0 02

0

02

0

02

04

a) b)

c)

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 oneof this jumps can be seen in (b). The velocity of the front –Fig. (d) and inset (c)– isnot defined in some regions where chaotic states are found. Period-doublings are markedwith circles and different states are labeled according to its symbolic code. Because ofthe 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 inFigs. 4.12 (c,d)). In addition, we find the well-known period-doubling routeto chaos, see the inset of Fig. 4.12 (c). In Fig. 4.14 three periodic orbitsand a chaotic one are shown. From right to left, a period-doubling cascadeleads to a type of chaos that is characterized by sustained propagation ofthe front with non-periodic velocity. This scenario, that generates chaoswith 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 ofDth, one expects to find, according to [Gle84], Shil’nikov wiggles for both,


60 80 100 120 140

<−

− T

ime

i

C−

C+

0 5 10 15 20−2

−1

0

1

2

ξ

η

a) b)

Figure 4.13: Chaotic motion of the front for r = 20 and D = 17.85 during 200 t.u.after a transient of 30000 t.u. a) x variable in gray scale where abcissa corresponds tothe oscillator number and time is running downwards. b) Cylindrical variables ξ and ηduring the 200 t.u. shown in the left figure. The interval chosen for variable ξ has beenshifted half a period, with respect to previous figures, to better observe the homoclinicchaos that is organized at the B-state.

0 10 20−1

0

1

2

ξ

period−one = 0

D = 17.785

0 10 20−1

0

1

2

ξ

period−two = 02

D = 17.783

0 10 20−1

0

1

2

ξ

period−four = 04

D = 17.782

0 10 20−1

0

1

2

ξ

η

chaos = 0∞

D = 17.7808


Figure 4.14: From right to left a period-doubling cascade leading to chaos. Values ofD belong to the interval shown in the inset of Fig. 4.12(c).


15 16 17 18 19 20 21 22 23 24 250.8

1

1.2

1.4

1.6

1.8

2

01

0, 1

0, 10212

02, 12

02, 12

0414

D

1/c T

1 /2

T/4 T/8

Figure 4.15: 1/c (solid line) and T1/2 (squares) as a function of D for r = 23. Twoperiodic states with codes 0212 (dots) and 0414 (diamonds), as well as a conjecturedunstable traveling solution (dashed line) are shown.

the two-lobed cycle (oscillating front) and the one-lobed cycle (travelingfront). Also we expect to find chaos of the type reported by Arneodo etal. [ACT81]; where the strange attractor organizes around a saddle-focuswith symmetry, like in Fig. 4.13.

However, we have seen in the previous subsection, that the local analysisis not always representative enough of the observable results that usuallycorrespond to the lowest branch (i.e. the smallest period). In fact, ourresults do not differ too much from those found for r = 20. In Fig. 4.15,oscillating –as well as traveling– solutions are arranged according to theirperiods (considering the hyper-cylindrical phase space).

The main oscillating solution (code 01) disappears in a saddle-nodebifurcation at D ≈ 18.04. Surely, oscillating solutions with 01 codeand larger period exist, but we have not been able to find them. Besidesthe standard oscillating solution, we find others labeled 0212 and 0414that correspond to more sophisticated oscillating regimes. Their dynamics


−20 0 20 40

−2

0

2

ξ

η

0212

D = 20

−60 −40 −20 0 20 40

−2

0

2

ξ

η

0414

D = 20.9

Figure 4.16: Periodic orbits 0212 and 0414 for r = 23 and two values of D. The cyclicrestriction of the variable ξ has been suppressed to better observe the periodic dynamics.

onto (ξ,η) is depicted in Fig. 4.16. Their existence can be attributed tosubsidiary homoclinic connections.

Concerning the propagating solutions (solid line in Fig. 4.15), the“period-one” propagating solutions (codes 0 and 1) undergo a period-doubling bifurcation at D ≈ 21.65. Nonetheless, this “period-one”solution reappears at D ≈ 19.43, so we conjecture, according to previousstudies [Gle84], that there exists an unstable solution (dashed line) linkingboth regions. The propagating solution corresponding to lower values of Dexhibits another period-doubling bifurcation at D ≈ 18.37.

For those intervals of D where no periodic orbits were found, chaoticbehavior (like that of Fig. 4.13) is found. Nonetheless, some very smallperiodic windows are found intermingled into the intervals with chaoticmotion.

4.6 The effect of parameter mismatch and asym-metry

Our system of coupled Lorenz oscillators is nothing but a mathematicalmodel, because it is clear that it is impossible to build an infinitelylarge experimental set-up with identical units, all of them being exactlysymmetric. This means that in a real system, no invariance undertranslation and under reflection exists.

Boundary effects can be significantly minimized by using a sufficientlylarge array, recall that the front is very localized, and only for large Da significant amount of oscillators is involved in the front. The effect ofusing non-identical units is difficult to be predicted but could manifest in aneighborhood of Dth by letting some localized oscillation to exist or making

4.7 Universality? 105

the front to reverse back at a given time. Nonetheless, there are lots of waysto make the oscillators non-identical and then giving rise to an enormousvariety of effects.

More tractable is the case where all the oscillators are identical, butasymmetric. In this situation, we should use the theory of imperfectgluing bifurcations. The larger is the asymmetry, the larger is theinterval of D at D ≈ Dth where new periodic orbits appear. Mainly,the case when the saddle value σ = λu + Re(λs) is negative has beenconsidered [TS87, GGT87, GGT88] showing that no more than two periodicorbits can coexist in the same parameter domain, and providing somesimple rules for the symbolic sequences of 0’s and 1’s of the cycles,based on a Farey tree structure. Experimental studies can be found inRefs. [RFH+95, HFP+98] and [GAM01]. The latter also considers the casewith σ > 0 (i.e. saddle-focus with δ < 1).

4.7 Universality?

4.7.1 Other couplings and systems

One may ask whether the symmetry breaking front bifurcationpresented here is universal or not; in other words, in what kind of systemcould one expect to find this transition?

We have found that other coupling matrices are able to induce frontpropagation in an array of coupled Lorenz oscillators. If one considerscoupling matrices with all elements zero except one, these two off-diagonalcouplings:

Γ = γkl = δk1δl2 (4.15)

Γ = γkl = δk2δl1 (4.16)

exhibit front propagation through a route that is similar to the oneexplained here. However, it may happen that the transition occurs insuch 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 apitchfork bifurcation again). Later, it is the B-state which undergoes a Hopfbifurcation, and finally the oscillating solution touches the A-state creatingthe traveling solutions. The logarithmic laws, Eqs. (4.9) and (4.10), arealso obtained as we shall see in the next chapter.


We have also checked our results in other two bistable systems: an arraywhose local dynamics is a truncation of the magnetohydrodynamic partialdifferential equations of a disc dynamo [CH80], and the FitzHugh-Nagumomodel [Fit61].

The dynamo model is similar to the Lorenz system and in someparameter range it exhibits symmetric bistability. The equations of themodel 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 stablefixed 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 theLorenz systems, for instance for the coupling matrix Γ = γkl = δk2δl2.Depending on the value of β the transition is smooth or chaotic. Whereasfor β = 6 we find a fine logarithmic profile of the front velocity, for β = 14the front velocity function is interrupted by a chaotic regime –like thatshown in Fig. 4.13– when approaching the threshold. Also, the oscillatingdynamics, 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 totraveling 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 ofoscillators. In some sense, the front becomes more ‘continuous’ (or lesssteep). 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, neighboringoscillators have similar (x, y, z) values. Therefore, once the Dos line iscrossed, a small increase of D (in comparison with Dth or Dos) is neededto achieve the multiple collision of cycles, i.e. the Dth line. As long asD is large, both A- and B-states are in a quasi-continuum and shouldexhibit similar eigenvalue spectra. Formally we can make A− and B−states coalesce, as intuitively occurs in the infinite diffusion limit, redefiningthe variable ξ: ξnew = ξ/Dγ , (γ > 0). In this way, the size of the cylindershrinks to zero at infinity.


So as r → r∞ ≈ 13.5, λu and λs of the B-state at Dth should meetthe pair of complex conjugate eigenvalues that characterizes the Hopfbifurcation 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). Wecontinue our discussion of the continuum limit with the FitzHugh-Nagumomodel in Sec. 4.7.3.b.

4.7.3 The discrete FitzHugh-Nagumo model

The FitzHugh-Nagumo (FHN) model was derived from the Hodgkin-Huxley model for nerve membranes [Mur89], with the aim of making itanalytically tractable. It has become one of the most important reaction-diffusion models. Depending on the parameters that control the localdynamics it may describe an excitable medium, a medium undergoing eithera Hopf or Turing bifurcation, and a bistable medium. This last possibility isof our interest, and in spite of being not so often considered as the excitablecase it may be relevant in some situations [GC77, ML81] .

4.7.3.a Transition to traveling front

The discrete FitzHugh-Nagumo model reads:

uj = uj − u3j − vj + D(uj+1 + uj−1 − 2uj)

vj = ε(uj − a1vj − 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 pointat the origin, and two odd symmetric stable fixed points (u±, v±) =

(±√

a1−1a1

,± 1a1

√a1−1

a1). As occurs in the continuous version [HM94],

propagation only succeeds for small ε (O(10−1)). For very small D, onlyone 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 bifurcationand the traveling solutions become the only stable ones. This route isapparently very different from those shown above, since a computation ofthe 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 acritical value DSN , two stable traveling solutions (both propagation senses)


0 0.5 1 1.50

2

4

6

8

10

12

DG

DSN

D

1/c

a)

0.1 0.2 0.3

4

6

8

10

12

DG

( DSN

, 1/c0 )

0 0.5 1 1.50

2

4

6

8

10

12

DG

DSN

D

1/c

a)

0.1 0.2 0.3

4

6

8

10

12

DG

( DSN

, 1/c0 )

−6 −4 −2 0

2

4

6 ~−λu−1

ln( D − DG

)

1/c

DG

≈0.125

b)

−15 −10 −5 0

−4

−2

0

2

ln( D − DSN

)

ln(1

/c0−

1/c)

DSN

=0.10782325

1/c0=11.971

~1/2c)

Figure 4.17: Velocity of the front for the discrete FitzHugh-Nagumo model (a). Internalparameters are a0=0, a1 = 2, and ε = 0.1 (see Eq. (4.18)). Data are shown with dotsand 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 theincrease 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, becausethe traveling solution is stable and the saddle index of the B-solution is approximately0.7 for the values of D considered. This implies that the gluing bifurcation may occurbetween unstable cycles only. Hence, the gluing mechanism involves the unstable travelingsolutions created at D = DSN . In Fig. (c) a log-log plot shows the dependence of 1/cas 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 saddle-node bifurcations. The unstable traveling solutions that appear in thesesaddle-node bifurcations become glued when D is slightly increased, atD = DG

2. In this way, the (unstable) oscillating solution that coalesceswith 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 ε =0.1. One may observe that within a range of values of D, 1/c exhibits alogarithmic 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)): (

1c− 1

c 0

)∝ ±(D −DSN )1/2. (4.19)

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

2Considering 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 changesfrom the A-state to a traveling state.


0.1 0.15 0.2 0.250.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ε

D

static

traveling

coexistence

DH

D

G

DSN

Figure 4.18: Diagram showing regions with different front dynamics in the discrete FHNmodel. At DSN , two pairs (stable-unstable) of counterpropagating traveling solutionsappear. At DG, both unstable traveling solutions become ‘glued’ to create an unstableoscillating 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 thesaddle index is not very small, δ ∼ 0.7, what makes possible the unstablecycles to disappear at DSN close to DG.

4.7.3.b The continuum limit

Let us consider consider now the continuous version of the FitzHugh-Nagumo (FHN) model:

∂tu(x, t) = u− u3 − v + ∂xxu

∂tv(x, t) = ε(u− a1v − a0) (4.20)

The diffusion constant of the variable u can be chosen as one without lossof generality. This may be done by rescaling the variable x: x → x/

√Du


(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±) asx → ∓∞.

As proved by Hagberg and Meron [HM94], the static front is stableabove a critical value of the parameter ε. At the critical value εc = 1/a2

1

the stationary front renders unstable through a pitchfork bifurcation, thatgives rise to two nonsymmetric (but symmetry-related) traveling solutions.The velocity of these solutions grows from zero following a square root law:c ≈ k(εc − ε)1/2

It would be interesting to compare these results with those obtainedfor discrete case. This is done by changing variables u(x, t) → uj(t) anddiscretizing the partial derivative such that

∂xxu(x, t) = limh→∞

u(x + h) + u(x− h)− 2u(x)h2

→ D(uj+1 + uj−1 − 2uj). (4.21)

Hence, the diffusion constant D is a measure of the spatial step, andtherefore the continuum limit (h = 0) is at D →∞.

Keeping the arguments exposed above in mind, we swept the parameterspace ε−D for a particular value of a1, and paying special attention to thelimit D → ∞. In Fig. 4.18, the regions in the ε − D plane with differentfront solutions are shown, for a1 = 2. The lines DSN,H divide the phasespace in three parts: static, traveling, and coexistence of both (that islocated 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, thatmediates the gluing of two unstable traveling solutions, occurs. Noticethat in Fig. 4.17, we studied the transitions along the line ε = 0.1. Themost relevant feature of Fig. 4.18 is the fact that all the lines diverge atε = 1/a2

1 = 1/4, which means that the continuum is effectively recoveredat D = ∞. In fact the coexistence region shrinks as D → ∞, whichagrees with the ‘soft’ transition from static to traveling front observed inthe continuous version.

As it occurred for the array of Lorenz units, the bifurcation lines divergeto infinity. Formally, the point located at D = ∞ –where the transitionbetween static and traveling front occurs– is a double zero-eigenvalue point.This kind of degeneracy is known as Takens-Bogdanov (see the Appendixfor an example).

Following the reasonings above, we are obliged to compare our resultswith the unfolding of a TB bifurcation corresponding to the case studied


here. Notice that the size of the cylinder is zero at D = ∞ (after therescaling of ξ proposed in Sec. 4.7.2), so D = ∞ is itself a bifurcation line ofcodimension one. The two counterpropagating traveling fronts that appearthrough a pitchfork bifurcation for the continuous version at ε = 1/a2

1, arethe limit case of the cyclic solutions observed in our cylinder when its widthapproaches zero.

Unfortunately, we have not been able to find in the literature anunfolding of the TB with periodic boundary conditions in one of itsvariables; and it seems not to be a straightforward task. Nonetheless, inthe book of Guckenheimer and Holmes [GH83] one can find the unfoldingof a TB bifurcation with “cubic” symmetry, or symmetry under rotationthrough π. This unfolding considers variables in R2, which is different toour case but satisfies the symmetry properties (ξ, η) → (−ξ,−η). Allowingtime reversal this unfolding presents only two cases, whose bifurcation lines(emanating from the TB point) strongly remind the scenarios observed forthe two systems studied in detail in this Chapter (the array of Lorenz unitsand the discrete FHN model). Thus, for the unfolding of the symmetricTB, and in the region of parameter space where three static solutions exist3,one follows one of these two routes:

1.

one stablestationarysolution

Hopf bif.−−−−−→

stableperiodicsolution

double heteroclinic connect.−−−−−−−−−−−−−−−−−−→

nostable

solution

2.

two stablestationarysolutions

Saddle-node−−−−−−−−−→

of cycles

coexistenceperiodic +

stationary sols.

gluing bif. +−−−−−−−−−−→subcr. Hopf bif.

stableperiodicsolution

We introduce now some comments about the relations between these routesand the transitions we have observed. Route 1 is related to the route inthe array of Lorenz oscillators, whereas route 2 presents strong analogieswith the discrete FHN model. First of all notice that as long as theseroutes 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 analogywith our periodic condition (due to the cylindrical geometry) stronger.

3Since a pitchfork bifurcation line passes through the TB point, there are threesolutions at one side of this line and one at the other. The latter is not of our interestbecause it is formally beyond D = ∞.


Thus in route 1, the double heteroclinic connection becomes a doublehomoclinic one. The final state does not reproduce our findings because thetwo counterpropagating solutions that turn around the cylinder have notanalogy, because of the different topology of R2. Something similar occurswith 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 solutionsthat 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 wouldcorrespond to the two counterpropagating solutions of the FHN model.Finally, the stationary solutions become unstable through a subcritical Hopfbifurcation.

4.8 Discussion

It has been shown that an array composed of (symmetric) bistableLorenz units undergoes a front bifurcation originating two counterprop-agating traveling solutions. The mechanism consists in a Hopf bifurcationof the static front, followed by a global bifurcation equivalent to a gluingbifurcation of cycles onto a cylindrical phase space. Accordingly, close tothe threshold, the period of oscillation of the front (T1) and the speed ofthe front (c) follow logarithmic laws. A saddle static solution mediatesthe gluing process, in such a way that the value of its unstable eigenvaluedetermines the rate of divergence of T1 and c−1. The transition is typicallydiscrete, and is possible thanks to the multi-variable and non-gradientnature of the local dynamics.

Also, it has been demonstrated that the gluing transition may bemediated by a saddle-focus point. In that case, when the value of thesaddle index (δ) is below or close to one, the transition becomes muchmore 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 bistablesystems, obtaining similar results (see also the next Chapter). The caseof the discrete FitzHugh-Nagumo model shows a different transition, inwhich a region of coexistence between static and traveling fronts is found.

For the system of Lorenz oscillators the oscillating-front region decreasesas the coupling D becomes larger, such that it becomes virtually zeroat 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 resultsindicate that the transition point between static and traveling fronts locatedat infinity can be considered a codimension-two point where the stable andthe unstable dislocation solutions coalesce (because of being at D = ∞).At the same time, they present a double zero eigenvalue (also known asTakens-Bogdanov degeneracy).

At present, a normal form does not exist for the Takens-Bogdanovbifurcation in so special a situation as the one considered here. Nonetheless,a comparison with the unfolding for the case when symmetry under rotationthrough π is imposed let us to make some conjectures. We believe that thetwo transitions shown here are the only ones to be expected (at least,at large diffusion) when considering one-dimensional reaction-diffusionsystems where the local dynamics presents symmetric bistability. Ofcourse, we are restricting to the case where the local dynamics exhibits one(centered) unstable fixed point. This is quite natural if one is consideringmulti-variable generalizations of the (one-variable) Nagumo equation.

Also, it is important to note that any numerical computation introducesa discretization of a system and therefore, one of the two transition typesshown here should exist in a small parameter region around the parity frontbifurcation point.

In our opinion, further investigation is needed, to know a priori whatlocal dynamics and what couplings (besides some trivial considerations) aresuitable to achieve propagation in a discrete symmetric bistable medium.

Chapter 5

Spatio-Temporal Patterns inan Array of Non-DiagonallyCoupled Lorenz Oscillators

Abstract. The effects of coupling strength and single-cell dynamics(SCD) on spatio-temporal pattern formation are studied in an array ofLorenz oscillators. Different spatio-temporal structures (stationary patterns,propagating wavefronts, short wavelength bifurcation) arise for bistable SCDand two well differentiated types of spatio-temporal chaos for chaotic SCD(in correspondence with the transition from stationary patterns to propagatingfronts). Front propagation in the bistable regime occurs through a route studiedin the previous chapter while the short wavelength pattern region emerges througha pitchfork bifurcation.

5.1 Model

Now, we consider an array of Lorenz oscillators coupled though a nearestneighbor scheme with a coupling matrix:

Γ =

0 1 00 0 00 0 0

(5.1)

We decided to focus with this coupling matrix since in this case, as itwill be shown later, the system undergoes a short wavelength bifurcation.

Thus, our dynamical equations are:

xj = σ(yj − xj) + D2 (yj+1 + yj−1 − 2yj)

yj = rxj − xjzj − yj j = 1, . . . , Nzj = xjyj − bzj

(5.2)

116 Spatio-Temporal Patterns in an Array of Non-Diagonally Coupled Lorenz ...

Off–diagonal diffusive type coupling appears in mechanical modelswith elastic junctions, like the stick-slip Burridge-Knopoff model (seee.g.[CHGP97] and references therein). Moreover, in terms of an electroniccaricature, this coupling represents an active transmission line where cellsare coupled through capacitances and inductances.

Both periodic and null-flow boundary conditions are used. Theparameters σ and b are fixed at the values σ = 10 and b = 8/3. The otherrelevant parameter besides the coupling strength D is r, that controls theinternal state (chaotic or bistable) of the oscillators. Recall that for thedynamics of a single unit there exists a critical value, which for the givenvalues of σ and b is rH ≈ 24.74. For 1 < r < rH the units are bistable, withtwo symmetric stable fixed points C±=(±

√b(r − 1), ±

√b(r − 1), r − 1)

and one unstable fixed point at the origin. For r > rH all the fixed pointsare unstable and the unit exhibits the well known Lorenz strange attractor.

A fourth-order Runge-Kutta method was used to integrate Eq. (5.2). Adiagram 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 therange (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 theone explained in the previous chapter. As the coefficient D is increased fromzero, boundaries between domains of both solutions (C+ and C−) becomesmoother than the step–like boundary observed for D = 0. That is, someoscillators move to the vicinity of C+ and C−. At a given value of D theboundaries start to oscillate, i.e. they undergo a Hopf bifurcation, but theydo not propagate, until finally over a threshold Dth propagation occurs. Ifthe array is open (null-flow boundary condition), the whole system collapsesfinally to one of the two stable solutions. On the other hand, for thecase of a ring (periodic boundary condition) stationary traveling wavefrontssolutions can be found.

1For values of r smaller than, but very close to rH , the chaotic attractor coexists withthe 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 Lorenzoscillators. Left regions correspond to r < rH where Lorenz oscillator is bistable, andright regions correspond to r > rH where Lorenz oscillator is chaotic. A representativetemporal evolution of a ring is located for each zone. They display in gray scalethe 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): shortwavelength ordering; Region V (dark yellow): spatio-temporal chaos; Region VI (darkgreen) : spatiotemporal chaos with front propagation; Region VII (dark blue): shortwavelength ordering with transient chaos; Region IV (red): spatio-temporal chaoswithout clusters. Solid lines correspond to theoretical results: DC , DN , Eqs. (5.4, 5.5)respectively.


−8 −4 0 4 8−8

−4

0

4

8

y

−8 −4 0 4 8−8

−4

0

4

8

−8 −4 0 4 8−8

−4

0

4

8

−8 −4 0 4 8−8

−4

0

4

8

−8 −4 0 4 8−8

−4

0

4

8

x

y

−8 −4 0 4 8−8

−4

0

4

8

x−8 −4 0 4 8

−8

−4

0

4

8

x−8 −4 0 4 8

−8

−4

0

4

8

x

D=7.4

D=8.9 D=8.8D=8.6D=8

D=6.93D=6 D=0

C+

C−

Figure 3D. Pazo et al. (2000)

Figure 5.2: Transition to propagation of a front from the trivial solution at D = 0 as Dincreases. A projection onto the x− y plane of the trajectory followed by the oscillatorsof the array is shown for r = 8 and different values of the coupling strength D. For0 < D < 6, 92 the system is odd symmetric in the x − y plane. For D ≈ 6.92 thissymmetry 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 startsto oscillate. A further increase of D enlarge the orbit of each oscillator. Finally, forD ≈ 8.85 all the orbits collide with their neighbor orbits. For a larger D propagationoccurs.

In order to get a more precise knowledge of the bifurcation arising in thesystem 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 ofthe array. For D = 0, oscillators are in a step-like configuration, asD increases, few cells in the neighborhood of the border of the step-likeinitial condition go to nearby points to C+ and C−. At D ≈ 6.92 thesymmetry of the system is broken through a pitchfork bifurcation (notethat it does not induce propagation), and is recovered when one of theoscillators reaches the origin at D ≈ 6.95. At D ≈ 7.5 the stationary frontsolution becomes unstable through a supercritical Hopf bifurcation. Thiscorresponds to the point where the boundary between both domains startsto oscillate. The amplitude of the oscillations grows with D. Finally forD = Dth ≈ 8.85 the orbit of each oscillator collides with the orbits of its

5.2 Bistable region 119

−1.5 −1 −0.5 0 0.5 1 1.51

2

3

4

5

6

7

8

D − Dth

(a)1/c T

1 /2

−1.5 −1 −0.5 0 0.5 1 1.510

20

30

40

50

D − Dth

(c)1/c T

1/2

−8 −6 −4 −2 01

2

3

4

5

6

7

8

ln | D−Dth

|

~ −λu−1

(b)

1/c T

1/2

−8 −6 −4 −2 010

20

30

40

50

ln | D−Dth

|

(d)

1/c T

1/2

Figure 2, D. Pazo & V. Perez−Muñuzuri (2001)

Figure 5.3: 1/c (solid line) and T1/2 (squares) as a function of D − Dth (a) andln(|D −Dth|) (b) for r = 8. D ≈ 8.841526 and λ−1

u ≈ 0.637.

neighbors2. This collision originates a multiple heteroclinic connection.As we know, if the system is infinite, in terms of global coordinates,the situation can be reduced to a pair of symmetry related homoclinicconnections in a cylindrical phase space. Thus, in the last picture of Fig. 5.2,when propagation occurs the oscillators follow a trajectory from C+ to C−close to the set of heteroclinic orbits. The opposite solution (from C− toC+) is also possible and is odd symmetric with respect to the one displayedin Fig. 5.2.

The front velocity (c), near the onset, obeys a logarithmic law as theone shown in the previous chapter (Eq. (4.10)). The results for r = 8 maybe seen in Fig. 5.3 where the oscillation period can also be observed. Theabrupt increasing from zero of the velocity of the front can be appreciatedin Fig. 5.4.

Finally, it must be pointed out that at larger values of r the transitionto traveling fronts shows an interval of D with chaotic propagation, due toShil’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 stableup to a critical value of the coupling parameter D. For this value, a shortwavelength bifurcation [HPC95] arises (see Fig. 5.1). This new pattern cancoexist with front propagation; now we observed a zig-zag pattern in eachdomain at both sides of the wavefront.

2We have measured such distance observing that the distances between neighboringorbits really fall to zero for D = Dth


Figure 5.4: Front propagationspeed c as a function of the couplingstrength D for r = 8.

9 9.5 10 10.5D

0

0.2

0.4

0.6

0.8

c (c

ells

/t.u

.)

Figure 4

D. Pazo et al. (2000)

0 0.1 0.2 0.3 0.4 0.5−5

−4

−3

−2

−1

0

1

λ max

k/N

D = 11

D = 1

Figure 5D. Pazó et al. (2000)

Figure 5.5: Largest eigenvalues at the uniform states in C± as a function of the wavenumber k for D = 1, 2, ..., 11 and r = 20.

5.3 Chaotic region 121

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 ofthe fixed points C±. The linearized equation for the Fourier modes (χk) is

χk = S(k)χk, S(k) =

−σ σ − 2D sin2(πkN ) 0

1 −1 ∓c±c ±c −b

(5.3)

where c =√

b(r − 1). The real part of the most unstable eigenvalue isplotted in Fig. 5.5 as a function of k for different values of D. For k = 0the stability is independent of D and the fixed point is a focus (becausethe less attracting eigenvalues are complex conjugates). However, for alarge enough value of D shortest wavelength Fourier modes (k ≈ N/2) aregoverned by the real eigenvalue. The maximum value of this eigenvalue isat k = N/2 (shortest wavelength) and crosses zero at D = DC . At thispoint, the determinant of the Jacobian matrix S(k = N/2) is zero. Thiscondition 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( 1N2 )

depending on the exact form of the boundary condition, see the book ofBarnett [Bar96] for details). Above this critical value other modes becomeunstable and the dynamics turns out to be more complex than the oneshown in the top pictures of Fig. 5.1.

5.3 Chaotic region

At r = rH the single–unit fixed points C± become unstable through asubcritical Hopf bifurcation. The behavior of an isolated Lorenz oscillatorthen becomes chaotic for r ≥ rH . The system does not possess any stablestationary or periodic state.

For D = 0 the oscillators are uncoupled and the whole system is highlychaotic. The situation persists for low values of D. Finally, as the couplingincreases, the system tends to form clusters around C+ and C−. Thedevelopment of clusters occurs in a very smooth way, hence a precise limitcannot be assigned to such transition. Anyway, an analytical calculationevidences a tendency to cluster formation as r grows. The latter proceedsthrough calculating the eigenvalues of the system at the uniform state (all


D = 0

D = 10 D = 11D =9D = 8D = 7D = 6

D = 1 D = 2 D = 3 D = 4 D = 5

Figure 5.6: Different behaviors for several couplings (D) for r = 28 (the same initialcondition was imposed in all cases). The behavior of the system changes drastically attwo values of D. One point separates propagation and no-propagation of fronts, and islocated between D = 6 and D = 7. The second point separates the zone where shortwavelength bifurcation arises, and is located just above D = σ = 10. In the picturecorresponding to D = 11 the system suffers a short wavelength bifurcation with twodislocations (located at the center and at the right-hand-side).


Figure 5.7: Dynamics of oneoscillator for D = 8 and r = 25 ina reference framework where phasecan be readily computed.

−2 −1 0 1 2−3

−2

−1

0

1

2

3

(x2+y2)1/2− (2b(r−1))1/2 z

− (

r−1)

Figure 7D. 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 increasesenough, there is a Fourier mode that becomes stable. This mode is k = N/2(the shortest wavelength mode); the fact that this mode becomes stablemakes the dynamics of neighboring oscillators not to diverge so strongly asbefore; this can be considered as the beginning of cluster formation. Thevalue 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]) showsa dependence with D equivalent to the behavior of the Lyapunov exponentsat the fixed points, but the mode k = N/2 becomes stable at a larger valueof D. The system prefers to form clusters around the fixed points than toform an extended coherent structure (consisting for example of some cellsof the array partially synchronized) with the Lorenz chaotic attractor asthe basis for each cell dynamics. Since our clusters are defined as long-lasting ensemble of cells close to C+ or C−, transverse Lyapunov exponentsaround the chaotic synchronized state do not give any new information asour 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 travelingwaves (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 newclusters through the appearance of two counter-propagating fronts. Theformation of new clusters becomes more frequent as r grows. So, for r


just above rH the whole array stays for a long time with all the oscillatorsturning around one of the fixed points (either C+ or C−), until two counter-propagating fronts are able to emerge. Recall that independently of theinitial conditions, the onset of traveling fronts makes the system collapsearound C+ or C−. For this situation, the phase (φ) and amplitude (A)can be properly defined [PROK97] by means of the projection shown inFig. 5.7. We chose to define phase and amplitude as follows3:

φ = arctan

(√x2 + y2 −

√2b(r − 1)

z − (r − 1)

)(5.6)

A2 =(| x | −

√b(r − 1)

)2+(

| y | −√

b(r − 1))2

+ (z − (r − 1))2 . (5.7)

There are domains where the phase of neighboring oscillators is highlycorrelated (see Fig. 5.8). The limits of these domains are oscillators whoseamplitude is zero or close to zero, and could be considered as defects(in analogy with the Ginzburg-Landau equation phenomenology). Thesubcritical Hopf bifurcation observed in the Lorenz oscillator does notinvolve 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 multistabilitybetween a limit cycle and a fixed point [DP90, NMV96]. Therefore thetransition observed in the array cannot be considered simply as a discretizedsubcritical-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 smallerand eventually it is not possible to distinguish between non–propagatingand propagating regions.

With respect to the short wavelength bifurcation arising in the bistableregion, this structure remains stable beyond rH . The (Hopf) instabilityof C± does not affect the zig-zag structures bifurcated from the uniformsolutions. An analytical calculation of the stability of these patterns is not

3A stricter way to define φ and A would be to pass to the two-dimensional manifoldwhere the subcritical Hopf bifurcation takes place. In this 2-D manifold a Ginzburg-Landau 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 fromthe G-L equation. Furthermore, it would be intriguing how to cast a definition valid forC+ and C−. Hence, the important features of the dynamics will be readily provided byour operational definitions.


r = 25

r = 28r = 27

r = 26

Figure 5.8: The system for D = 8 and r = 25, 26, 27, 28. Each value of r shows threepictures, from left to right: x variable, amplitude and two-color-discretized phase. Aslong as r increases phase correlation diminishes because of the appearance of more frontsand defects (small amplitude, i.e. white color).


feasible because the solutions depend, both on r and D. We can howeverargue why the result obtained in Eq. (5.4) can be extended successfully tothe chaotic region (Fig. 5.1). In this region the mode k = 0 is unstable, andas 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 thanin the bistable one, because there are more than one unstable modes. Thedynamics of the system is determined by the behavior of each Fouriermode. Whereas k = 0 is unstable (independently of D), long wavelengthmodes stabilize as D increases. On the other hand, short wavelength modesbecome stable for low D but turn unstable again when D approaches DC .Therefore, the shortest wavelength modes govern the dynamics at highvalues of D. Due to the chaotic dynamics existing for r > rH the system iseventually close enough to one of the uniform solutions, and then, is shiftedto one of the “frozen” zig-zag patterns.

5.4 Discussion

This chapter studies an array of oscillators where wavefront solutionsarise. In the bistable region the transition has been demonstrated to beequivalent to that shown in the previous chapter for on-diagonal coupledoscillators. Accordingly, logarithmic profiles for the oscillation period andthe velocity of the front are obtained.

We have extended our study to the parameter range where the fixedpoints become unstable (r > rH). For D < DC , the system exhibits twotypes of spatio-temporal chaos depending on whether exists or not frontpropagation. In the propagating region we find two characteristic processes:spontaneous creation of counter-propagating fronts and front reversal. Theboundary that separated propagating from non-propagating regions in thebistable case separates now these two kinds of spatio-temporal chaos.

We have also found that over a certain coupling the system undergoesa short wavelength bifurcation. This kind of bifurcation is observed indiscrete systems only (as well as the onset of propagating fronts by theroute explained in this thesis). The preponderance of the short wavelengthmodes at large D suggests the lack of a continuum limit for the kind ofcoupling studied in this chapter. We have also observed that this shortwavelength pattern inhibits the spatio-temporal chaos beyond rH , givingrise to an ordered pattern.

All the necessary conditions to achieve the behaviors described aboveare still a matter of future research.

Chapter 6

Conclusions and Outlook

Interaction between oscillators is an issue of interest for diversedisciplines ranging from biology to engineering. Throughout this workseveral techniques have been used in order to elucidate different phenomenaoccurring in this kind of systems: synchronization, transition to high-dimensional chaos, wavefronts, ...

In the first part of this thesis the problem of the onset of phase and lagsynchronization between chaotic oscillators has been addressed. To get adeeper insight on the transitions occurring in the system, we have focusedon the invariants inside the attractor. The stabilization of the unstableperiodic orbits allowed us to grasp the profound changes occurring in theinvariant set when transiting between different states. In particular, theobservation of intermittent exotic lag configurations just before perfect lagsynchronization settles, was clarified. It is explained by the approach ofthe trajectory to an UPO, that was born at an out-of-phase locking in thetransition to phase synchronization.

The second chapter deals with a transition to high-dimensional chaosoccurring in a ring of three unidirectionally coupled Lorenz oscillators.Although the system exhibits three-frequency quasiperiodicity, it is a globalbifurcation between cycles which creates the chaotic set; and, accordingly,a chaotic transient is observed. The mean chaotic transient divergesfollowing a power law when approaching the point where the chaotic setbecomes attracting through a boundary crisis. In this transition, thereexists a double heteroclinic connection to a pair of unstable three-frequencyquasiperiodic tori. Finally, after a small parameter range of coexistencebetween chaos and quasiperiodicity, the mentioned pair of unstable 3-tori collide simultaneously with the two stable 3-tori through saddle-

128 Conclusions and Outlook

node bifurcations. The (remaining) chaotic attractor has an informationdimension, obtained by the Kaplan-Yorke formula, above four. Finally,the robustness and generality of the route to high-dimensional chaos isdiscussed along one section.

The largest part of this thesis is devoted to the onset of traveling frontsin discrete bistable media. We consider reaction-diffusion systems wherethe local dynamics is bistable and symmetric, so there is not a preferredstate. We have observed two characteristic routes leading to travelingfronts. In the first one, the static front becomes unstable through a Hopfbifurcation giving rise to a oscillating front. Finally, a global bifurcationconsisting on two homoclinic loops creates two counterpropagating travelingsolutions. The velocity of these traveling solutions obeys a logarithmic lawwith regard to the distance to the critical point, which is visible as anabrupt increase from zero at that point. We have also observed that, insome circumstances, the double homoclinic loop connects a saddle-focusequilibrium. In this case it is possible to find a much more convolutedtransition between oscillating and traveling front, such that at some valuesof the coupling the front propagates in an irregular manner which is amanifestation of the underlying Shi’lnikov-type chaos. For the discreteFitzHugh-Nagumo model the transition is different. Instead of a regimewith oscillating front, there exists a region of coexistence between staticand traveling fronts. In this route –as well as in the route that involvesan oscillating front– the intermediate region shrinks to zero in the infinitecoupling limit, that corresponds to the continuum limit. We have seen thatthe point located at infinity, where the transition occurs, is a double zero-eigenvalue point and therefore the bifurcation lines emerging from thatpoint should be predicted by the normal form of such codimension-twobifurcation. Since our case is geometrically somewhat special, we have notfound in the literature the corresponding normal form. Nonetheless, thenormal form for the double-zero eigenvalue with symmetry (in the plane)presents two characteristic cases that present strong analogies with thetransitions we have observed in discrete bistable media. Therefore weconjecture that only two possible scenarios are expected to be found inthe infinite coupling limit when considering the transition to propagationin discrete bistable media.

The last part of this thesis studies the transitions occurring in an arrayof non-diagonal coupled Lorenz oscillators. In this case we consideredbistable as well as chaotic Lorenz oscillators. In the first case, thereexist the transition to traveling front through an oscillating front; but

129

furthermore, for large coupling the system undergoes a short wavelengthbifurcation. In the chaotic case, the transition between oscillating andtraveling fronts becomes a well-differentiated transition between two typesof spatio-temporal chaos. Also, the short wavelength bifurcation inhibitsspatio-temporal chaos giving rise to an ordered pattern.

Outlook

Regarding the onset of chaotic phase synchronization, it would bedesirable to develop an efficient technique to find unstable tori (somethinglike the ones we have used to find unstable periodic orbits). This wouldmake possible to get a finer description of the onset of chaotic phasesynchronization.

Most theoretical works are interested in the rigorous proof of theexistence (or not) of a horseshoe, under a given global bifurcation. Onthe other hand, other scientists do not try to give a geometric view oftheir transitions to chaos and quite often the computation of the Lyapunovexponents is their only interest. It is to be expected that a strongercooperation between pure mathematicians and more applied scientists willallow to elucidate the general mechanisms leading to high-dimensionalchaos.

In what concerns the transition to traveling front in bistable media, itis 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 shouldbe established.

Appendix A

The FitzHugh-Nagumo cell:a case of a Takens-Bogdanovcodimension-two point

The aim of this Appendix is to describe the parameter space of a cellof the FitzHugh-Nagumo type. This system contains two variables (u, v)that obey the following differential equations:

u = u− u3 − v (A.1)

v = ε(u− a1v − a0) (A.2)

The parameter ε measures the different time scales of both variablesand, like in Chapter 4, we restrict here to the case a0 = 0, what makes oursystem symmetric under reflection (u, v) → (−u,−v).

The trivial solution (0, 0) undergoes a pitchfork bifurcation at a1 = 1giving rise for a1 > 1 to two symmetry related solutions (u±, v±) =

(±√

a1−1a1

,± 1a1

√a1−1

a1). For a system with two variables the stability of

a fixed point may be studied considering the values of trace (tr) anddeterminant (∆) of the Jacobian matrix J at that point. In our systemJ takes the form:

J =(

1− 3u2 −1ε −a1ε

)(A.3)

132 Appendix A

If the determinant is negative the fixed point is a saddle with twoeigenvalues λu > 0 and λs < 0 (notice that the trace is equal to λu +λs andtherefore depending on the sign of the trace either the stable or the unstableeigenvalue 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, thesolution at the origin undergoes a pitchfork bifurcation at this line whichmakes 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 bifurcationlines 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 existsat the left of this line and three at its right. The red line (G) is the locus of the gluingbifurcation between two unstable cycles. The green line (SN) corresponds to a saddle-node bifurcation of cycles. The tr±=0, G and SN lines are born at the codimension-twopoint located at (a1, ε)=(1,1). Between tr±=0 and SN, a stable limit cycle and twosymmetry related stable stationary solutions coexist.

A positive determinant offers a larger amount of characteristic dynamicsdepending on the values of the trace. Thus, if the trace is positive (resp.negative) the fixed point is unstable (resp. stable). If ∆ < tr2/4 theequilibrium is a node whereas if ∆ > tr2/4 it is a focus. Hence, if ∆ > 0the 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 bifurcationprovided the system to have nonlinearities (otherwise at this line the systemwould 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 andblue colors respectively.

It is easy to check that the origin has two degenerate vanishingeigenvalues at (a1, ε) = (1, 1). This is a (local) codimension-two pointthat comprises the Takens-Bogdanov bifurcation; in fact, the pitchfork andthe 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, thatinvolves a symmetry under reflection, is to our knowledge only considered inthe book of Guckenheimer and Holmes [GH83]. Every system undergoinga Takens-Bogdanov bifurcation may be reduced, by successive changesof variables, to its corresponding normal form (this is a mathematicallyconvoluted although somewhat mechanical at the same time). Dependingon the system, the TB with symmetry is found to have two differentrealizations (allowing time reversal). We observe one of both here, andit is characterized by three bifurcation lines (see Fig. A.1) that emanatefrom 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 gluingbifurcation between unstable cycles occurs.

Finally, it is to note that at (a1, ε) = (1.5, 0). there exist anothercodimension two point. This point is quite special because it is thesimultaneous occurrence of a TB bifurcation for the two symmetry relatedfixed points (u±, v±).

1If a0 were considered as a third parameter we could say that we are studying acodimension-three point, and therefore letting a0 to vary around zero would give a picturemuch 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. Pazo, I. P. Marino, V. Perez-Villar, and V. Perez-Munuzuri,“Transition chaotic phase synchronization through random phasejumps”, Int. J. Bif. and Chaos 10, 2533-2539 (2000).

II. D. Pazo, N. Montejo, and V. Perez-Munuzuri, “Wave fronts andspatiotemporal chaos in an array of coupled Lorenz oscillators”, Phys.Rev. E 63, 066206(1-7) (2001).

III. D. Pazo, E. Sanchez, and M. A. Matıas, “Transition to high-dimensional chaos through quasiperiodic motion”, Int. J. Bif. andChaos 11, 2683-2688 (2001).

IV. D. Pazo and V. Perez-Munuzuri, “Onset of wavefronts in a discretebistable medium”, Phys. Rev. E 64, 065203[R] (2001).

V. D. Pazo, M. Zaks, and J. Kurths, “Role of unstable periodic orbitsin phase and lag synchronization between coupled chaotic oscillators”,Chaos 13, 309-318 (2003).

VI. D. Pazo and V. Perez-Munuzuri, “Traveling fronts in an array ofsymmetric bistable units”, (submitted to Chaos).

VII. D. Pazo and M. A. Matıas, “Transition to high-dimensional chaosthrough a global bifurcation” (submitted to Phys. Rev. Lett.).

VIII. E. Sanchez, D. Pazo, and M. A. Matıas, “Experimental study of thetransitions between synchronous chaos and a periodic rotating wave”(submitted to Phys. Rev. E).

Index

ansatz, 61

Birman-Williams theorem, 73bistability, 86boundary crisis, 76

chaotic rotating wave, 54chaotic transients, 76cluster formation, 121continuum limit, 109cylindrical coordinates, 91

dynamo model, 106

eyelet intermittency, 37

FitzHugh-Nagumo model, 86,107, 131

Ginzburg-Landau equation, 124gluing bifurcation, 92, 107, 133

sadle focus, 97

heteroclinic explosion, 72high-dimensional chaos, 52hyperchaos, 52

Kaplan-Yorke conjecture, 52

logarithmic divergence, 93logistic map, 31Lorenz oscillator, 87Lyapunov spectrum, 56

Newhouse-Ruelle-TakensTheorem, 65

pendulum, 92period-doubling cascade, 101periodic rotating wave, 55, 62phase locking, 33

Rossler oscillator, 29reaction-diffusion equation, 85, 87return map, 77

saddle index, 99Shil’nikov wiggle, 99short wavelength bifurcation, 119,

124slave locking, 71synchronization, 27

lag, 28, 42phase, 28, 36

Takens-Bogdanov bifurcation,110, 133

UPO, 28

coupled oscillators: chaotic synchronization, high-dimensional

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