Phase synchronization of chaotic systems with small phase diffusion.

The geometric theory of phase locking between periodic oscillators is extended to phase coherent chaotic systems. This approach explains the qualitative features of phase locked chaotic systems and provides an analytical tool for a quantitative description of the phase locked states. Moreover, this geometric viewpoint allows us to identify obstructions to phase locking even in systems with negligible phase diffusion, and to provide sufficient conditions for phase locking to occur. We apply these techniques to the Rössler system and a phase coherent electronic circuit and find that numerical results and experiments agree well with theoretical predictions.

[1]  Tong Kun Lim,et al.  PHASE JUMPS NEAR A PHASE SYNCHRONIZATION TRANSITION IN SYSTEMS OF TWO COUPLED CHAOTIC OSCILLATORS , 1998 .

[2]  Mark Hess,et al.  TRANSITION TO PHASE SYNCHRONIZATION OF CHAOS , 1998 .

[3]  J. Kurths,et al.  Phase Synchronization of Chaotic Oscillators by External Driving , 1997 .

[4]  J. Kurths,et al.  Phase synchronization of chaotic oscillations in terms of periodic orbits. , 1997, Chaos.

[5]  R. Llinás,et al.  Experimentally determined chaotic phase synchronization in a neuronal system. , 1998, Proceedings of the National Academy of Sciences of the United States of America.

[6]  Richard J. Fitzgerald.  Phase Synchronization May Reveal Communication Pathways in Brain Activity , 1999 .

[7]  Bernd Blasius,et al.  Complex dynamics and phase synchronization in spatially extended ecological systems , 1999, Nature.

[8]  A. Selverston,et al.  Synchronous Behavior of Two Coupled Biological Neurons , 1998, chao-dyn/9811010.

[9]  J Kurths,et al.  Phase synchronization in the forced Lorenz system. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[10]  K. Wong,et al.  Properties of phase locking with weak phase-coherent attractors , 2001 .

[11]  R. Jensen.  Synchronization of randomly driven nonlinear oscillators , 1998 .

[12]  T. Carroll.  A simple circuit for demonstrating regular and synchronized chaos , 1995 .

[13]  Polina S. Landa,et al.  Synchronizing the chaotic oscillations by external force , 1991 .

[14]  Arkady Pikovsky,et al.  On the interaction of strange attractors , 1984 .

[15]  G. Ermentrout,et al.  Oscillator death in systems of coupled neural oscillators , 1990 .

[16]  Colin Sparrow,et al.  Chaos in a three-dimensional single loop feedback system with a piecewice linear feedbackk function , 1981 .

[17]  E. Stone,et al.  Frequency entrainment of a phase coherent attractor , 1992 .

[18]  T. Carroll,et al.  VOLUME-PRESERVING AND VOLUME-EXPANDING SYNCHRONIZED CHAOTIC SYSTEMS , 1997 .

[19]  O. Rössler.  An equation for continuous chaos , 1976 .

[20]  Kurths,et al.  Phase synchronization of chaotic oscillators. , 1996, Physical review letters.

[21]  Neil Fenichel.  Persistence and Smoothness of Invariant Manifolds for Flows , 1971 .

[22]  S Boccaletti,et al.  Unifying framework for synchronization of coupled dynamical systems. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  Marco Monti,et al.  Phase Synchronization of Chaos in a plasma discharge tube , 2000, Int. J. Bifurc. Chaos.

[24]  Krešimir Josić,et al.  Synchronization of chaotic systems and invariant manifolds , 2000 .

[25]  Ying-Cheng Lai,et al.  Noise scaling of phase synchronization of chaos , 2000 .

[26]  Jürgen Kurths,et al.  Detection of n:m Phase Locking from Noisy Data: Application to Magnetoencephalography , 1998 .

[27]  Ying-Cheng Lai,et al.  PHASE CHARACTERIZATION OF CHAOS , 1997 .