A rigorous mathematical treatment of chaotic phase synchronization is still lacking, although it has been observed in many numerical and experimental studies. In this article we address the extension of results on phase synchronization in periodic oscillators to systems with phase coherent chaotic attractors with small phase diffusion. As models of such systems we consider special flows over diffeomorphisms in which the neutral direction is periodically perturbed. A generalization of the Averaging Theorem for periodic systems is used to extend Kuramoto’s geometric theory of phase locking in periodically forced limit cycle oscillators to this class of systems. This approach results in reduced equations describing the dynamics of the phase difference between drive and response systems over long time intervals. The reduced equations are used to illustrate how the structure of a chaotic attractor is important in its response to a periodic perturbation, and to conclude that chaotic phase coherent systems may not always be treated as noisy periodic oscillators in this context. Although this approach is strictly justified for periodic perturbations affecting only the phase variable of a chaotic oscillator, we argue that these ideas are applicable much more generally.
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March 2003
Research Article|
February 21 2003
A geometric theory of chaotic phase synchronization
Margaret Beck;
Margaret Beck
Department of Mathematics and Statistics and Center for BioDynamics, Boston University, Boston, Massachusetts 02215
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Krešimir Josić
Krešimir Josić
Department of Mathematics and Statistics and Center for BioDynamics, Boston University, Boston, Massachusetts 02215
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Chaos 13, 247–258 (2003)
Article history
Received:
April 11 2002
Accepted:
July 17 2002
Citation
Margaret Beck, Krešimir Josić; A geometric theory of chaotic phase synchronization. Chaos 1 March 2003; 13 (1): 247–258. https://doi.org/10.1063/1.1505812
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