An Isaac Newton Institute Workshop

Theory and Applications of Coupled Cell Networks

High-dimensional chaos versus synchronization in the Kuramoto model

Author: Yuri Maistrenko (Forschungszentrum Juelich )

Abstract

A complex high-dimensional chaotic behavior is found in the finite-dimensional Kuramoto model of globally coupled phase oscillators. This type of chaos extends over small and intermediate coupling strength up to the synchronization transition, and is characterized by half of the spectrum of Lyapunov exponents being positive and the Lyapunov dimension equaling almost the total system dimension. We analyze the case of uniform distribution of the natural frequencies and find that the intensity of the phase chaos, as given by the maximal Lyapunov exponent, decays quadratically with coupling strength and inverse proportionally to ensemble size. Intriguingly, strongest phase chaos occurs for intermediate-size ensembles.

We argue that phase chaos is caused by intrinsic nonlinear phase interactions and is a common property of networks of oscillators of very different nature, such as phase oscillators with different coupling matrix and different frequency distributions, e.g. Gaussian. The phenomenon is also inherent in networks of limit-cycle oscillators and chaotic oscillators, e.g., Roessler systems. In the case of coupled chaotic oscillators, phase chaos manifests itself in the appearance of additional chaotic "dimensions", where additional positive LEs (with respect to those of individual oscillators) emerge.