From chimeras to extensive chaos in networks of heterogeneous Kuramoto oscillator populations

TL;DR

This paper analyzes how networks of heterogeneous Kuramoto oscillators can exhibit complex behaviors like chimera states and extensive chaos, providing a unified framework for understanding these phenomena across different network topologies.

Contribution

It introduces an analytical framework using Ott-Antonsen equations and stability analysis to explain the emergence of chimera states and extensive chaos in large oscillator networks.

Findings

Instability in two-population models leads to chaos in large networks.

Homogeneous solutions can become unstable, resulting in complex dynamics.

Extensive space-time chaos scales linearly with system size.

Abstract

Populations of coupled oscillators can exhibit a wide range of complex dynamical behavior, from complete synchronization to chimera and chaotic states. We can thus expect complex dynamics to arise in networks of such populations. Here we analyze the dynamics of networks of populations of heterogeneous mean-field coupled Kuramoto-Sakaguchi oscillators, and show that the instability that leads to chimera states in a simple two-population model also leads to extensive chaos in large networks of coupled populations. Formally, the system consists of a complex network of oscillator populations whose mesoscopic behavior evolves according to the Ott-Antonsen equations. By considering identical parameters across populations, the system contains a manifold of homogeneous solutions where all populations behave identically. Stability analysis of these homogeneous states provided by the master stability function formalism shows that non-trivial dynamics might emerge on a wide region of the parameter space for arbitrary network topologies. As examples, we first revisit the two-population case, and provide a complete bifurcation diagram. Then, we investigate the emergent dynamics in large ring and Erd\"os-R\'enyi networks. In both cases, transverse instabilities lead to extensive space-time chaos, i.e., irregular regimes whose complexity scales linearly with the system size. Our work provides a unified analytical framework to understand the emergent dynamics of networks of oscillator populations, from chimera states to robust high-dimensional chaos.