Chimera states through invariant manifold theory

TL;DR

This paper proves the existence and stability of chimera states in a network of coupled oscillators using invariant manifold theory, showing their persistence and stability under various coupling conditions.

Contribution

It introduces a novel analytical framework combining invariant manifold theory and averaging techniques to establish chimera states in symmetric star networks.

Findings

Chimera states can be metastable or asymptotically stable.

Persistence of chimera states depends on intra-star coupling strength.

Sparse intra-star coupling leads to asymptotic stability.

Abstract

We establish the existence of chimera states, simultaneously supporting synchronous and asynchronous dynamics, in a network consisting of two symmetrically linked star subnetworks consisting of identical oscillators with shear and Kuramoto--Sakaguchi coupling. We show that the chimera states may be metastable or asymptotically stable. If the intra-star coupling strength is of order $\varepsilon$, the chimera states persist on time scales at least of order $1/\varepsilon$ in general, and on time-scales at least of order $1/\varepsilon^2$ if the intra-star coupling is of Kuramoto--Sakaguchi type. If the intra-star coupling configuration is sparse, the chimeras are asymptotically stable. The analysis relies on a combination of dimensional reduction using a M\"obius symmetry group and techniques from averaging theory and normal hyperbolicity.