Weak chimeras in minimal networks of coupled phase oscillators

TL;DR

This paper introduces the concept of weak chimeras in minimal networks of identical phase oscillators, demonstrating their existence in specific network configurations and analyzing the influence of Kuramoto-Sakaguchi coupling on their stability.

Contribution

It defines weak chimeras as invariant sets with partial frequency synchronization and shows they cannot occur in globally coupled or very small networks, providing explicit examples.

Findings

Weak chimeras exist in networks of 4, 6, and 10 oscillators.

Kuramoto-Sakaguchi coupling can produce neutrally stable weak chimeras.

Weak chimeras are absent in globally coupled or very small networks.

Abstract

We suggest a definition for a type of chimera state that appears in networks of indistinguishable phase oscillators. Defining a "weak chimera" as a type of invariant set showing partial frequency synchronization, we show that this means they cannot appear in phase oscillator networks that are either globally coupled or too small. We exhibit various networks of four, six and ten indistinguishable oscillators where weak chimeras exist with various dynamics and stabilities. We examine the role of Kuramoto-Sakaguchi coupling in giving degenerate (neutrally stable) families of weak chimera states in these example networks.