Generalized splay states in phase oscillator networks

TL;DR

This paper introduces generalized m-splay states in phase oscillator networks, providing explicit stability conditions that are simple, observable-based, and applicable to large and complex networks, including those with inertia or adaptive coupling.

Contribution

It defines generalized splay states, derives explicit stability criteria in terms of observables, and extends the analysis to networks with inertia and adaptive coupling.

Findings

Explicit stability conditions for splay states in terms of order parameters.

Conditions are simple and applicable to large networks.

Results extended to oscillators with inertia and adaptive coupling.

Abstract

Networks of coupled phase oscillators play an important role in the analysis of emergent collective phenomena. In this article, we introduce generalized $m$-splay states constituting a special subclass of phase-locked states with vanishing $m$th order parameter. Such states typically manifest incoherent dynamics, and they often create high-dimensional families of solutions (splay manifolds). For a general class of phase oscillator networks, we provide explicit linear stability conditions for splay states and exemplify our results with the well-known Kuramoto-Sakaguchi model. Importantly, our stability conditions are expressed in terms of just a few observables such as the order parameter or the trace of the Jacobian. As a result, these conditions are simple and applicable to networks of arbitrary size. We generalize our findings to phase oscillators with inertia and adaptively coupled phase oscillator models.