On the Stability of a Chain of Phase Oscillators

TL;DR

This paper analyzes the stability of traveling wave states in a chain of phase oscillators with asymmetric coupling, revealing that only near in-phase states are stable and identifying conditions for phase slipping.

Contribution

It provides a stability analysis of traveling wave solutions in oscillator chains with asymmetric coupling, highlighting the stability of in-phase states and conditions for phase slipping.

Findings

Only in-phase synchronization is locally stable.

Number of unstable dimensions relates to oscillators near phase difference π.

Phase slipping occurs despite Lyapunov functional presence.

Abstract

We study a chain of $N+1$ phase oscillators with asymmetric but uniform coupling. This type of chain possesses $2^{N}$ ways to synchronize in so-called travelling wave states, i.e. states where the phases of the single oscillators are in relative equilibrium. We show that the number of unstable dimensions of a travelling wave equals the number of oscillators with relative phase close to $\pi$. This implies that only the relative equilibrium corresponding to approximate in-phase synchronization is locally stable. Despite the presence of a Lyapunov-type functional periodic or chaotic phase slipping occurs. For chains of length 3 and 4 we locate the region in parameter space where rotations (corresponding to phase slipping) are present.