Multiscale Dynamics in Communities of Phase Oscillators

TL;DR

This paper studies the complex multiscale dynamics of coupled phase oscillator communities with mixed attractive and repulsive interactions, using dimensional reduction and slow-fast analysis to understand stability and group behavior.

Contribution

It introduces a reduced model for multigroup oscillator systems with mixed coupling, analyzing stability and dynamics in symmetric and asymmetric cases.

Findings

Neutrally stable manifold of equilibria identified for symmetric systems

Unstable nature of other equilibria established

Slow evolution equations derived for asymmetric deviations

Abstract

We investigate the dynamics of systems of many coupled phase oscillators with het- erogeneous frequencies. We suppose that the oscillators occur in M groups. Each oscillator is connected to other oscillators in its group with "attractive" coupling, such that the coupling promotes synchronization within the group. The coupling between oscillators in different groups is "repulsive"; i.e., their oscillation phases repel. To address this problem, we reduce the governing equations to a lower-dimensional form via the ansatz of Ott and Antonsen . We first consider the symmetric case where all group parameters are the same, and the attractive and repulsive coupling are also the same for each of the M groups. We find a manifold L of neutrally stable equilibria, and we show that all other equilibria are unstable. For M \geq 3, L has dimension M - 2, and for M = 2 it has dimension 1. To address the general asymmetric case, we then introduce small deviations from symmetry in the group and coupling param- eters. Doing a slow/fast timescale analysis, we obtain slow time evolution equations for the motion of the M groups on the manifold L. We use these equations to study the dynamics of the groups and compare the results with numerical simulations.