Heteroclinic Dynamics of Localized Frequency Synchrony: Stability of Heteroclinic Cycles and Networks

TL;DR

This paper analyzes the stability of heteroclinic cycles and networks in coupled oscillator populations, revealing how coupling parameters influence sequential frequency transitions relevant for neural oscillator functionality.

Contribution

It provides explicit stability results and indices for heteroclinic cycles and networks in coupled phase oscillators, advancing understanding of their dynamics.

Findings

Stability indices depend on coupling parameters.

Heteroclinic cycles facilitate sequential frequency switching.

Dynamics are relevant for neural oscillator functions.

Abstract

In the first part of this paper, we showed that three coupled populations of identical phase oscillators give rise to heteroclinic cycles between invariant sets where populations show distinct frequencies. Here, we now give explicit stability results for these heteroclinic cycles for populations consisting of two oscillators each. In systems with four coupled phase oscillator populations, different heteroclinic cycles can form a heteroclinic network. While such networks cannot be asymptotically stable, the local attraction properties of each cycle in the network can be quantified by stability indices. We calculate these stability indices in terms of the coupling parameters between oscillator populations. Hence, our results elucidate how oscillator coupling influences sequential transitions along a heteroclinic network where individual oscillator populations switch sequentially between a high and a low frequency regime; such dynamics appear relevant for the functionality of neural oscillators.