A global bifurcation organizing rhythmic activity in a coupled network

TL;DR

This paper investigates how a heteroclinic bifurcation in coupled phase oscillators explains a phase transition in collective rhythmic activity, using mathematical reduction techniques to identify changes in limit cycle topology.

Contribution

It identifies a heteroclinic bifurcation as the organizing principle behind the transition in collective dynamics in a coupled oscillator system.

Findings

Heteroclinic bifurcation separates two families of limit cycles.

Both limit cycle families are stable in the model.

The bifurcation explains the qualitative change in rhythmic activity.

Abstract

We study a system of coupled phase oscillators near a saddle-node on an invariant circle bifurcation and driven by random intrinsic frequencies. Under the variation of control parameters, the system undergoes a phase transition changing the qualitative properties of collective dynamics. Using the Ott-Antonsen reduction and geometric techniques for ordinary differential equations, we identify a heteroclinic bifurcation in a family of vector fields on a cylinder, which explains the change in collective dynamics. Specifically, we show that the heteroclinic bifurcation separates two topologically distinct families of limit cycles: contractible limit cycles before the bifurcation from noncontractibile ones after the bifurcation. Both families are stable for the model at hand.