Bifurcation From Networks of Unstable Attractors to Heteroclinic Switching

TL;DR

This paper introduces a novel bifurcation in dynamical systems where networks of unstable attractors transition into heteroclinic cycles, demonstrated through pulse-coupled oscillator networks with delayed interactions.

Contribution

It reveals a new bifurcation mechanism where unstable attractors become heteroclinically connected saddle states as system non-invertibility is removed.

Findings

Unstable attractors can form heteroclinic cycles in oscillator networks.

The transition is analytically and numerically characterized.

This represents a new bifurcation type in dynamical systems.

Abstract

We present a dynamical system that naturally exhibits two unstable attractors that are completely enclosed by each others basin volume. This counter-intuitive phenomenon occurs in networks of pulse-coupled oscillators with delayed interactions. We analytically and numerically investigate this phenomenon and clarify the mechanism underlying it: Upon continuously removing the non-invertibility of the system, the set of two unstable attractors becomes a set of two non-attracting saddle states that are heteroclinically connected to each other. This transition from a network of unstable attractors to a heteroclinic cycle constitutes a new type of bifurcation in dynamical systems.