Bifurcation From Networks of Unstable Attractors to Heteroclinic Switching

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.