Unstable attractors induce perpetual synchronization and desynchronization

TL;DR

This paper demonstrates that unstable attractors in a network of pulse-coupled oscillators cause perpetual switching between synchronized and desynchronized states, challenging the usual stability assumption in nonlinear systems.

Contribution

It introduces a model where unstable attractors naturally occur and dominate dynamics, revealing their role in perpetual synchronization-desynchronization cycles.

Findings

Unstable attractors lead to ongoing switching between states.

Weak noise induces desynchronization from synchronized clusters.

Unstable attractors dominate large network dynamics.

Abstract

Common experience suggests that attracting invariant sets in nonlinear dynamical systems are generally stable. Contrary to this intuition, we present a dynamical system, a network of pulse-coupled oscillators, in which \textit{unstable attractors} arise naturally. From random initial conditions, groups of synchronized oscillators (clusters) are formed that send pulses alternately, resulting in a periodic dynamics of the network. Under the influence of arbitrarily weak noise, this synchronization is followed by a desynchronization of clusters, a phenomenon induced by attractors that are unstable. Perpetual synchronization and desynchronization lead to a switching among attractors. This is explained by the geometrical fact, that these unstable attractors are surrounded by basins of attraction of other attractors, whereas the full measure of their own basin is located remote from the attractor. Unstable attractors do not only exist in these systems, but moreover dominate the dynamics for large networks and a wide range of parameters.