Unstable Attractors: Existence and Robustness in Networks of Oscillators With Delayed Pulse Coupling

摘要

We consider unstable attractors; Milnor attractors $A$ such that, for someneighbourhood $U$ of $A$, almost all initial conditions leave $U$. Previousresearch strongly suggests that unstable attractors exist and even occurrobustly (i.e. for open sets of parameter values) in a system modellingbiological phenomena, namely in globally coupled oscillators with delayed pulseinteractions. In the first part of this paper we give a rigorous definition of unstableattractors for general dynamical systems. We classify unstable attractors intotwo types, depending on whether or not there is a neighbourhood of theattractor that intersects the basin in a set of positive measure. We giveexamples of both types of unstable attractor; these examples havenon-invertible dynamics that collapse certain open sets onto stable manifoldsof saddle orbits. In the second part we give the first rigorous demonstration of existence androbust occurrence of unstable attractors in a network of oscillators withdelayed pulse coupling. Although such systems are technically hybrid systems ofdelay differential equations with discontinuous `firing' events, we show thattheir dynamics reduces to a finite dimensional hybrid system system after afinite time and hence we can discuss Milnor attractors for this reduced finitedimensional system. We prove that for an open set of phase resetting functionsthere are saddle periodic orbits that are unstable attractors.