4 - Effect of noise on discrete dynamical systems with multiple attractors

Summary

Introduction

The isolation of a physical system from its environment is a frequent approximation which leads to the idea of a deterministic conservative system. All physical systems are, however, coupled to the outside world, giving rise to the related phenomena of fluctuations and dissipation. In a dissipative dynamical system volumes in phase space contract onto attractors. Fluctuations allow the system to escape from attractors, rendering all attractors metastable. The long time behavior of a noisy dissipative system is thus intermittent, consisting of motion near the various attractors of the system alternating with transitions between attractors. In the limit of small noise, the times spent on the attractors become longer, and the transitions rarer. In this chapter we describe some recent work on stochastic dynamical systems with multiple attractors, with particular emphasis on systems possessing multiple limit cycles.

Although there has been much interest in noisy iterated maps, to our knowledge no one has actually derived a noisy map from a stochastic differential equation. In Section 4.2 we discuss some of the considerations that must go into any such derivation. As an example, we examine how noise affects a driven oscillator in both the phase-locked and unlocked regimes. The details of the noisy dynamics play an important role in determining how noise must affect the resulting map. Section 4.3 contains a formal derivation of a noisy iterated map from a linearized stochastic differential equation.