Knowing when to stop: how noise frees us from determinism

Abstract

Deterministic chaotic dynamics presumes that the state space can be partitioned arbitrarily finely. In a physical system, the inevitable presence of some noise sets a finite limit to the finest possible resolution that can be attained. Much previous research deals with what this attainable resolution might be, all of it based on global averages over a stochastic flow. We show how to compute the locally optimal partition, for a given dynamical system and given noise, in terms of local eigenfunctions of the Fokker-Planck operator and its adjoint. We first analyze the interplay of the deterministic dynamics with the noise in the neighborhood of a periodic orbit of a map, by using a discretized version of Fokker-Planck formalism. Then we propose a method to determine the 'optimal resolution' of the state space, based on solving Fokker-Planck's equation locally, on sets of unstable periodic orbits of the deterministic system. We test our hypothesis on unimodal maps.