Optimal resolution of the state space of a chaotic flow in presence of noise

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 a global averages over stochastic flow. What is novel here is that we show how to compute the locally optimal partition, for a given dynamical system and given noise, in terms of local eigenfunctions of the forward-backward actions 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 in the non-wandering set (i.e. near the unstable periodic orbits) of the deterministic system, and test our hypothesis on unimodal maps.