Knowing when to stop: how noise frees us from determinism

TL;DR

This paper introduces a method to determine the optimal resolution of state space in noisy dynamical systems by computing local eigenfunctions of the Fokker-Planck operator, balancing deterministic chaos and noise effects.

Contribution

It presents a novel approach to compute the locally optimal partition of state space using local Fokker-Planck eigenfunctions, advancing understanding of noise influence on chaos.

Findings

Method effectively computes local eigenfunctions for given noise levels.

Optimal resolution varies near unstable periodic orbits.

Tested on unimodal maps with promising results.

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.