How well can one resolve the state space of a chaotic map?

TL;DR

This paper investigates the limits of resolving the state space of chaotic maps under noise, proposing a method to determine the finest partition by analyzing the Fokker-Planck operator's eigenfunctions, with numerical validation.

Contribution

It introduces a novel approach to find the optimal state space partition for noisy chaotic systems using eigenfunctions of the adjoint Fokker-Planck operator.

Findings

The method successfully identifies the finest attainable partition in a one-dimensional chaotic system.

Eigenfunction overlaps serve as a criterion for optimal partitioning.

Numerical tests support the hypothesis that this approach captures the resolution limits imposed by noise.

Abstract

All physical systems are affected by some noise that limits the resolution that can be attained in partitioning their state space. For chaotic, locally hyperbolic flows, this resolution depends on the interplay of the local stretching/contraction and the smearing due to noise. We propose to determine the `finest attainable' partition for a given hyperbolic dynamical system and a given weak additive white noise, by computing the local eigenfunctions of the adjoint Fokker-Planck operator along each periodic point, and using overlaps of their widths as the criterion for an optimal partition. The Fokker-Planck evolution is then represented by a finite transition graph, whose spectral determinant yields time averages of dynamical observables. Numerical tests of such `optimal partition' of a one-dimensional repeller support our hypothesis.