Mapping densities in a noisy state space

TL;DR

This paper investigates how weak noise affects the structure of chaotic state spaces, introducing local invariants to partition the space based on the interplay of noise and deterministic dynamics.

Contribution

It presents a novel state space partitioning algorithm utilizing asymptotic eigenfunctions for noisy dynamics in hyperbolic systems.

Findings

Weak noise smooths fractal structures in chaotic spaces

Introduces a maximum resolution limit due to noise

Develops a new partitioning method based on local invariants

Abstract

Weak noise smooths out fractals in a chaotic state space and introduces a maximum attainable resolution to its structure. The balance of noise and deterministic stretching/contraction in each neighborhood introduces local invariants of the dynamics that can be used to partition the state space. We study the local discrete-time evolution of a density in a two-dimensional hyperbolic state space, and use the asymptotic eigenfunctions for the noisy dynamics to formulate a new state space partition algorithm.