Neighborhoods of periodic orbits and the stationary distribution of a noisy chaotic system

TL;DR

This paper derives explicit formulas for neighborhoods of hyperbolic periodic orbits in noisy chaotic systems, providing a basis for the stationary distribution essential for long-term statistical analysis.

Contribution

It introduces a method to compute neighborhoods of periodic orbits in noisy chaotic systems using explicit formulas, advancing understanding of stationary distributions.

Findings

Explicit formulas for neighborhood widths of hyperbolic periodic orbits.

Neighborhoods form a basis for functions on the attractor.

Stationary distribution can be expressed in this basis.

Abstract

The finest state space resolution that can be achieved in a physical dynamical system is limited by the presence of noise. In the weak-noise approximation the neighborhoods of deterministic periodic orbits can be computed as distributions stationary under the action of a local Fokker-Planck operator and its adjoint. We derive explicit formulae for widths of these distributions in the case of chaotic dynamics, when the periodic orbits are hyperbolic. The resulting neighborhoods form a basis for functions on the attractor. The global stationary distribution, needed for calculation of long-time expectation values of observables, can be expressed in this basis.