Perturbing Chaos with Cycle Expansions

TL;DR

This paper demonstrates that cycle expansions can be used to extend perturbation theory into chaotic regimes by preserving analyticity of spectral functions through symbolic dynamics, enabling better analysis of chaotic systems.

Contribution

It introduces a method to maintain analyticity of spectral functions in chaotic systems using cycle expansions, allowing perturbative analysis despite bifurcations.

Findings

Spectral functions remain analytic when symbolic dynamics is preserved.

A subset of unstable periodic orbits can be selected to maintain analyticity.

Perturbation theory can be applied to chaotic regimes using cycle expansions.

Abstract

Due to existence of periodic windows, chaotic systems undergo numerous bifurcations as system parameters vary, rendering it hard to employ an analytic continuation, which constitutes a major obstacle for its effective analysis or computation. In this manuscript, however, based on cycle expansions we found that spectral functions and thus dynamical averages are analytic, if symbolic dynamics is preserved so that a perturbative approach is indeed possible. Even if it changes, a subset of unstable periodic orbits (UPOs) can be selected to preserve the analyticity of the spectral functions. Therefore, with the help of cycle expansions, perturbation theory can be extended to chaotic regime, which opens a new avenue for the analysis and computation in chaotic systems.