Accelerating cycle expansions by dynamical conjugacy

TL;DR

This paper introduces a coordinate transformation technique to improve the convergence of cycle expansions in nonlinear systems by removing measure singularities, demonstrated on one-dimensional maps.

Contribution

The paper proposes a novel method of dynamical conjugacy via coordinate transformation to accelerate cycle expansion convergence in non-hyperbolic systems.

Findings

Convergence of cycle expansions is slowed by measure singularities.

Coordinate transformations can remove singularities and restore fast convergence.

Method is validated on several one-dimensional maps.

Abstract

Periodic orbit theory provides two important functions---the dynamical zeta function and the spectral determinant for the calculation of dynamical averages in a nonlinear system. Their cycle expansions converge rapidly when the system is uniformly hyperbolic but greatly slowed down in the presence of non-hyperbolicity. We find that the slow convergence can be associated with singularities in the natural measure. A properly designed coordinate transformation may remove these singularities and results in a dynamically conjugate system where fast convergence is restored. The technique is successfully demonstrated on several examples of one-dimensional maps and some remaining challenges are discussed.