Wielding Intermittency with Cycle Expansions

TL;DR

This paper introduces a combined approach using cycle expansions and perturbation theory to improve the computation of dynamical averages in intermittent systems, addressing limitations of traditional periodic orbit theory.

Contribution

It proposes a novel scheme that integrates cycle expansion and perturbation theory, specifically tailored for intermittent maps with singularities, to enhance accuracy.

Findings

More precise computation of natural measures near singularities.

Effective handling of intermittency in 1D maps with a single singularity.

Improved dynamical averages compared to traditional methods.

Abstract

As periodic orbit theory works badly on computing the observable averages of dynamical systems with intermittency, we propose a scheme to cooperate with cycle expansion and perturbation theory so that we can deal with intermittent systems and compute the averages more precisely. Periodic orbit theory assumes that the shortest unstable periodic orbits build the framework of the system and provides cycles expansion to compute dynamical quantities based on them, while the perturbation theory can locally analyze the structure of dynamical systems. The dynamical averages may be obtained more precisely by combining the two techniques together. Based on the integrability near the marginal orbits and the hyperbolicity in the part away from the singularities in intermittent systems, the chief idea of this paper is to revise intermittent maps and maintain the natural measure produced by the original maps. We get the natural measure near the singularity through the Taylor expansions and periodic orbit theory captures the natural measure in the other parts of the phase space. We try this method on 1-dimensional intermittent maps with single singularity, and more precise results are achieved.