Proportions of Incommensurate, Resonant, and Chaotic Orbits for Torus Maps

TL;DR

This paper analyzes the distribution of incommensurate, resonant, and chaotic orbits in one- and two-dimensional torus maps, providing precise calculations and revealing different behaviors across map categories using advanced numerical methods.

Contribution

It offers a more accurate calculation of the power law coefficients for Arnold's circle map and shows the absence of a universal law in two-dimensional torus maps, identifying categories with similar behaviors.

Findings

Universal power law for nonresonant orbits in Arnold's circle map

No universal law for orbit classes in two-dimensional torus maps

Effective numerical methods for classifying dynamical behaviors

Abstract

This paper focuses on distinguishing classes of dynamical behavior for one- and two-dimensional torus maps, in particular between orbits that are incommensurate, resonant, periodic, or chaotic. We first consider Arnold's circle map, for which there is a universal power law for the fraction of nonresonant orbits as a function of the amplitude of the nonlinearity. Our methods give a more precise calculation of the coefficients for this power law. For two-dimensional torus maps, we show that there is no such universal law for any of the classes of orbits. However, we find different categories of maps with qualitatively similar behavior. Our results are obtained using three fast and high precision numerical methods: weighted Birkhoff averages, Farey trees, and resonance orders.