Phase-Space Volume of Regions of Trapped Motion: Multiple Ring Components and Arcs

TL;DR

This paper investigates how stability resonances influence the phase-space volume of trapped motion, revealing mechanisms for the formation of multiple ring components and arcs in dynamical systems with multiple degrees of freedom.

Contribution

It extends the understanding of phase-space structures by analyzing the excitation of resonances in systems with more than two degrees of freedom, explaining the formation of rings and arcs.

Findings

Resonance-induced gaps reduce trapped phase-space regions.

Multiple ring components can form due to resonance excitation.

Arcs occur when additional resonance conditions are satisfied.

Abstract

The phase--space volume of regions of regular or trapped motion, for bounded or scattering systems with two degrees of freedom respectively, displays universal properties. In particular, sudden reductions in the phase-space volume or gaps are observed at specific values of the parameter which tunes the dynamics; these locations are approximated by the stability resonances. The latter are defined by a resonant condition on the stability exponents of a central linearly stable periodic orbit. We show that, for more than two degrees of freedom, these resonances can be excited opening up gaps, which effectively separate and reduce the regions of trapped motion in phase space. Using the scattering approach to narrow rings and a billiard system as example, we demonstrate that this mechanism yields rings with two or more components. Arcs are also obtained, specifically when an additional (mean-motion) resonance condition is met. We obtain a complete representation of the phase-space volume occupied by the regions of trapped motion.