Asymptotically isochronous systems

TL;DR

This paper explores mechanisms behind dynamical systems whose solutions become asymptotically periodic, focusing on two general mechanisms exemplified by integrable and nonintegrable many-body models with isochronous behavior.

Contribution

It identifies and illustrates two general mechanisms leading to asymptotically isochronous systems, including a new class of solvable many-body problems and deformations of known models.

Findings

Identification of two mechanisms for asymptotic isochrony

Examples include a solvable many-body problem

Deformation of known models to achieve asymptotic periodicity

Abstract

Mechanisms are elucidated underlying the existence of dynamical systems whose generic solutions approach asymptotically (at large time) isochronous evolutions: all their dependent variables tend asymptotically to functions periodic with the same fixed period. We focus on two such mechanisms, emphasizing their generality and illustrating each of them via a representative example. The first example belongs to a recently discovered class of integrable indeed solvable many-body problems. The second example consists of a broad class of (generally nonintegrable) models obtained by deforming appropriately the well-known (integrable and isochronous) many-body problem with inverse-cube two-body forces and a one-body linear ("harmonic oscillator") force.