Non-twist tori in conformally symplectic systems

TL;DR

This paper defines twist and non-twist invariant tori in conformally symplectic systems, develops algorithms to compute non-twist tori, and explores their breakdown mechanisms through implementation in 2D systems.

Contribution

It introduces proper definitions for twist and non-twist tori in conformally symplectic systems and provides computational algorithms for non-twist tori.

Findings

Algorithms successfully compute non-twist tori.

Definitions clarify the distinction between twist and non-twist tori.

Exploration of breakdown mechanisms of non-twist tori.

Abstract

Dissipative mechanical systems on the torus with a friction that is proportional to the velocity are modeled by conformally symplectic maps on the annulus, which are maps that transport the symplectic form into a multiple of itself (with a conformal factor smaller than 1). It is important to understand the structure and the dynamics on the attractors. With the aid of parameters, and under suitable non-degeneracy conditions, one can obtain that, by adjusting parameters, there is an attractor that is an invariant torus whose internal dynamics is conjugate to a rotation [CCdlL13]. By analogy with symplectic dynamics, there have been some debate in establishing appropriate definitions for twist and non-twist invariant tori (or systems). The purpose of this paper is two-fold: (a) to establish proper definitions of twist and non-twist invariant tori in families of conformally symplectic systems; (b) to derive algorithms of computation of non-twist invariant tori. The last part of the paper is devoted to implementations of the algorithms, illustrating the definitions presented in this paper, and exploring the mechanisms of breakdown of non-twist tori. For the sake of simplicity we have considered here 2D systems, i.e. defined in the 2D annulus, but generalization to higher dimensions is straightforward.