Non-twist invariant circles in conformally symplectic systems

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. It is well-known that, with the aid of parameters, and under suitable non-degeneracy conditions, one can obtain that there is an attractor that is an invariant torus whose internal dynamics is conjugate to a rotation. By analogy with symplectic dynamics, a natural question is establishing appropriate definitions for twist and non-twist invariant tori in conformally symplectic systems. The main goals of this paper are: (a) to establish proper definitions of twist and non-twist invariant tori in families of conformally symplectic systems; (b) to interpret these definitions in terms of dynamical properties; (c) to derive algorithms to compute twist and non-twist invariant tori; (d) to implement these algorithms in examples; (e) to explore the mechanisms of breakdown of twist and non-twist invariant tori. Hence, the last part of the paper is devoted to implementations of the algorithms, illustrating the definitions presented in this paper, and studying robustness properties of invariant tori.