Attractive invariant circles \`a la Chenciner

TL;DR

This paper investigates the persistence and properties of invariant quasi-periodic circles in dissipative twist maps under perturbations, using normal form and elimination of parameters techniques to analyze their existence and hyperbolicity.

Contribution

It introduces a normal form approach combined with parameter elimination to establish the existence and hyperbolicity of invariant circles in perturbed dissipative twist maps.

Findings

Existence of a Cantor set of invariant circles for small dissipation.

Persistence of invariant circles in a neighborhood of the Cantor set.

Normal hyperbolicity holds for dissipation of order rom the perturbation size.

Abstract

Studying general perturbations of a dissipative twist map depending on two parameters, a frequency $\nu$ and a dissipation $\eta$, the existence of a Cantor set $\mathcal C$ of curves in the $(\nu,\eta)$ plane such that the corresponding equation possesses a Diophantine quasi-periodic invariant circle can be deduced, up to small values of the dissipation, as direct consequence of a normal form Theorem in the spirit of R\"ussmann and the ``elimination of parameters" technique. These circles are normally hyperbolic as soon as $\eta\not=0$, which implies that the equation still possesses a circle of this kind for values of the parameters belonging to a neighborhood $\mathcal V$ of this set of curves. Obviously, the dynamics on such invariant circles is no more controlled and may be generic, but the normal dynamics is controlled in the sense of their basins of attraction. As it is expected, by classical graph-transform method we are able to determine a first rough region where the normal hyperbolicity prevails and a circle persists, for a strong enough dissipation $\eta\sim O(\sqrt{\epsilon}),$ $\epsilon$ being the size of the perturbation. Then, through normal-form techniques, we shall enlarge such regions and determine such a (conic) neighborhood $\mathcal{V}$, up to values of dissipation of the same order as the perturbation, by using the fact that the proximity of the set $\mathcal{C}$ allows, thanks to R\"ussmann's translated curve theorem, to introduce local coordinates of the type (dissipation, translation) similar to the ones introduced by Chenciner in 1985.