The parameters space of the spin-orbit problem I. Normally hyperbolic invariant circles

TL;DR

This paper explores the parameter space of the dissipative spin-orbit problem in Celestial Mechanics, focusing on regions with normally hyperbolic invariant circles to understand the dynamics and their persistence under perturbations.

Contribution

It introduces a global analysis of the parameter space for the dissipative spin-orbit problem, applying Rüssmann's translated curve theorem to identify regions with persistent invariant circles.

Findings

Identification of parameter regions with normally hyperbolic invariant circles

Application of Rüssmann's theorem to the spin-orbit problem

Insights into the stability and bifurcations of invariant circles

Abstract

In this paper we start a global study of the parameter space (dissipation, perturbation, frequency) of the dissipative spin-orbit problem in Celestial Mechanics with the aim of delimiting regions where the dynamics, or at least some of its important features, is determined. This is done via a study of the corresponding family of time $2\pi$-maps. In the same spirit as Chenciner in his 1895 article on bifurcations of elliptic fixed points, we are at first interested in delimiting regions where the normal hyperbolicity is sufficiently important to guarantee the persistence of an invariant attractive (resp. repulsive) circle under perturbation. As a tool, we use an analogue for diffeomorphisms in this family of R\"ussmann's translated curve theorem in analytic category.