Invariant manifolds of parabolic fixed points (I). Existence and dependence on parameters

TL;DR

This paper investigates the existence, regularity, and parameter dependence of stable manifolds near parabolic fixed points using the parametrization method, providing new results on approximation and regularity loss.

Contribution

It introduces an a posteriori approach for true invariant manifolds near parabolic points, extending previous polynomial approximations to more general cases.

Findings

Existence of invariant manifolds near parabolic fixed points confirmed.

Regularity loss identified in the differentiable case.

Application to the elliptic three-body problem demonstrates practical relevance.

Abstract

In this paper we study the existence and regularity of stable manifolds associated to fixed points of parabolic type in the differentiable and analytic cases, using the parametrization method. The parametrization method relies on a suitable approximate solution of a functional equation. In the case of parabolic points, if the manifolds have dimension two or higher, in general this approximation cannot be obtained in the ring of polynomials but as a sum of homogeneous functions and it is given in~\cite{BFM2015b}. Assuming a sufficiently good approximation is found, here we provide an "a posteriori" result which gives a true invariant manifold close to the approximated one. In the differentiable case, in some cases, there is a loss of regularity. We also consider the case of parabolic periodic orbits of periodic vector fields and the dependence of the manifolds on parameters. Examples are provided. We apply our method to prove that in several situations, namely, related to the parabolic infinity in the elliptic spatial three body problem, these invariant manifolds exist and do have polynomial approximations.