Contribution to the study of invariant manifolds and the splitting of separatrices of parabolic points

[eng] In general, when beginning to explore any scientific field, one focuses on the generic situations; that is, one centers on the behaviours that appear in “most” of the cases encountered in practice. This methodology allows an easier understanding of the problem, since the non-generic (or degenerate) cases are left out (at least a priori) in a first approach. This way, the casuistic is simpler and the general theory can be developed more easily. Although this is a good scientific procedure, the aim of Science is to explain reality in the most complete way possible. So, when the general case has been already described (perhaps not completely, but at least in a good part), one should study the non-generic cases: the exceptions. It should not be forgotten that, in nature, not all the processes follow a general rule. The exceptional cases often provide new types of behaviour. Therefore, a lot can be learned from the exceptions, as much at an intrinsic level (situations that differ from the general qualitative behaviour) as for the new techniques that are developed in order to understand them. In certain contexts, it is generic to encounter degenerate cases. Let us think, for instance, about the case of parametric families, f(mi), which describe different behaviours depending on the value of mi. In this situation, it is generic (that is, it occurs in most of the families) to find values of the parameter f(mi)(0) for which the behaviour of f(mi)(0) is degenerate

Structural stability (QQMDS)

2021/20221r quadrimestr

Gevrey estimates for one dimensional parabolic invariant manifolds of non-hyperbolic fixed points

We study the Gevrey character of a natural parameterization of one dimensional invariant manifolds associated to a parabolic direction of fixed points of analytic maps, that is, a direction associated with an eigenvalue equal to $1$. We show that, under general hypotheses, these invariant manifolds are Gevrey with type related to some explicit constants. We provide examples of the optimality of our results as well as some applications to celestial mechanics, namely, the Sitnikov problem and the restricted planar three body problem

Homoclinic Points (QQMDS)

2021/20221r quadrimestr

Exponentially small splitting of invariant manifolds of parabolic points

We consider families of one and a half degrees of freedom Hamiltonians with high frequency periodic dependence on time, which are perturbations of an autonomous system. We suppose that the origin is a parabolic xed point with non-diagonalizable linear part and that the unperturbed system has a homoclinic connexion associated to it. We provide a set of hypotheses under which the splitting is exponentially small and is given by the Poincaré-Melnikov function

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

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 [BFM]. 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

A rigorous derivation of the asymptotic wavenumber of spiral wave solutions of the complex Ginzburg-Landau equation

In this work n-armed Archimedian spiral wave solutions of the complex Ginzburg-Landau equation are considered. These solutions are showed to depend on two characteristic parameters, the so called twist parameter and the asymptotic wavenumber. The existence and uniqueness of the value of the asymptotic wavenumber, depending on the twist parameter, for which n-armed Archimedian spiral wave solutions exist is a classical result, obtained back in the 80s by Kopell and Howard. In this work we deal with a different problem, that is, the asymptotic expression of the asymtptotic wavenumer for small values of the twist parameter. Since the eighties, different heuristic perturbation techniques, like formal asymptotic expansions, have conjectured an asymptotic expression of which is exponentially small with respect to the twist parameter. However, the validity of this expression has remained opened until now, despite of the fact that it has been widely used for more than 40 years. In this work, using a functional analysis approach, we finally prove the validity of the asymptotic formula, providing a rigorous bound for its relative error

Invariant manifolds of parabolic fixed points (II). Approximations by sums of homogeneous functions.

We study the computation of local approximations of invariant manifolds of parabolic fixed points and parabolic periodic orbits of periodic vector fields. If the dimension of these manifolds is two or greater, in general, it is not possible to obtain polynomial approximations. Here we develop an algorithm to obtain them as sums of homogeneous functions by solving suitable cohomological equations. We deal with both the differentiable and analytic cases. We also study the dependence on parameters. In the companion paper [BFM] these approximations are used to obtain the existence of true invariant manifolds close by. Examples are provided