We study the problem of exponentially small splitting of separatrices of one degree of freedom classical Hamiltonian systems with a non-autonomous perturbation fast and periodic in time. We provide a result valid for general systems which are algebraic or trigonometric polynomials in the state variables. It consists on obtaining a rigorous proof of the asymptotic formula for the measure of the splitting. We obtain that the splitting has the asymptotic behavior K ε β e −a/ε , identifying the constants K , β, a in terms of the system features. We consider several cases. In some cases, assuming the perturbation is small enough, the values of K , β coincide with the classical Melnikov approach. We identify the limit size of the perturbation for which this theory holds true. However for the limit cases, which appear naturally both in averaging and bifurcation theories, we encounter that, generically, K and β are not well predicted by Melnikov theory.

Abstract. We consider families of one and a half degrees of freedom rapidly forced Hamiltonian system which are perturbations of one degree of freedom Hamiltonians having a homoclinic connection. We derive the inner equation for this class of Hamiltonian system which is expressed as the Hamiltonian-Jacobi equation of one a half degrees of freedom Hamiltonian. The inner equation depends on a parameter not necessarily small.We prove the existence of special solutions of the inner equation with a given behavior at infinity. We also compute the asymptotic expression for the difference between these solutions. In some perturbative cases, this asymptotic expression is strongly related with the Melnikov function associated to our initial Hamiltonian.

Abstract. We use the parameterization method to prove the existence and properties of one-dimensional submanifolds of the center manifold associated to the fixed point of C r maps with linear part equal to the identity. We also provide some numerical experiments to test the method in these cases.1. Introduction. We consider C r maps of R 1+n having a parabolic fixed point and study the existence of one-dimensional invariant manifolds passing through this fixed point.We assume that the fixed point is the origin and that the linear part of the map at the fixed point is the identity. Then a whole neighborhood of the origin is a center manifold. However there may exist invariant submanifolds of points which go to the origin by the iteration of the map. In this setting we refer to such submanifolds as stable manifolds. In the same way we can speak of unstable manifolds.These problems appear naturally in Celestial Mechanics. In these applications, often the fixed point is the image of infinity under a suitable transformation and the invariant manifolds are the separation from bounded and unbounded motions. See for example, [23,16,19,6,15] for studies of parabolic invariant manifolds in R 2 and applications to Celestial Mechanics.

Summary. In this paper we study the exponentially small splitting of a heteroclinic connection in a one-parameter family of analytic vector fields in R 3 . This family arises from the conservative analytic unfoldings of the so-called Hopf zero singularity (central singularity). The family under consideration can be seen as a small perturbation of an integrable vector field having a heteroclinic orbit between two critical points along the z axis. We prove that, generically, when the whole family is considered, this heteroclinic connection is destroyed. Moreover, we give an asymptotic formula of the distance between the stable and unstable manifolds when they meet the plane z = 0. This distance is exponentially small with respect to the unfolding parameter, and the main term is a suitable version of the Melnikov integral given in terms of the Borel transform of some function depending on the higher-order terms of the family. The results are obtained in a perturbative setting that does not cover the generic unfoldings of the Hopf singularity, which can be obtained as a singular limit of the considered family. To deal with this singular case, other techniques are needed. The reason to study the breakdown of the heteroclinic orbit is that it can lead to the birth of some homoclinic connection to one of the critical points in the unfoldings of the Hopf-zero singularity, producing what is known as a Shilnikov bifurcation.

We consider maps defined on an open set of R nþm having a fixed point whose linear part is the identity. We provide sufficient conditions for the existence of a stable manifold in terms of the nonlinear part of the map.These maps arise naturally in some problems of Celestial Mechanics. We apply the results to prove the existence of parabolic orbits of the spatial elliptic three-body problem. r