© 2015, Springer-Verlag Berlin Heidelberg. The restricted three body problem models the motion of a massless body under the influence of the Newtonian gravitational force caused by two other bodies called the primaries. When they move along circular Keplerian orbits and the third body moves in the same plane, one has the restricted planar circular three body problem (RPC3BP). In suitable coordinates, it is a Hamiltonian system of two degrees of freedom. The conserved energy is usually called the Jacobi constant. Llibre and Simó [Math Ann 248(2):153–184, 1980] proved the existence of oscillatory motions for this system. That is, orbits which leave every bounded region but which return infinitely often to some fixed bounded region. To prove their existence they had to assume the ratio between the masses of the primaries to be small enough. In this paper we prove the existence of such motions for any value of the mass ratio(Formula presented.) closing the problem of existence of oscillatory motions in the RPC3BP. To obtain such motions, we restrict ourselves to the level sets of the Jacobi constant. We show that, for any value of the mass ratio and for large values of the Jacobi constant, there exist transversal intersections between the stable and unstable manifolds of infinity in these level sets. These transversal intersections guarantee the existence of a symbolic dynamics that creates the oscillatory orbits. The main achievement is to prove the existence of these orbits without assuming the mass ratio (Formula presented.) small. When (Formula presented.) is not small, this transversality can not be checked by means of classical perturbation theory. Since our method is valid for all values of v, we are able to detect a curve in the parameter space, formed by (Formula presented.) and the Jacobi constant, where cubic homoclinic tangencies between the invariant manifolds of infinity appear.Postprint (published version

We consider weakly coupled map lattices with a decaying interaction. That is, we consider systems which consist of a phase space at every site such that the dynamics at a site is little affected by the dynamics at far away sites. We develop a functional analysis framework which formulates quantitatively the decay of the interaction and is able to deal with lattices such that the sites are manifolds. This framework is very well suited to study systematically invariant objects. One obtains that the invariant objects are essentially local. We use this framework to prove a stable manifold theorem and show that the manifolds are as smooth as the maps and have decay properties (i.e. the derivatives of one of the coordinates of the manifold with respect to the coordinates at far away sites are small). Other applications of the framework are the study of the structural stability of maps with decay close to uncoupled possessing hyperbolic sets and the decay properties of the invariant manifolds of their hyperbolic sets, in the companion paper by Fontich et al. (2011) [10].

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.

We consider di erentiable maps having an invariant torus with normal behavior having a central part. We p r o ve the existence and regularity of pseudostable manifolds and regularity with respect to parameters. Then we p r o ve a l a m bda lemma in this setting for C 2 maps.

The McMillan map is a one-parameter family of integrable symplectic\ud maps of the plane, for which the origin is a hyperbolic xed point\ud with a homoclinic loop, with small Lyapunov exponent when the parameter is\ud small. We consider a perturbation of the McMillan map for which we show\ud that the loop breaks in two invariant curves which are exponentially close one\ud to the other and which intersect transversely along two primary homoclinic\ud orbits. We compute the asymptotic expansion of several quantities related to\ud the splitting, namely the Lazutkin invariant and the area of the lobe between\ud two consecutive primary homoclinic points. Complex matching techniques are\ud in the core of this work. The coe cients involved in the expansion have a\ud resurgent origin, as shown inPeer ReviewedPostprint (published version