Pau Martín

Hamiltonian systems with orbits covering densely submanifolds of small codimension

The existence of a transition chain in a Hamiltonian system leads to the existence of orbits shadowing it, if some lambda lemma can be applied. This fact has been used to prove the existence of diffusion in perturbations of integrable a priori stable systems. We prove that, under suitable conditions, there are orbits which cover densely codimension two submanifolds of the (2n - 1)-dimensional energy level. We also construct examples, near general integrable systems, where the computations to verify the sufficient conditions for the appearance of such phenomenon can be performed.

Exponentially Small Asymptotic Formulas for the Length Spectrum in Some Billiard Tables

ABSTRACT Let q ≥ 3 be a period. There are at least two (1, q)-periodic trajectories inside any smooth strictly convex billiard table. We quantify the chaotic dynamics of axisymmetric billiard tables close to their boundaries by studying the asymptotic behavior of the differences of the lengths of their axisymmetric (1, q)-periodic trajectories as q → +∞. Based on numerical experiments, we conjecture that, if the billiard table is a generic axisymmetric analytic strictly convex curve, then these differences behave asymptotically like an exponentially small factor q−3e−rq times either a constant or an oscillating function, and the exponent r is half of the radius of convergence of the Borel transform of the well-known asymptotic series for the lengths of the (1, q)-periodic trajectories. Our experiments are focused on some perturbed ellipses and circles, so we can compare the numerical results with some analytical predictions obtained by Melnikov methods. We also detect some non-generic behaviors due to the presence of extra symmetries. Our computations require a multiple-precision arithmetic and have been programmed in PARI/GP.

Homoclinic Solutions to Infinity and Oscillatory Motions in the Restricted Planar Circular Three Body Problem

The circular restricted three body problem models the motion of a massless body under the influence of the Newtonian gravitational force caused by two other bodies, the primaries, which describe circular planar Keplerian orbits. The system has a first integral, the Jacobi constant. The existence of oscillatory motions for the restricted planar circular three body problem, that is, of orbits which leave every bounded region but which return infinitely often to some fixed bounded region, was proved by Llibre and Simo [18] in 1980. However, their proof only provides such orbits for values of the ratio between the masses of the two primaries exponentially small with respect to the Jacobi constant. In the present work, we extend their result proving the existence of oscillatory motions for any value of the mass ratio. The existence of these motions is a consequence of the transversal intersection between the stable and unstable manifolds of infinity, which guarantee the existence of a symbolic dynamics that creates the oscillatory orbits. We show that this intersection does happen for any value of the mass ratio and for big values of the Jacobi constant. We remark that, since in our setting the mass ratio is no longer small, this transversality cannot be checked by means of classical perturbation theory respect to the mass ratio. Furthermore, since our method is valid for all values of mass ratio, we are able to detect a curve in the parameter space, formed by the mass ratio and the Jacobi constant, where cubic homoclinic tangencies between the invariant manifolds of infinity appear.