Une méthode numérique pour la recherche de solutions périodiques des systèmes hamiltoniens

Abstract

Starting from a problem given by the physicists we give a numerical method for finding periodical solutions of non-linear differential systems. This method is based upon the minimization of functional on the flow. Appliyng the Largrange multipliers permits to replace this last problem by the resolution of an algebraic system using a Newton like method. We give the algorithm permitting the computation of solutions η, T of the problem, i.e. the initial conditions η leading to T – periodic orbits. Two examples are given; one concerned with a two-degree of freedom non integrable conservative hamiltonian system, the second with a one-degree of freedom system with a forcing term. In the former case a nice relationship between periods of the two principal periodic families and the structure function of the hamiltonian could be conjectured.