De la bifurcació de Hopf al teorema del centre de Lyapunov. Aplicacions al problema restringit dels tres cossos

Abstract

[en] Lyapunov center theorem provides the conditions in order to ensure the existence of periodic orbits in the surroundings of a fixed point of a Hamiltonian system, when its lineal problem already contain periodic orbits. Therefore, by studying a simpler system, it is possible to obtain information about the nonlinear dynamical system. In this final project, we will state and study this theorem, providing its proof (using Hopf bifurcation theorem), and we will also give an insight of Hamiltonian mechanics and the restricted three body problem in its circular version. In the latter, we will see that the Lyapunov theorem can be applied to some fixed points, and in a computational way, we will present explicitly some periodic solutions around the equilibrium point called $L_1$.