Orbites hétérocliniques au voisinage d'une bifurcation de Poincaré dégénérée pour les champs de vecteurs de ℝ²

Abstract

We consider a ℝ²-ordinary differential equation, where the fixed point (0, 0) presents a degenerate Poincaré-bifurcation of resonancek(=2k′) and dominanced(=k−1). We prove the existence of a 2-dimensional linear manifoldV in the parameter space. Λ, on which the perturbed dominant differential system (SD) possesses heteroclinic orbits between fixed points. The numerical continuation of the local stable or unstable manifolds of the saddle fixed points shows that for any neighborhood, in Λ, of a point ofV corresponding to a saddle heteroclinic orbit, there exists only one stable (resp. unstable) periodic orbit close to the stable — in the Andronov sense [1]-(resp. unstable) heteroclinic orbit. Applications are given fork=4 andk=6.