Orbites hétéroclinique au voisinage d'une bifurcation de Poincaré dégénérée pour les champs de vecteurs de \({\mathbb{R}}^ 2\). (Heteroclinic orbits near a degenerate Poincaré-bifurcation for \({\mathbb{R}}^ 2\) vector fields) (Q749737)

We consider an \({\mathbb{R}}^ 2\)-ordinary differential equation, where the fixed point (0,0) presents a degenerate Poincaré-bifurcation of resonance \(k(=2k')\) and dominance \(d(=k-1)\). We prove the existence of a 2-dimensional linear manifold V in the parameter space \(\Lambda\), on which the perturbed dominant differential system (SD) possesses heteroclinic orbits between fixed points. The continuation of the local stable or unstable manifolds of the saddle fixed points shows that for any neighbourhood, in \(\Lambda\), of a point of V corresponding to a saddle heteroclinic orbit, there exists only one stable (resp. unstable) periodic orbit close to the stable - in the Andronov sense - (resp. unstable) heteroclinic orbit. Applications are given for \(k=4\) and \(k=6\).