Homoclinic orbits to invariant sets of quasi-integrable exact maps

Patrick Bernard (1, 2)

1 IF - Institut Fourier
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

The resonant tori of an integrable system are destroyed by a perturbation. If the Hamiltonian is convex, they give rise to hyperbolic lower-dimensional invariant tori or to Aubry–Mather invariant sets. Bolotin has proved the existence of homoclinic orbits to the hyperbolic tori but not to the Aubry–Mather invariant sets. We solve this problem and obtain, for each resonant frequency, the existence of an invariant set with homoclinic orbits.

Domaines: Systèmes dynamiques [math.DS]

Patrick Bernard. Homoclinic orbits to invariant sets of quasi-integrable exact maps. Ergodic Theory and Dynamical Systems, 2000, 20 (6), pp.1583-1601. ⟨hal-01251209⟩