Repulsion from resonances
[Répulsion par les résonances]
Mémoires de la Société Mathématique de France, no. 128 (2012) , 125 p.

Nous considérons des systèmes « lents-rapides », dont le mouvement rapide est périodique et le mouvement lent intégrable, en présence de résonances faibles ou fortes. En supposant que les phases initiales sont aléatoires et que certaines conditions de non-dégénérescence sont satisfaites, nous démontrons que l’évolution effective des invariants adiabatiques est donnée par un processus de Markov. Ce processus de Markov consiste en un mouvement le long des trajectoires d’un champ de vecteurs qui peut présenter des sauts occasionnels. Le générateur du processus limite est calculé à partir de la dynamique du système au voisinage des résonances fortes.

We consider slow-fast systems with periodic fast motion and integrable slow motion in the presence of both weak and strong resonances. Assuming that the initial phases are random and that appropriate non-degeneracy assumptions are satisfied we prove that the effective evolution of the adiabatic invariants is given by a Markov process. This Markov process consists of the motion along the trajectories of a vector field with occasional jumps. The generator of the limiting process is computed from the dynamics of the system near strong resonances.

DOI : 10.24033/msmf.439
Classification : 34C29, 70K65, 37D25, 60F17
Keywords: Averaging, Slow-Fast Systems, Markov Processes, Invariant Cones, Resonances
Mot clés : Moyennisation, Systèmes lents-rapides, Processus de Markov, Cônes invariants, Résonances 
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Dolgopyat, Dmitry. Repulsion from resonances. Mémoires de la Société Mathématique de France, Série 2, no. 128 (2012), 125 p. doi : 10.24033/msmf.439. http://numdam.org/item/MSMF_2012_2_128__1_0/

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