Stability Islands in Domains of Separatrix Crossings in Slow--Fast Hamiltonian Systems

A. A. Vasil'ev ; A. I. Neishtadt ; C. Simó ; D. V. Treschev

Informatics and Automation, Analysis and singularities. Part 2, Tome 259 (2007), pp. 243-255

Résumé

We consider a two-degrees-of-freedom Hamiltonian system with one degree of freedom corresponding to fast motion and the other corresponding to slow motion. The ratio of typical velocities of changes of the slow and fast variables is the small parameter $\varepsilon$ of the problem. At frozen values of the slow variables, there is a eparatrix on the phase plane of the fast variables, and there is a region in the phase space (the domain of separatrix crossings) where the projections of phase points onto the plane of the fast variables repeatedly cross the separatrix in the process of evolution of the slow variables. Under a certain symmetry condition, we prove the existence of many (of order $1/\varepsilon$) stable periodic trajectories in the domain of separatrix crossings. Each of these trajectories is surrounded by a stability island whose measure is estimated from below by a value of order $\varepsilon$. So, the total measure of the stability islands is estimated from below by a value independent of $\varepsilon$. The proof is based on an analysis of asymptotic formulas for the corresponding Poincaré map.