KAM in Generalized Hamiltonian Systems with Multi-Scales.

The paper concerns the persistence of invariant tori in nearly integrable generalized Hamiltonian systems with multi-scales in the integrable part, i.e. the integrable part of the Hamiltonian involves several time scales and at each time scale the corresponding Hamiltonian depends on only part of the action variables. Such Hamiltonian systems arise frequently in problems of celestial mechanics and statistical physics holding volume-preserving structures. Our main motivation comes from the persistence of invariant tori including resonant ones in volume-preserving dynamical systems such as steady Euler flows. We introduce a multi-scale Diophantine condition and via a quasilinear KAM method prove a multi-scale KAM theorem. As applications, we discuss the KAM persistence for multi-scale volume-preserving dynamical systems. [ABSTRACT FROM AUTHOR]