Chaotic Jumping Near Resonances: Finite-Dimensional Systems

The evolution of physical problems can be decomposed into slow and fast components near resonances. The slow variables are typically slowly varying amplitudes and nearly resonant phase combinations, while the fast variables describe the remaining degrees of freedom. In many cases there are sets of solutions on which the fast oscillatory components vanish. These solutions then form a slow manifold, whose geometry and stability determines the nature of the dynamics near the resonance. The slow variation on this manifold is due to some small detuning or perturbation from the exact resonant states, which form a resonant,or critical, manifold. Viewing slow manifolds as small perturbations of critical manifolds is the main idea of geometric singular perturbation theory (cf. Section 1.18), and this is the approach we shall take in this book.