Weak instabilities in many-dimensional Hamiltonian systems

The problem of long-time predictions in many-dimensional Hamiltonian systems is examined. Some geometrical methods of nonlinear dynamics are reviewed, and applied to the study of a class of instabilities that are peculiar of systems with more than two degrees of freedom. These are called "weak instabilities," since they manifest themselves only after a long time. A qualitative analysis of the weak instability induced by slow parametric modulation (the modulational diffusion) is developed. The relevance of these phenomena to the problem of stability of charged particles in accelerators and storage rings is discussed.