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4. On the structure of near integrable n-d.o.f. Hamiltonian systems

A framework for understanding the global structure of near integrable $n$ d.o.f. systems is proposed. The goal is to reach a similar situation to the near integrable 1.5 d.o.f. systems, where one is able in a glance of the integrable phase portrait, understand where instabilities are expected to arise. It is suggested that the main tool for understanding the system structure is an energy-momentum diagram and generalized Fomenko graphs. The appropriate choice of co-ordinates for the diagram is determined, to leading order, by the form of the perturbations. Then, the relation between bifurcations in such plots and resonances is established, as is the relation to the presentation of the energy surfaces in the frequency space. It is demonstrated that for some systems this procedure is sufficient for achieving a full qualitative description of the near-integrable dynamics. In particular, the persistent appearance of instabilities associated with resonant lower dimensional tori will be discussed.

Joint work with A. Litvak-Hinnenzon.

Michael Grinfeld 2002-10-23