The goal of the talk is to present some results in the study of Hamiltonian
dynamics in a neighborhood of homoclinic orbits to different type of equilibria,
periodic orbits and invariant Diophantine tori and to demonstrate how these
investigations lead to some problems in symplectic topology. The latters appear
first when we try to transform the system locally near the basic invariant set
(equilibrium, periodic orbit or invariant torus) to some convenient form to
understand the behavior of the system near this set. The another source is
the global behavior of the system under consideration when one studies the
global dynamics near a homoclinic orbit.
Here the first problem is to distinguish the genericity conditions. This
leads to some problems of symplectic topology, in particular, about the
existence and the number of intersection points under linear symplectic
transformations of Lagrangian tori. The connections with the classical
scattering theory for linear nonautonomous differential systems are also
discussed.
Results on which this talk is based are obtained under a partial support of
the RFBR grant 00-01-00905.
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