In this talk consider analytic Hamiltonian slow-fast systems with finitely many slow degrees of freedom. We allow for infinitely many fast degrees of freedom. We present a result on the existence of an almost invariant symplectic slow manifold for which the error field is exponentially small. The method we use is motivated by a paper of MacKay from 2004.
The method does not notice resonances, and therefore we do not pose any restrictions on the motion normal to the slow manifold other than it being fast and analytic. We also present a stability result and obtain a generalization of a result of Gelfreich and Lerman on an invariant slow manifold to (finitely) many fast degrees of freedom. This is joint work with Kristian Kristiansen (Copenhagen).
In part 1 I only talked about slow manifolds of general, i.e. ,non-Hamiltonian systems. In part 2 I will very briefly review part 1 and then focus on symplectic slow manifolds for Hamiltonian systems. So if you have not attended part 1, but are interested in Hamiltonian systems then you are welcome to join.