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Recursive behaviors of Hamiltonian dynamical systems in finite and infinite dimension
Abstract
Nonlinear dynamical systems both in finite and infinite dimension are a fundamental instrument to understand/modelize physical phenomena, which often have recursive/undulatory nature: the rotation of a satellite, the behavior of a planetary system, the motion of the sea, the deflection of a beam, electromagnetic waves (light, radio waves)... Many of these are modeled by Hamiltonian differential equations (ODEs in finite dimension or PDEs in the infinite case) and their mathematical description is often extremely complicated, characte- rized by a non-trivial interplay between stable and chaotic behaviors. A paradigmatic approach consists in studying the existence, stability, robustness and genericity of invariant manifolds that support a global dynamics which can be explicitly described. In the nearly integrable finite dimensional case, these objects are typically tori, have almost full measure, and support a Kronecker flow; this is a rather well established subject. In the infinite dimensional setting, very little is known and the results that one can obtain are strongly related to the boundary conditions of the PDE at stake. In this talk I shall give an overview of this fascinating subject and focus on some specific results regarding the existence of solutions and long time stability.
