Tere M. Seara --- An introduction to geometric methods in Arnold diffusion
Amphithéâtre Charles Hermite
IHP - Bâtiment Borel
In this seminar, I will provide an overview of the so-called geometric methods in Arnold diffusion. Arnold diffusion occurs in perturbations of integrable systems. In such systems, when written in action-angle variables, the action does not evolve under the flow (it is a first integral of motion). Roughly speaking, a system undergoes Arnold diffusion when, for arbitrarily small "typical" perturbations, the change in actions becomes of order one (independent of the parameter values). Several methods have been used to detect this phenomenon. In this talk, we will review the geometric methods, introducing concepts such as Normally Hyperbolic Invariant Manifolds (NHIM) and their stable and unstable manifolds. When these manifolds intersect along a "homoclinic channel", one can define the "Scattering map", which encodes the heteroclinic connections between points on the NHIM. By combining iterations of the Scattering map with the internal dynamics of the NHIM, we will find "pseudo-orbits" along which the action increases. These orbits will be followed by real orbits where the action also increases. Finally, we will apply this methodology to the famous example provided by Arnold in 1964.