A General Mechanism of Instability in Hamiltonian Systems

CDSNS Colloquium
Monday, January 30, 2017 - 11:00
1 hour (actually 50 minutes)
Skiles 005
Univ. Polit. Catalunya
We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable  Hamiltonian systems. Our approach relies on  successive applications of the `outer dynamics' along homoclinic orbits to a normally hyperbolic invariant manifold. The information on the outer dynamics is encoded by a geometrically defined map, referred to as the  `scattering map'. We find pseudo-orbits of the scattering map that keep moving in some privileged  direction. Then we use the recurrence property of the `inner dynamics', restricted to the normally hyperbolic invariant manifold, to return to those pseudo-orbits. Finally, we apply topological methods  to show the existence of true orbits that follow the successive applications of the  two dynamics. This method  differs, in several crucial aspects,  from earlier works.  Unlike the well known `two-dynamics' approach, the method relies heavily on the outer dynamics alone. There are virtually no assumptions on the inner dynamics, as its invariant objects  (e.g., primary and secondary tori, lower dimensional hyperbolic tori and their stable/unstable manifolds, Aubry-Mather sets) are not used at all. The method applies to unperturbed  Hamiltonians of arbitrary degrees of freedom  that are not necessarily convex. In addition, this mechanism is easy to verify (analytically or numerically)  in concrete examples, as well as to establish diffusion in generic systems.