Professor: Amadeu Delsham

Dates: August 26 - 30, 2013

The phenomenon of diffusion in phase space for Hamiltonian systems is

important for applications and has attracted a great deal of attention

both in the mathematical and in the physical literature. The rather

loose (and somewhat inadequate) name diffusion captures the intuition

that there are trajectories that wander widely and explore large

regions of phase space.

In the course, we will introduce a geometrical mechanism for diffusion

in a priori unstable nearly integrable Hamiltonian systems. Such

systems are a perturbation of an integrable Hamiltonian systems

possessing a normally hyperbolic invariant manifold (NHIM) whose

associated unstable and stable invariant manifolds coincide along a

separatrix. To establish diffusion close to the NHIM, the key point is

to notice that close to the NHIM there are more than one dynamics, the

inner one inside the NHIM, an the outer one encoded by the scattering

map(s) associated to the transversal homoclinic manifolds of the

perturbed NHIM. To detect such diffusion, one can compute the

scattering map and check that its invariant sets differ from the ones

of the inner dynamics of the NHIM.

Several scenarios will be presented in the course, like the periodic

perturbation of one pendulum and one rotor, the multidimensional

versions with several pendula and rotors, and some other problems from

Celestial Mechanics, like the spatial Hill problem or the elliptic

restricted three body problem, based on papers of the lecturer and

coauthors that can be found in his web page

http://www.ma1.upc.edu/~amadeu