Global instability and Geometrical Mechanism for Diffusion in Hamiltonian systems

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