We introduce a geometric mechanism for di?usion in a priori unstable
nearly integrable dynamical systems. It is based on the observation that
resonances, besides destroying the primary KAM tori, create secondary tori
and tori of lower dimension. We argue that these objects created by resonances
can be incorporated in transition chains taking the place of the
destroyed primary KAM tori.
We establish rigorously the existence of this mechanism in a simple
model that has been studied before. The main technique is to develop a
toolkit to study, in a unified way, tori of different topologies and their invariant
manifolds, their intersections as well as shadowing properties of these
bi-asymptotic orbits. This toolkit is based on extending and unifying standard
techniques. A new tool used here is the scattering map of normally
hyperbolic invariant manifolds.
The model considered is a one-parameter family, which for " = 0 is an
integrable system. We give a small number of explicit conditions the jet of
order 3 of the family that, if verified imply diffusion. The conditions are
just that some explicitely constructed functionals do not vanish identically
or have non-degenerate critical points, etc.
An attractive feature of the mechanism is that the transition chains are
shorter in the places where the heuristic intuition and numerical experimentation
suggests that the diffusion is strongest.
