Lindstedt series and Hamilton--Jacobi equation for hyperbolic tori in three time scales problems

Interacting systems consisting of two rotators and a pendulum are considered, in a case in which the uncoupled systems have three very different characteristic time scales. The abundance of unstable quasi periodic motions in phase space is studied via Lindstedt series. The result is a strong improvement, compared to our previous results, on the domain of validity of bounds that imply existence of invariant tori, large homoclinic angles, long heteroclinic chains and drift--diffusion in phase space.