Periodic orbits and Arnold diffusion

Abstract

We consider three degrees of freedom initially hyperbolic Hamiltonian systems $H_\mu$, where $0<\mu <$$<1$ is the perturbing parameter. We prove that, under some technical assumptions, the Arnold diffusion time can be of order $(1/\mu)$log$(1/\mu)$, as conjectured by P. Lochak.<br> Our method is based on the construction of a dual chain of hyperbolic periodic orbits surrounding a given transition chain of partially hyperbolic tori, whose parameters (angles, periods) can be related to parameters (diophantine condition, angles) of the original chain of tori. Using Easton's method of windows, we give a general formula for the time of drift along such a chain of hyperbolic periodic orbits. We then deduce the result for chain of partially hyperbolic tori.