Georgia Institute of Technology College of Sciences

Consider a generic perturbation of a nearly integrable system of {\it arbitrary degrees of freedom $n\ge 2$ system} $$H_0(p)+\eps H_1(\th,p,t),\quad \th\in \T^n,\ p\in B^n,\ t\in \T=\R/\T,$$ with strictly convex $H_0$. Jointly with P.Bernard and K.Zhang we prove existence of orbits $(\th,p)(t)$ exhibiting Arnold diffusion $$\|p(t)-p(0) \| >l(H_1)>0 \quad \textup{independently of }\eps.$$ Action increment is independent of size of perturbation$\eps$, but does depend on a perturbation $\eps H_1$.This establishes a weak form of Arnold diffusion. The main difficulty in getting rid of $l(H_1)$ is presence of strong double resonances. In this case for $n=2$we prove existence of normally hyperbolic invariant manifolds passing through these double resonances. (joint with P. Bernard and K. Zhang)