Arnold diffusion in nearly integrable Hamiltonian systems

Abstract

In this paper, Arnold diffusion is proved to be generic phenomenon in nearly integrable convex Hamiltonian systems with three degrees of freedom: $$H(x,y)=h(y)+\epsilon P(x,y),x\in {\mathbb{T}}^3,y\in {\mathbb{R}}^3.$$ Under typical perturbation $\epsilon P$, the system admits "connecting" orbit that passes through any two prescribed small balls in the same energy level $H^{-1}(E)$ provided $E$ is bigger than the minimum of the average action, namely, $E>\min \alpha$.