### Speed of Arnold diffusion for analytic Hamiltonian systems

Ke Zhang

Abstract.
For quasi-convex analytic Hamiltonians that is $\epsilon-$close to
integrable, the Nekhoroshev theorem predicts that its action variable is
stable for time of order $\exp({C\epsilon^{-a}})$, here we call $a$ the
stability exponent. This exponent predicts a lower bound on the time it
takes for Arnold diffusion to happen. For $n-$degrees of freedom, the
stability exponent $a=\frac{1}{2n}$ in general, and $a=\frac{1}{2(n-m)}$
if the orbit is close to m-resonances. We construct an orbit of a
generalized version of Arnold's example, such that Arnold diffusion
happens in $\exp({C\epsilon^{-\frac{1}{2(n-2)}}})$-time. The exponent
$\frac{1}{2(n-2)}$ is optimal in the class considered as the orbit
passes close enough to a double resonance.