Amadeu Delshams, Gemma Huguet
Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems
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ABSTRACT. In the present paper we consider the case of a general $\cont{r+2}$
perturbation, for $r$ large enough, of an a priori unstable
Hamiltonian system of $2+1/2$ degrees of freedom, and we provide
explicit conditions on it, which turn out to be $\cont{2}$ generic
and are verifiable in concrete examples, which guarantee the
existence of Arnold diffusion.
This is a generalization of the result in Delshams et al.,
\emph{Mem. Amer. Math. Soc.}, 2006, where it was considered the case
of a perturbation with a finite number of harmonics in the angular
variables.
The method of proof is based on a careful analysis of the geography
of resonances created by a generic perturbation and it contains a
deep quantitative description of the invariant objects generated by
the resonances therein. The scattering map is used as an essential
tool to construct transition chains of objects of different
topology. The combination of quantitative expressions for both the
geography of resonances and the scattering map provides, in a
natural way, explicit computable conditions for instability.