Arnold diffusion in arbitrary degrees of freedom and crumpled 3-dimensional normally hyperbolic invariant cylinders

2011/12/13 by Bernard, Patrick, Kaloshin, Vadim, Zhang, Ke
#37J40 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1112.2773

Abstract

In the present paper we prove a form of Arnold diffusion. The main result says that for a "generic" perturbation of a nearly integrable system of arbitrary degrees of freedom n≥ 2 H₀(p)+\eps H₁(þ,p,t), þ∈ \Tⁿ, p∈ Bⁿ, t∈ \T=\R/\T, with strictly convex H₀ there exists an orbit (þ_\e,p_e)(t) exhibiting Arnold diffusion in the sens that [sup_t>0‖p(t)-p(0) ‖ >l(H₁)>0] where l(H₁) is a positive constant independant of \e. Our proof is a combination of geometric and variational methods. We first build 3-dimensional normally hyperbolic invariant cylinders of limited regularity, but of large size, extrapolating on \citeBe3 and \citeKZZ. Once these cylinders are constructed we use versions of Mather variational method developed in Bernard \citeBe1, Cheng-Yan \citeCY1, CY2.

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