Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders

By P. Bernard, Vadim Kaloshin, Ke Zhang. 2016

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AbstractWe prove a form of Arnold diffusion in the a-priori stable case. Let $$H_{0}(p)+\epsilon H_{1}(\theta,p,t),\quad \theta \in {\mathbb{T}^{n}},\,p \in B^{n},\,t \in \mathbb{T}= \mathbb{R}/\mathbb{T},$$H0(p)+ϵH1(θ,p,t),θ∈Tn,p∈Bn,t∈T=R/T,be a nearly integrable system of arbitrary degrees of freedom $${n \geqslant 2}$$n⩾2 with a strictly convex H0. We show that for a “generic” $${\epsilon H_1}$$ϵH1, there exists an orbit $${(\theta,p)}$$(θ,p) satisfying $$\|p(t)-p(0)\| > l(H_{1}) > 0,$$‖p(t)-p(0)‖>l(H1)>0,where $${l(H_1)}$$l(H1) is independent of $${\epsilon}$$ϵ. The diffusion orbit travels along a codimension-1 resonance, and the only obstruction to our construction is a finite set of additional resonances.For the proof we use a combination of geometric and variational methods, and manage to adapt tools which have recently been developed in the a-priori unstable case.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/9db987b30c7956b8f013698172c32b56e6b63138