Acta Mathematica

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

Patrick Bernard, Vadim Kaloshin, and Ke Zhang

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We prove a form of Arnold diffusion in the a-priori stable case. Let H0(p)+ϵH1(θ,p,t),θTn,pBn,tT=R/T,be a nearly integrable system of arbitrary degrees of freedom n2 with a strictly convex H0. We show that for a “generic” ϵH1, there exists an orbit (θ,p) satisfying p(t)-p(0)>l(H1)>0,where l(H1) is independent of 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.

Article information

Acta Math., Volume 217, Number 1 (2016), 1-79.

Received: 4 April 2013
Revised: 28 September 2016
First available in Project Euclid: 22 February 2017

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2017 © Institut Mittag-Leffler


Bernard, Patrick; Kaloshin, Vadim; Zhang, Ke. Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders. Acta Math. 217 (2016), no. 1, 1--79. doi:10.1007/s11511-016-0141-5.

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