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 $H_{0}(p)+\epsilon H_{1}(\theta,p,t),\quad \theta \in {\mathbb{T}^{n}},\,p \in B^{n},\,t \in \mathbb{T}= \mathbb{R}/\mathbb{T},$be a nearly integrable system of arbitrary degrees of freedom ${n \geqslant 2}$ with a strictly convex H0. We show that for a “generic” ${\epsilon H_1}$, there exists an orbit ${(\theta,p)}$ satisfying $\|p(t)-p(0)\| > l(H_{1}) > 0,$where ${l(H_1)}$ is independent of ${\epsilon}$. 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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