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

Bernard P, Kaloshin V, Zhang K. 2016. Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders. Acta Mathematica. 217(1), 1–79.

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Journal Article | Published | English
Author
Bernard, Patrick; Kaloshin, VadimIST Austria ; Zhang, Ke
Abstract
We prove a form of Arnold diffusion in the a-priori stable case. Let 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⩾2 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.
Publishing Year
Date Published
2016-09-28
Journal Title
Acta Mathematica
Volume
217
Issue
1
Page
1-79
ISSN
IST-REx-ID

Cite this

Bernard P, Kaloshin V, Zhang K. Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders. Acta Mathematica. 2016;217(1):1-79. doi:10.1007/s11511-016-0141-5
Bernard, P., Kaloshin, V., & Zhang, K. (2016). Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders. Acta Mathematica. Institut Mittag-Leffler. https://doi.org/10.1007/s11511-016-0141-5
Bernard, Patrick, Vadim Kaloshin, and Ke Zhang. “Arnold Diffusion in Arbitrary Degrees of Freedom and Normally Hyperbolic Invariant Cylinders.” Acta Mathematica. Institut Mittag-Leffler, 2016. https://doi.org/10.1007/s11511-016-0141-5.
P. Bernard, V. Kaloshin, and K. Zhang, “Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders,” Acta Mathematica, vol. 217, no. 1. Institut Mittag-Leffler, pp. 1–79, 2016.
Bernard P, Kaloshin V, Zhang K. 2016. Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders. Acta Mathematica. 217(1), 1–79.
Bernard, Patrick, et al. “Arnold Diffusion in Arbitrary Degrees of Freedom and Normally Hyperbolic Invariant Cylinders.” Acta Mathematica, vol. 217, no. 1, Institut Mittag-Leffler, 2016, pp. 1–79, doi:10.1007/s11511-016-0141-5.

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