Arnold diffusion: a variational construction
type: Item
instance of: scholarly article
title: Arnold diffusion: a variational construction
zbMATH Open document ID: 0910.58015
author: Zhihong Xia
published in: Documenta Mathematica
publication date: 1998-08-06T00:00:00Z
full work available at URL: https://eudml.org/doc/229520
full work available at URL: Xia.MAN.html
review text: The author proposes a new method for studying the phenomenon of Arnold diffusion for perturbed integrable Hamiltonian systems. Usually, one tries to construct (lower dimensional) transition tori to create order one changes in the action variable for arbitrary small perturbations, but nearby resonances may cause breaking of the transition chain. The author's variational approach is based on a generalization of the Aubry-Mather theory for twist maps and Mather's theory on connecting orbits of action-minimizing sets. In such an approach, also cantori are used to describe transition and diffusion. To construct connecting orbits between action-minimizing orbits, a generalization is introduced of Peierl's barrier functions, measuring the splitting of the stable and unstable manifold of normally hyperbolic invariant tori and cantori, and whose gradient (approximately) is the Poincaré-Melnikov function. The present paper does not contain the proofs of the theorems; they are announced for publication elsewhere.
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 70H05
Mathematics Subject Classification ID: 70H30
zbMATH DE Number: 1184295
zbMATH Keywords: Arnold diffusion
zbMATH Keywords: variational methods
zbMATH Keywords: KAM theory
reviewed by: W. Sarlet
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1184295