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March  2003, 9(2): 451-470. doi: 10.3934/dcds.2003.9.451

Periodic orbits and Arnold diffusion

1. 

Institut de Mathématiques de Jussieu, 175 rue du Chevaleret, 75013 Paris, France

2. 

Equipe de Topologie, Université de Bourgogne, Dijon, France

Received  August 2001 Revised  July 2002 Published  December 2002

We consider three degrees of freedom initially hyperbolic Hamiltonian systems $H_\mu$, where $0<\mu <$$<1$ is the perturbing parameter. We prove that, under some technical assumptions, the Arnold diffusion time can be of order $(1/\mu)$log$(1/\mu)$, as conjectured by P. Lochak.
Our method is based on the construction of a dual chain of hyperbolic periodic orbits surrounding a given transition chain of partially hyperbolic tori, whose parameters (angles, periods) can be related to parameters (diophantine condition, angles) of the original chain of tori. Using Easton's method of windows, we give a general formula for the time of drift along such a chain of hyperbolic periodic orbits. We then deduce the result for chain of partially hyperbolic tori.
Citation: Jacky Cresson, Christophe Guillet. Periodic orbits and Arnold diffusion. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 451-470. doi: 10.3934/dcds.2003.9.451
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