Normally Hyperbolic Invariant Laminations and diffusive behaviour for the generalized Arnold example away from resonances

Abstract

In this paper we study existence of Normally Hyperbolic Invariant Laminations (NHIL) for a nearly integrable system given by the product of the pendulum and the rotator perturbed with a small coupling between the two. This example was introduced by Arnold. Using a {\it separatrix map}, introduced in a low dimensional case by Zaslavskii-Filonenko and studied in a multidimensional case by Treschev and Piftankin, for an open class of trigonometric perturbations we prove that NHIL do exist. Moreover, using a second order expansion for the separatrix map from [GKZ], we prove that the system restricted to this NHIL is a skew product of nearly integrable cylinder maps. Application of the results from [CK] about random iteration of such skew products show that in the proper $\epsilon$-dependent time scale the push forward of a Bernoulli measure supported on this NHIL weakly converges to an Ito diffusion process on the line as $\epsilon$ tends to zero.

arXiv admin note: text overlap with arXiv:1110.2117 by other authors

References in corpus (2)

• Global instability in the elliptic restricted three body problem
• Dynamics of the dominant Hamiltonian, with applications to Arnold diffusion

Cited by in corpus (5)

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• Random Iteration of Cylinder Maps and diffusive behavior away from resonances
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