Splitting of separatrices in the resonances of nearly integrable Hamiltonian systems of one and a half degrees of freedom

Marcel Guardia

doi:10.3934/dcds.2013.33.2829

Article Contents

2013, Volume 33, Issue 7: 2829-2859. Doi: 10.3934/dcds.2013.33.2829

This issue Previous Article Lipschitz metric for the Camassa--Holm equation on the line Next Article Bifurcation of isolated closed orbits from degenerated singularity in $\mathbb{R}^{3}$

Splitting of separatrices in the resonances of nearly integrable Hamiltonian systems of one and a half degrees of freedom

Marcel Guardia^1,

1.
School of Mathematics, Institute for Advanced Study, Einstein Drive, Simonyi Hall, Princeton, New Jersey, 08540

Received: March 2012

Revised: April 2012

Published online: January 2013

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Abstract

In this paper we consider general nearly integrable analytic Hamiltonian systems of one and a half degrees of freedom which are a trigonometric polynomial in the angular state variable. In the resonances of these systems generically appear hyperbolic periodic orbits. We study the possible transversal intersections of their invariant manifolds, which is exponentially small, and we give an asymptotic formula for the measure of the splitting. We see that its asymptotic first order is of the form $K \varepsilon^{\beta} \text{e}^{-a/\varepsilon}$ and we identify the constants $K,\beta,a$ in terms of the system features. We compare our results with the classical Melnikov Theory and we show that, typically, in the resonances of nearly integrable systems Melnikov Theory fails to predict correctly the constants $K$ and $\beta$ involved in the formula.

Keywords:

Nearly integrable Hamiltonian systems,
exponentially small splitting of separatrices,
Melnikov method,
complex matching.

Mathematics Subject Classification: 37J40, 37J45.

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