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

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.

Highlights

In this paper we consider nearly integrable analytic Hamiltonian systems of one and a half degrees of freedom
For δ = 0 the phase space is foliated by invariant tori I = I∗ with frequency vector ω(I∗) = (∂I h0(I∗), 1)
Which tori persist and which break down, and, what new kinds of dynamics appear in the regions where the unperturbed invariant tori break down

Summary

Introduction

In this paper we consider nearly integrable analytic Hamiltonian systems of one and a half degrees of freedom. For a generic trigonometric polynomial V , the associated Hamiltonian system has a hyperbolic critical point, such that its invariant manifolds coincide forming a separatrix which is a graph with respect to x and satisfies Hypothesis HP6.

Objectives

Results

Conclusion