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

TL;DR

This paper analyzes the exponentially small splitting of invariant manifolds in nearly integrable Hamiltonian systems with one and a half degrees of freedom, providing an asymptotic formula and comparing it with Melnikov Theory.

Contribution

It derives an asymptotic formula for the measure of splitting in resonances and identifies key constants, highlighting limitations of Melnikov Theory in these scenarios.

Findings

The splitting measure is asymptotically proportional to $K ext{varepsilon}^eta e^{-a/ extvarepsilon}$.

Constants $K$, $eta$, and $a$ are explicitly identified in terms of system features.

Melnikov Theory often fails to accurately predict the constants in the splitting formula.

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, tipically, in the resonances of nearly integrable systems Melnikov Theory fails to predict correctly the constants $K$ and $\beta$ involved in the formula.