Exponentially small splitting of separatrices beyond Melnikov analysis: rigorous results

TL;DR

This paper rigorously analyzes the exponentially small splitting of separatrices in non-autonomous Hamiltonian systems, extending beyond Melnikov theory to provide precise asymptotic formulas and identify system-specific constants.

Contribution

It offers a rigorous proof of the asymptotic measure of splitting for general algebraic or trigonometric polynomial systems, surpassing classical Melnikov predictions in certain cases.

Findings

Splitting behaves as $K ext{e}^{-a/ ext{epsilon}}$ with system-dependent constants.

Melnikov theory accurately predicts constants for small perturbations.

In limit cases, Melnikov theory does not reliably predict the splitting constants.

Abstract

We study the problem of exponentially small splitting of separatrices of one degree of freedom classical Hamiltonian systems with a non-autonomous perturbation fast and periodic in time. We provide a result valid for general systems which are algebraic or trigonometric polynomials in the state variables. It consists on obtaining a rigorous proof of the asymptotic formula for the measure of the splitting. We obtain that the splitting has the asymptotic behavior $K \varepsilon^{\beta} \text{e}^{-a/\varepsilon}$, identifying the constants $K,\beta,a$ in terms of the system features. We consider several cases. In some cases, assuming the perturbation is small enough, the values of $K,\beta$ coincide with the classical Melnikov approach. We identify the limit size of the perturbation for which this theory holds true. However for the limit cases, which appear naturally both in averaging and bifurcation theories, we encounter that, generically, $K$ and $\beta$ are not well predicted by Melnikov theory.