Splitting of separatrices for rapid degenerate perturbations of the classical pendulum

TL;DR

This paper investigates the exponentially small splitting of invariant manifolds in a perturbed pendulum system, especially when the first harmonic of the perturbation vanishes, using Hamilton-Jacobi methods and an inner equation approach.

Contribution

It develops a novel method to analyze splitting when the Melnikov function fails, providing an algorithm to compute the leading exponentially small term in such cases.

Findings

Splitting occurs at order μ^n, with n depending on the perturbation harmonics.

The Melnikov function is not accurate when the first harmonic coefficient vanishes.

An explicit algorithm for computing the splitting term is provided.

Abstract

In this work we study the splitting distance of a rapidly perturbed pendulum $H(x,y,t)=\frac{1}{2}y^2+(\cos(x)-1)+\mu(\cos(x)-1)g\left(\frac{t}{\varepsilon}\right)$ with $g(\tau)=\sum_{|k|>1}g^{[k]}e^{ik\tau}$ a $2\pi$-periodic function and $\mu,\varepsilon \ll 1$. Systems of this kind undergo exponentially small splitting and, when $\mu\ll 1$, it is known that the Melnikov function actually gives an asymptotic expression for the splitting function provided $g^{[\pm 1]}\neq 0$. Our study focuses on the case $g^{[\pm 1]}=0$ and it is motivated by two main reasons. On the one hand the general understanding of the splitting, as current results fail for a perturbation as simple as $g(\tau)=\cos(5\tau)+\cos(4\tau)+\cos(3\tau)$. On the other hand, a study of the splitting of invariant manifolds of tori of rational frequency $p/q$ in Arnold's original model for diffusion leads to the consideration of pendulum-like Hamiltonians with $ g(\tau)=\sin\left(p\cdot\frac{t}{\varepsilon}\right)+\cos\left(q\cdot\frac{t}{\varepsilon}\right), $ where, for most $p, q\in\mathbb{Z}$ the perturbation satisfies $g^{[\pm 1]}\neq 0$. As expected, the Melnikov function is not a correct approximation for the splitting in this case. To tackle the problem we use a splitting formula based on the solutions of the so-called inner equation and make use of the Hamilton-Jacobi formalism. The leading exponentially small term appears at order $\mu^n$, where $n$ is an integer determined exclusively by the harmonics of the perturbation. We also provide an algorithm to compute it.