Splitting of separatrices for rapid degenerate perturbations of the classical pendulum

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.