In this work we study the splitting distance of a rapidly perturbed pendulum
H(x,y,t)=21y2+(cos(x)−1)+μ(cos(x)−1)g(εt)
with g(τ)=∑∣k∣>1g[k]eikτ a 2π-periodic function and
μ,ε≪1. Systems of this kind undergo exponentially small
splitting and, when μ≪1, it is known that the Melnikov function actually
gives an asymptotic expression for the splitting function provided g[±1]=0. Our study focuses on the case g[±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(τ)=cos(5τ)+cos(4τ)+cos(3τ). 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(τ)=sin(p⋅εt)+cos(q⋅εt), where, for most p,q∈Z the perturbation satisfies g[±1]=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 μn, where n is an integer determined exclusively by the harmonics
of the perturbation. We also provide an algorithm to compute it