On the Rapidly Forced Pendulum

Abstract

The rapidly forced pendulum equation $\mathop x\limits^{..} + \sin x = \delta \sin t/\varepsilon$, with δ = δ ₀ ε ^p, p ≥ −2 and δ ₀, ε sufficiently small, is considered. We sketch our proof that the term proportional to δ ² in the splitting distance d(t ₀) in the ($\mathop x\limits^.$,t) plane has the form

$$ \delta ( - \frac{\pi }{4}\varepsilon \sec h\frac{\pi }{{2\varepsilon }} + O({\varepsilon ^2}\exp ( - \pi /2\varepsilon )))\sin \frac{{2{t_0}}}{\varepsilon. }$$

From this it follows that for −2 ≤ p < −1, the Melnikov term πδ sin $\frac{1}{e}$ t ₀ sech $\frac{1}{{2e}}\pi$ is dominated by the O(δ ²)term as ε ↓0.