Exponentially Small Expressions for Separatrix Splittings

Abstract

Exponentially small expressions for the splitting of separatrices are provided for second order equations with a rapidly forced perturbation term

$$\ddot x + f(x) = \mu {\varepsilon ^p}g\left( {\frac{t}{\varepsilon }} \right),$$

where μ and ε are independent small parameters. These asymptotical expressions coincide with the ones predicted by the Poincaré-Melnikov theory, and therefore their size is $O(\mu {\varepsilon ^{p - r}}{e^{ - \frac{e}{\varepsilon }}})$, where ai is the pole of the derivative of the homoclinic solution of the unperturbed equation, and r its order. The main ideas of the proof of these asymptotic formulas are presented, assuming p ≥ r — 1, and that the first Fourier coefficients of the Poincaré-Melnikov function M(s,ε) are different from zero.