Splitting of separatrices in Hamiltonian systems with one and a half degrees of freedom

Abstract

The splitting of separatrices for Hamiltonians with $1{1\over 2}$ degrees of freedom $$ h(x,t /\varepsilon ) = h^{0}(x) + \mu \varepsilon ^{p} h^{1}(x,t /\varepsilon ) $$ is measured. We assume that $ h^{0}(x)= h^{0}(x_{1},x_{2})= x_{2}^{2}/2+V(x_{1})$ has a separatrix $x^{0}(t)$, $ h^{1}(x,\theta )$ is $2\pi $-periodic in $\theta $, $\mu $ and $\varepsilon >0 $ are independent small parameters, and $p\ge 0$. Under suitable conditions of meromorphicity for $x_{2}^{0}( u )$ and the perturbation $ h^{1}(x^{0}( u ),\theta )$, the order $ \ell $ of the perturbation on the separatrix is introduced, and it is proved that, for $ p \ge \ell $, the splitting is exponentially small in $\varepsilon $, and is given in first order by the Melnikov function.