Amadeu Delshams, Tere M. Seara
Splitting of separatrices in Hamiltonian systems with
one and a half degrees of freedom
(116K, LaTeX 2.09)
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. Here
$ h^{0}(x)= h^{0}(x_{1},x_{2})= x_{2}^{2}/2+V(x_{1})$ has an associated
separatrix $x^{0}(t)$, $ h^{1}(x,\theta )$ is $2\pi $-periodic in
$\theta =t /\varepsilon $, $\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 given in first order by
the Melnikov function.