MPEJ Volume 3, No.4, 40pp
Received: July 28, 1997, Revised: Sep 10, 1997, Accepted: Sep 30, 1997
Amadeu Delshams, Tere M. Seara
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.