Amadeu Delshams, Rafael Ramirez-Ros
Exponentially small splitting of separatrices for standard-like maps
(405K, LaTeX 2.09)

ABSTRACT.  The splitting of separatrices for the standard-like maps
$$
 F(x,y) = (y,-x + 2\mu y/(1+y^{2}) + \varepsilon V'(y)),
\qquad
 \mu=\cosh h, h>0, \varepsilon \in {\bf R},
$$
is measured.
For even entire perturbative potentials $V(y) = \sum_{n\ge 2} V_{n} y^{2n}$
such that $\widehat{V}(2\pi) \neq 0$, where
$\widehat{V}(\xi) = \sum_{n\ge 2} V_{n} \xi^{2n-1}/(2n-1)!$
is the Borel transform of $V(y)$, the following asymptotic formula for the
area $A$ of the lobes between the perturbed separatrices is established
$$
 A = 8 \pi \widehat{V}(2\pi) \varepsilon e^{-\pi^{2}/h} [1 + O(h^{2})]
\qquad
 (\varepsilon = o(h^{6} \ln^{-1}h), h \to 0^{+}).
$$
This formula agrees with the one provided by the Melnikov theory, which
cannot be applied directly, due to the exponentially small size of $A$
with respect to $h$.