Amadeu Delshams, Rafael Ramirez-Ros
Singular separatrix splitting and Melnikov method: An experimental study
(400K, PostScript)
ABSTRACT. We consider families of analytic area-preserving maps depending on two
parameters: the perturbation strength $\varepsilon$ and the characteristic
exponent $h$ of the origin.
For $\varepsilon=0$, these maps are integrable with a separatrix to the
origin, whereas they asymptote to flows with homoclinic connections as
$h\rightarrow 0^{+}$.
For fixed $\varepsilon\neq 0$ and small $h$, we show that these connections
break up.
The area of the lobes of the resultant turnstile is given asymptotically by
$\varepsilon \exp(-\pi^{2}/h)\Theta^{\varepsilon} (h)$, where
$\Theta^{\varepsilon} (h)$ is an even Gevrey-1 function such that
$\Theta^{\varepsilon} (0)\neq 0$ and the radius of convergence of its Borel
transform is $2\pi^{2}$.
As $\varepsilon\rightarrow 0$, the function $\Theta^{\varepsilon} $ tends to
an entire function $\Theta^{0} $.
This function $\Theta^{0} $ agrees with the one provided by the Melnikov
theory, which cannot be applied directly, due to the exponentially small
size of the lobe area with respect to $h$.
These results are supported by detailed numerical computations; we use an
expensive multiple-precision arithmetic and expand the local invariant curves
up to very high order.