TITLE: Singular separatrix splitting and the Melnikov method: An experimental study AUTHORS: Amadeu Delshams, Rafael Ramirez-Ros Departament de Matematica Aplicada I Universitat Politecnica de Catalunya Diagonal 647, 08028 Barcelona, Spain E-mails: amadeu@ma1.upc.es, rafael@tere.upc.es SOFTWARE AVAILABILITY: The zipped version includes the C-programs that produce the numerical results in the preprint. It is also included a brief guide (README.tex) to know: 1) what the programs accomplish, and 2) how they can be created from the C-files. The programs have been written in standard ANSI-C code. 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. KEYWORDS: Area-preserving map, singular separatrix splitting, Melnikov method, numerical experiments MSC numbers: 34C37, 34E05, 34E15, 65L12