# Analysis and its Applications Seminar

### Exponentially Small Splitting: A Direct Approach

### Abstract

This talk is on a study of the splitting distance of the stable and unstable manifold of a Dung equation subject to a time-periodic non-Hamiltonian perturbation. We introduce a simple recursion to write high order splitting distances as collections of certain well-structured multiple integrals, which we name as high order Melnikov integrals. Let ! be the forcing frequency. We also develop an analytic scheme to evaluate high order Melnikov integrals to extract an exponentially small factor out of splitting distances of all order for large !. We reveal that exponentially small splitting is not a phenomenon tied exclusively to Hamiltonian perturbations. It is rather induced by a certain symmetry embedded in the kernel functions of high order Melnikov integrals. This symmetry is beheld by many non-Hamiltonian equation