Differences equations and splitting of separatrices

The purpose of this dissertation is to study how the discretization of a differential equation affects, the stable and unstable manifolds in two concrete examples: the logistic equation and the pendulum equation. The logistic equation is equivalent to a system with two fixed points A and B. It is known that the stable manifolds at A coincides with the unstable manifold at B. By improving some results of A. Fruchard and R. Schafke, we show that the two manifolds do not coincide any more in the discretezed equation. The proof is a modification of an approach introduced by R. Schafke and H. Volkmer. First, we build a formal solution with polynomial coefficients. Then we give an asymptotic approximation of these coefficients. From these estimates we can obtain a quasi-solution, that is, a function which satisfies the difference equation except for an exponentially small error; moreover we can evaluate the asymptotic behavior of the distance between the two manifolds. To conclude, we show that some constant alpha appearing in the dominant term of the distance between the manifolds is non zero, and we further give a precise approximation for it. The second part of the thesis is dedicated to an analogous study regarding the pendulum equation and its discretization (Standard mapping). Similar results were obtained by Lazutkin et al., but our proof is completely different. This case is harder than the previous one, for we deal with a second order equation.