The Separatrix Map

Abstract

The separatrix map was invented to study dynamical systems near asymptotic man-ifolds. It was introduced by Zaslavsky and Filonenko [147] (see also [32, 49]) for near-integrable Hamiltonian systems with one-and-a-half degrees of freedom and independently by Shilnikov [123] in generic systems. The main difference between these two approaches is as follows. The Zaslavsky separatrix map determines the dynamics globally near the unperturbed separatrices, but needs the system to be near-integrable. The Shilnikov separatrix map does not need any closeness to inte-grability, but deals with the dynamics in a neighborhood of a homoclinic orbit. In the present chapter we obtain explicit formulas for the Zaslavsky separatrix maps. These results are used in Chap. 5 for studying of the dynamics in the stochas-tic layer.