Dynamics in chaotic regions of area-preserving maps and the separatrix map

Abstract

Transversal homoclinic intersections lead to chaotic behaviours, and understanding them is the aim of our work. We will consider area and orientation preserving maps, such as the standard map, to study these intersections of the invariant manifolds of a saddle point and their splitting due to perturbation. S.Smale contributed to understanding transversal homoclinic intersections with the horseshoe map, which helps us describe the characteristics of the orbits when such intersections occur. The result is the presence of infinitely many periodic and non-periodic orbits, studied with the help of bi-infinite sequences. To study the dynamics near the intersection of these invariant manifolds, we will consider the separatrix map and derive it for the Chirikov standard map.