About Some Theorems by L.P. Sil'nikov

Abstract

Some theorems by L.P. Sil'nikov, which describe the dynamics in the neighborhood of homoclinic orbits, bi-asymptotic to a saddle focus, and initially proved for real analytic vector fields are collected here. Recent results in dynamical systems theory allow us to precise some of the conclusions and to generalize these theorems to the C1,1 class. Certain heteroclinic loops involving a saddle focus are also considered.