GLOBAL ASPECTS OF HOMOCLINIC BIFURCATIONS OF VECTOR-FIELDS - INTRODUCTION

In this paper we investigate a class of smooth one parameter families of vector fields on some n-dimensional manifold, exhibiting a homoclin ic bifurcation. That is, we consider generic families X(mu), where X(0 ) has a distinguished hyperbolic singularity p and a homoclinic orbit; an orbit converging to p both for positive and negative time. We assu me that this homoclinic orbit is of saddle-saddle type, characterized by the existence of well defined directions along which it converges t o the singularity p. We do not confine our study to a small. neighbour hood of the homoclinic orbit. Instead, we incorporate the position of the stable and unstable set of the homoclinic orbit in our study and s how that homoclinic bifurcations can lead to complicated bifurcations and dynamics, including phenomena like intermittency and annihilation of suspended horseshoes.