Homoclinic Bifurcations

1. Introduction. We say that a one-parameter family of diffeomorphisms ip^: M — • M, p G R, has a homoclinic bifurcation, or a homoclinic tangency, for p = 0 if ipo has an orbit of nontransverse intersection of a stable and an unstable manifold, both of the same hyperbolic fixed point (or periodic point), which splits, for p > 0, into two orbits of transverse intersection of these stable and unstable manifolds. Definitions will be recalled in § §2 and 3. These orbits of intersection of stable and unstable manifolds of the same hy-perbolic fixed points, or homoclinic orbits, often imply or are implied by complex dynamic behavior. So one may expect that at or near homoclinic bifurcations one will have transitions from simple to complex dynamic behavior and also (discontinuous) transitions between different kinds of complex dynamics. These transitions form the subject of this paper. This is a survey of recent work which was carried out mainly in collaboration with J. Palis and which is a continuation of the earlier work of S. Newhouse and J. Palis.