Explosions of chaotic sets

Large-scale invariant sets such as chaotic attractors undergo bifurcations as a parameter is varied. These bifurcations include sudden changes in the size and/or type of the set. An explosion is a bifurcation in which new rec urrent points suddenly appear at a non-zero distance from any pre-existing recurrent points. We discuss the following. In a generic one-parameter fami ly of dissipative invertible maps of the plane there are only four known me chanisms through which an explosion can occur: (1) a saddle-node bifurcatio n isolated from other recurrent points, (2) a saddle-node bifurcation embed ded in the set of recurrent points, (3) outer homoclinic tangencies, and (4 ) outer heteroclinic tangencies. (The term "outer tangency" refers to a par ticular configuration of the stable and unstable manifolds at tangency.) In particular, we examine different types of tangencies of stable and unstabl e manifolds from orbits of pre-existing invariant sets. This leads to a gen eral theory that unites phenomena such as crises, basin boundary metamorpho ses, explosions of chaotic saddles, etc. We illustrate this theory with num erical examples. (C) 2000 Elsevier Science B.V. All rights reserved.