Explosions in Dimensions One through Three

Crises are discontinuous changes in the size of a chaotic attractor as a parameter is varied. A special type of crisis is an explosion, in which the new points of the attractor form far from any previously recurrent points. This article summarizes new results in explosions in dimension one, and surveys previous results in dimensions two and three. Explosions can be the result of homoclinic and heteroclinic bifurcations. In dimensions one and two, homoclinic and heteroclinic bifurcations occur at tangencies. We give a classification of onedimensional explosions through homoclinic tangency. We describe our previous work on the classification of planar explosions through heteroclinic tangencies. Three-dimensional heteroclinic bifurcations can occur without tangencies. We describe our previous work, which gives an example of such a bifurcation and explains why three-dimensional crossing bifurcations exhibit unstable dimension variability, a type of non-hyperbolic behavior which results in a breakdown of shadowing. In addition, we give details for a new scaling law for the parameter-dependent variation of the density of the new part of the chaotic attractor.