Ergod. Th. & Dynam. Sys. (2002), 22, 953–972 c  2002 Cambridge University Press DOI: 10.1017/S0143385702000615 Printed in the United Kingdom Explosions: global bifurcations at heteroclinic tangencies K. ALLIGOOD†, E. SANDER† and J. YORKE‡ † Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA (e-mail: alligood@gmu.edu, esander@gmu.edu) ‡ Institute for Physical Sciences and Technology, Department of Mathematics and Department of Physics, University of Maryland, College Park, MD 20742, USA (e-mail: yorke@ipst.umd.edu) (Received 5 November 2000 and accepted in revised form 21 September 2001) Abstract. We investigate bifurcations in the chain recurrent set for a particular class of one-parameter families of diffeomorphisms in the plane. We give necessary and sufficient conditions for a discontinuous change in the chain recurrent set to occur at a point of heteroclinic tangency. These are also necessary and sufficient conditions for an -explosion to occur at that point. 1. Introduction Large scale invariant sets of planar diffeomorphisms can vary discontinuously in size with changes in parameters. Global bifurcations of observable sets, such as crises of attractors or metamorphoses of basin boundaries, are the most easily detected and probably the most often described in scientific literature. (For a partial list of references, see [12].) For example, Figure 1 shows a jump in the size of a chaotic attractor of the Ikeda map, as the attractor merges with an unstable invariant Cantor set. At the bifurcation parameter, it is not just that two invariant sets merge to form a larger attractor. Gaps in the unstable invariant set fill in suddenly as the bifurcation parameter is passed. These gaps do not fill in gradually: they are of positive width larger than a uniform constant for every parameter prior to bifurcation. For a further discussion of this example, see Robert et al [11]. The key to the investigation of changes in invariant sets is the set of recurrent points, i.e. points x such that x ∈ ω(x). The discontinuous appearance of new recurrent points at global bifurcations, as occurs in the gaps in the example above, is called an explosion in the recurrent set. Explosions in the non-wandering set, called -explosions, are described in [8]. In our context, it is more natural to work with the set of chain recurrent points, which includes both the set of non-wandering points and the set of recurrent points.