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Collision, explosion and collapse of homoclinic classes

Homoclinic classes of generic C1-diffeomorphisms are maximal transitive sets and pairwise disjoint. We present here a model explaining how two different homoclinic classes may intersect, failing to be disjoint. For that we construct a one-parameter family of diffeomorphisms (gs)sϵ[−1,1] with hyperbolic points P and Q having non-trivial homoclinic classes such that for s < 0 the classes of P and Q are disjoint; for s = 0 the classes collide and their intersection is a saddle-node and for s > 0, after an explosion, the two classes are equal. Our constructions involve bifurcations through heterodimensional and saddle-node cycles.