About Two Mechanisms of Reunion of Chaotic Attractors

Considering a family of two-dimensional piecewise linear maps, we discuss two different mechanisms of reunion of two (or more) pieces of cyclic chaotic attractors into a one-piece attracting set, observed in several models. It is shown that, in the case of so-called contact bifurcation of the 2 nd kind, the reunion occurs immediately due to homoclinic bifurcation of some saddle cycle belonging to the basin boundary of the attractor. In the case of so-called contact bifurcation of the 1 st kind, the reunion is a result of a contact of the attractor with its basin boundary which is fractal, including the stable set of a chaotic invariant hyperbolic set appeared after the homoclinic bifurcation of a saddle cycle on the basin boundary