Bifurcations of Chaotic Attractors in One-Dimensional Piecewise Smooth Maps

It is well known that dynamical systems defined by piecewise smooth functions exhibit several phenomena which cannot occur in smooth systems, such as for example, border collision bifurcations, sliding, chattering, etc. [di Bernardo et al., 2008]. One such phenomenon is the persistence of chaotic attractors under parameter perturbations, referred to as robust chaos [Banerjee et al., 1998]. In this case, there is an open region in the parameter space, called chaotic domain or domain of robust chaos, corresponding to chaotic attractors only. It is wellknown that these attractors may undergo bifurcations (frequently referred to as crises) leading to a sudden change of the shape of the attractor, or to a change of the number of its bands (connected components). Such bifurcations may be organized in complex bifurcation scenarios [Avrutin & Schanz, 2008; Avrutin & Sushko, 2012]. At the boundary of a chaotic domain similar bifurcations may lead to a transformation of a chaotic attractor into a chaotic repeller. Although such bifurcations were intensively studied by many authors, there is not a unique widely accepted terminology in this field. In particular, interior crises [Grebogi et al., 1982, 1983, 1987; Ott, 1993] can be seen as a special class of contact bifurcations [Mira & Narayaninsamy, 1993; Gardini et al., 1996; Mira et al., 1996a; Mira et al., 1996b; Fournier-Prunaret et al., 1997; Maistrenko et al., 1998] called also explosions of