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Unfolding the Cusp-Cusp Bifurcation of Planar Endomorphisms

In many applications of practical interest, for example, in control theory, economics, electronics, and neural networks, the dynamics of the system under consideration can be modeled by an endomorphism, which is a discrete smooth map that does not have a uniquely defined inverse; one also speaks simply of a noninvertible map. In contrast to the better known case of a dynamical system given by a planar diffeomorphism, many questions concerning the possible dynamics and bifurcations of planar endomorphisms remain open. In this paper we make a contribution to the bifurcation theory of planar endomorphisms. Namely, we present the unfoldings of a codimension-two bifurcation, which we call the cusp-cusp bifurcation, that occurs generically in families of endomorphisms of the plane. The cusp-cusp bifurcation acts as an organizing center that involves the relevant codimension-one bifurcations. The central singularity is an interaction of two different types of cusps. First, an endomorphism typically folds the pha...