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 phase space along curves $J_0$ where the Jacobian of the map is zero. The image $J_1$ of $J_0$ may contain a cusp point, which persists under perturbation; the literature also speaks of a map of type $Z_1 <Z_3$. The second type of cusp occurs when a forward invariant curve $W$, such as a segment of an unstable manifold, crosses $J_0$ in a direction tangent to the zero eigenvector. Then the image of $W$ will typically contain a cusp. This situation is of codimension one and generically leads to a loop in the unfolding. The central singularity that defines the cusp-cusp bifurcation is, hence, defined by a tangency of an invariant curve $W$ with $J_0$ at the preimage of the cusp point on $J_1$. We study the bifurcations in the images of $J_0$ and the curve $W$ in a neighborhood of the parameter space of the organizing center—where both images have a cusp at the same point in the phase space. To this end, we define a suitable notion of equivalence that distinguishes between the different possible local phase portraits of the invariant curve relative to the cusp on $J_1$. Our approach makes use of local singularity theory to derive and analyze completely a normal form of the cusp-cusp bifurcation. In total we find eight different two-parameter unfoldings of the central singularity. We illustrate how our results can be applied by showing the existence of a cusp-cusp bifurcation point in an adaptive control system. We are able to identify the associated two-parameter unfolding for this example and provide all the different phase portraits.