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 centerâ€”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.
Publisher: SIAM (Society for Industrial and Applied Mathematics)