Unfolding the cusp-cusp bifurcation of planar endomorphisms

B Krauskopf, HM Osinga, B Peckham, Bruc

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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 modelled 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 organising center that involves the relevant codimension-one bifurcations. The central singularity involves an interaction of two different types of cusps. Firstly, 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_3 <Z_1. 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 pre-image 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 different phase portraits
Original languageEnglish
Publication statusPublished - Oct 2006

Bibliographical note

Sponsorship: B.K. and H.M.O. were both supported by EPSRC Advanced Research Fellowship grants.
B.B.P. acknowledges support from the National Science Foundation (grant #DMS-


  • codimension-two bifurcation
  • invariant curve
  • discrete-time system
  • noninvertible planar map
  • unstable manifold

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