## Abstract

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 endomor-phism, 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 en-domorphism 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.

Original language | English (US) |
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Pages (from-to) | 403-440 |

Number of pages | 38 |

Journal | SIAM Journal on Applied Dynamical Systems |

Volume | 6 |

Issue number | 2 |

DOIs | |

State | Published - 2007 |

Externally published | Yes |

## Keywords

- Codimension-two bifurcation
- Discrete-time system
- Invariant curve
- Noninvertible planar map
- Unstable manifold