### Abstract

In a one-parameter study of a noninvertible family of maps of the plane arising in the context of a numerical integration scheme, Lorenz studied a sequence of transitions from an attracting fixed point to "computational chaos". As part of the transition sequence, he proposed the following as a possible scenario for the breakup of an invariant circle (IC): the IC develops regions of increasingly sharper curvature until at a critical parameter value it develops cusps; beyond this parameter value, the IC fails to persist, and the system exhibits chaotic behavior on an invariant set with loops [Computational chaos - a prelude to computational instability, Physica D 35 (1989) 299]. We investigate this problem in more detail and show that the IC is actually destroyed in a global bifurcation before it has a chance to develop cusps. Instead, the global unstable manifolds of saddle-type periodic points are the objects which develop cusps and subsequently "loops" or "antennae". The one-parameter study is better understood when embedded in the full two-parameter space and viewed in the context of the two-parameter Arnold horn structure. Certain elements of the interplay of noninvertibility with this structure, the associated ICs, periodic points and global bifurcations are examined.

Original language | English (US) |
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Pages (from-to) | 101-121 |

Number of pages | 21 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 177 |

Issue number | 1-4 |

DOIs | |

State | Published - Mar 15 2003 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics

### Keywords

- Bifurcation
- Chaos
- Integration
- Invariant circles
- Noninvertible maps

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## Cite this

*Physica D: Nonlinear Phenomena*,

*177*(1-4), 101-121. https://doi.org/10.1016/S0167-2789(02)00751-0