### Abstract

In this paper we are concerned with the dynamics of noninvertible transformations of the plane. Three examples are explored and possibly a new bifurcation, or "eruption," is described. A fundamental role is played by the interactions of fixed points and singular curves. Other critical elements in the phase space include periodic points and an invariant line. The dynamics along the invariant line, in two of the examples, reduces to the one-dimensional Newton's method which is conjugate to a degree two rational map. We also determine, computationally, the characteristic exponents for all of the systems. An unexpected coincidence is that the parameter range where the invariant line becomes neutrally stable, as measured by a zero Lyapunov exponent, coincides with the merging of a periodic point with a point on a singular curve.

Original language | English |
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Pages (from-to) | 108-120 |

Number of pages | 13 |

Journal | Chaos |

Volume | 6 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 1996 |

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*Chaos*,

*6*(2), 108-120. https://doi.org/10.1063/1.166158

}

*Chaos*, vol. 6, no. 2, pp. 108-120. https://doi.org/10.1063/1.166158

**On noninvertible mappings of the plane : Eruptions.** / Billings, Lora; Curry, James H.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On noninvertible mappings of the plane

T2 - Eruptions

AU - Billings, Lora

AU - Curry, James H.

PY - 1996/1/1

Y1 - 1996/1/1

N2 - In this paper we are concerned with the dynamics of noninvertible transformations of the plane. Three examples are explored and possibly a new bifurcation, or "eruption," is described. A fundamental role is played by the interactions of fixed points and singular curves. Other critical elements in the phase space include periodic points and an invariant line. The dynamics along the invariant line, in two of the examples, reduces to the one-dimensional Newton's method which is conjugate to a degree two rational map. We also determine, computationally, the characteristic exponents for all of the systems. An unexpected coincidence is that the parameter range where the invariant line becomes neutrally stable, as measured by a zero Lyapunov exponent, coincides with the merging of a periodic point with a point on a singular curve.

AB - In this paper we are concerned with the dynamics of noninvertible transformations of the plane. Three examples are explored and possibly a new bifurcation, or "eruption," is described. A fundamental role is played by the interactions of fixed points and singular curves. Other critical elements in the phase space include periodic points and an invariant line. The dynamics along the invariant line, in two of the examples, reduces to the one-dimensional Newton's method which is conjugate to a degree two rational map. We also determine, computationally, the characteristic exponents for all of the systems. An unexpected coincidence is that the parameter range where the invariant line becomes neutrally stable, as measured by a zero Lyapunov exponent, coincides with the merging of a periodic point with a point on a singular curve.

UR - http://www.scopus.com/inward/record.url?scp=21344444686&partnerID=8YFLogxK

U2 - 10.1063/1.166158

DO - 10.1063/1.166158

M3 - Article

AN - SCOPUS:21344444686

VL - 6

SP - 108

EP - 120

JO - Chaos

JF - Chaos

SN - 1054-1500

IS - 2

ER -