On noninvertible mappings of the plane

Eruptions

Lora Billings, James H. Curry

Research output: Contribution to journalArticleResearchpeer-review

15 Citations (Scopus)

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 languageEnglish
Pages (from-to)108-120
Number of pages13
JournalChaos
Volume6
Issue number2
DOIs
StatePublished - 1 Jan 1996

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volcanic eruptions
Singular Curve
Periodic Points
exponents
Newton methods
Invariant
Line
curves
Newton-Raphson method
Merging
Characteristic Exponents
Rational Maps
Coincidence
Newton Methods
Lyapunov Exponent
Phase Space
Bifurcation
Fixed point
interactions
Zero

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Billings, Lora ; Curry, James H. / On noninvertible mappings of the plane : Eruptions. In: Chaos. 1996 ; Vol. 6, No. 2. pp. 108-120.
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On noninvertible mappings of the plane : Eruptions. / Billings, Lora; Curry, James H.

In: Chaos, Vol. 6, No. 2, 01.01.1996, p. 108-120.

Research output: Contribution to journalArticleResearchpeer-review

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