A Closer Look at a Fibonacci-like Iterated Nonlinear Map: More Chaos and Order

P.R.J. Asveld

    Research output: Book/ReportReportOther research output

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    Abstract

    We continue our study of a Fibonacci-like analogue of the well-known iterated nonlinear map $x_{n+1}=\lambda x_n(1-x_n)$;cf. [1]. This second-order variant contains two parameters: viz. one which corresponds to $\lambda$, and a second one --denoted by $\alpha$-- that determines the relative influence of the two previous values. As in [1] we fix $\lambda$ to 3.9999 and we vary the parameter $\alpha$ from 0 to 1. We consider many subintervals of $\alpha$ --not mentioned in [1]-- that exhibit interesting types of behavior (regular patterns, periodic limit cycles, "intermediate stages", et cetera). The effect of the initial conditions on the ultimate behavior of this iterated mapping is also discussed. Keywords and phrases: dynamical system, nonlinearity, chaos, order, second-order difference equation, bifurcation. [1] P.R.J. Asveld, A Fibonacci-like Iterated Nonlinear Map or Order from Chaos in the Region of the Tiny Tornados, Memorandum Informatica 90-10 (1990), Dept. of Comp. Sci., Twente University of Technology.
    Original languageEnglish
    Place of PublicationEnschede
    PublisherUniversity of Twente
    Number of pages66
    Publication statusPublished - 1991

    Publication series

    NameMemoranda informatica
    PublisherUniversity of Twente
    No.91-70
    ISSN (Print)0924-3755

    Fingerprint

    Nonlinear Map
    Chaos
    Second-order Difference Equations
    Limit Cycle
    Two Parameters
    Initial conditions
    Continue
    Bifurcation
    Dynamical system
    Vary
    Nonlinearity
    Analogue
    Influence

    Keywords

    • HMI-SLT: Speech and Language Technology
    • Dynamical system
    • Nonlinearity
    • Chaos
    • Order
    • Second-order difference equation
    • Bifurcation
    • Initial conditions
    • Phase-plane portraits

    Cite this

    Asveld, P. R. J. (1991). A Closer Look at a Fibonacci-like Iterated Nonlinear Map: More Chaos and Order. (Memoranda informatica; No. 91-70). Enschede: University of Twente.
    Asveld, P.R.J. / A Closer Look at a Fibonacci-like Iterated Nonlinear Map : More Chaos and Order. Enschede : University of Twente, 1991. 66 p. (Memoranda informatica; 91-70).
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    abstract = "We continue our study of a Fibonacci-like analogue of the well-known iterated nonlinear map $x_{n+1}=\lambda x_n(1-x_n)$;cf. [1]. This second-order variant contains two parameters: viz. one which corresponds to $\lambda$, and a second one --denoted by $\alpha$-- that determines the relative influence of the two previous values. As in [1] we fix $\lambda$ to 3.9999 and we vary the parameter $\alpha$ from 0 to 1. We consider many subintervals of $\alpha$ --not mentioned in [1]-- that exhibit interesting types of behavior (regular patterns, periodic limit cycles, {"}intermediate stages{"}, et cetera). The effect of the initial conditions on the ultimate behavior of this iterated mapping is also discussed. Keywords and phrases: dynamical system, nonlinearity, chaos, order, second-order difference equation, bifurcation. [1] P.R.J. Asveld, A Fibonacci-like Iterated Nonlinear Map or Order from Chaos in the Region of the Tiny Tornados, Memorandum Informatica 90-10 (1990), Dept. of Comp. Sci., Twente University of Technology.",
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    author = "P.R.J. Asveld",
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    language = "English",
    series = "Memoranda informatica",
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    Asveld, PRJ 1991, A Closer Look at a Fibonacci-like Iterated Nonlinear Map: More Chaos and Order. Memoranda informatica, no. 91-70, University of Twente, Enschede.

    A Closer Look at a Fibonacci-like Iterated Nonlinear Map : More Chaos and Order. / Asveld, P.R.J.

    Enschede : University of Twente, 1991. 66 p. (Memoranda informatica; No. 91-70).

    Research output: Book/ReportReportOther research output

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    T1 - A Closer Look at a Fibonacci-like Iterated Nonlinear Map

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    N2 - We continue our study of a Fibonacci-like analogue of the well-known iterated nonlinear map $x_{n+1}=\lambda x_n(1-x_n)$;cf. [1]. This second-order variant contains two parameters: viz. one which corresponds to $\lambda$, and a second one --denoted by $\alpha$-- that determines the relative influence of the two previous values. As in [1] we fix $\lambda$ to 3.9999 and we vary the parameter $\alpha$ from 0 to 1. We consider many subintervals of $\alpha$ --not mentioned in [1]-- that exhibit interesting types of behavior (regular patterns, periodic limit cycles, "intermediate stages", et cetera). The effect of the initial conditions on the ultimate behavior of this iterated mapping is also discussed. Keywords and phrases: dynamical system, nonlinearity, chaos, order, second-order difference equation, bifurcation. [1] P.R.J. Asveld, A Fibonacci-like Iterated Nonlinear Map or Order from Chaos in the Region of the Tiny Tornados, Memorandum Informatica 90-10 (1990), Dept. of Comp. Sci., Twente University of Technology.

    AB - We continue our study of a Fibonacci-like analogue of the well-known iterated nonlinear map $x_{n+1}=\lambda x_n(1-x_n)$;cf. [1]. This second-order variant contains two parameters: viz. one which corresponds to $\lambda$, and a second one --denoted by $\alpha$-- that determines the relative influence of the two previous values. As in [1] we fix $\lambda$ to 3.9999 and we vary the parameter $\alpha$ from 0 to 1. We consider many subintervals of $\alpha$ --not mentioned in [1]-- that exhibit interesting types of behavior (regular patterns, periodic limit cycles, "intermediate stages", et cetera). The effect of the initial conditions on the ultimate behavior of this iterated mapping is also discussed. Keywords and phrases: dynamical system, nonlinearity, chaos, order, second-order difference equation, bifurcation. [1] P.R.J. Asveld, A Fibonacci-like Iterated Nonlinear Map or Order from Chaos in the Region of the Tiny Tornados, Memorandum Informatica 90-10 (1990), Dept. of Comp. Sci., Twente University of Technology.

    KW - HMI-SLT: Speech and Language Technology

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    KW - Initial conditions

    KW - Phase-plane portraits

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    Asveld PRJ. A Closer Look at a Fibonacci-like Iterated Nonlinear Map: More Chaos and Order. Enschede: University of Twente, 1991. 66 p. (Memoranda informatica; 91-70).