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

P.R.J. Asveld

Research output: Book/ReportReportOther research output

### 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 language English Enschede University of Twente 66 Published - 1991

### Publication series

Name Memoranda informatica University of Twente 91-70 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.",
keywords = "HMI-SLT: Speech and Language Technology, Dynamical system, Nonlinearity, Chaos, Order, Second-order difference equation, Bifurcation, Initial conditions, Phase-plane portraits",
author = "P.R.J. Asveld",
year = "1991",
language = "English",
series = "Memoranda informatica",
publisher = "University of Twente",
number = "91-70",

}

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

Research output: Book/ReportReportOther research output

TY - BOOK

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

T2 - More Chaos and Order

AU - Asveld, P.R.J.

PY - 1991

Y1 - 1991

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

KW - Dynamical system

KW - Nonlinearity

KW - Chaos

KW - Order

KW - Second-order difference equation

KW - Bifurcation

KW - Initial conditions

KW - Phase-plane portraits

M3 - Report

T3 - Memoranda informatica

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

PB - University of Twente

CY - Enschede

ER -

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).