Periodicity and convergence for \(x_{n+1}=| x_{n}-x_{n-1}|\).
type: Item
instance of: scholarly article
title: Periodicity and convergence for \(x_{n+1}=| x_{n}-x_{n-1}|\).
zbMATH Open document ID: 1041.39008
author: Hassan Sedaghat
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-03-14T00:00:00Z
review text: The author proves that each solution \(\{x_n\}\) of the equation \(x_{n+1}=| x_{n}-x_{n-1}| \) is either eventually periodic with period 3 or converges to zero. The case depends on whether the ratio \(x_0/x_{-1}\) of the initial values is rational or irrational. The sequence of ratios \(r_n=x_n/x_{n-1}\), \(n=0,1,\ldots\), satisfies the difference equation \(r_{n+1}=\phi(r_n)=| 1/r_n-1| \) that has \(p\)-cycles for all integer \(p\neq 3\). Each cycle is explicitly determined by Fibonacci numbers. A unique positive fixed point of \(\phi\) is shown to be a snap-back repeller. According to the result by \textit{F. R. Marotto} [J. Math. Anal. Appl. 63, 199--223 (1978; Zbl 0381.58004)] it means that the irrational ratios \(r_n\) behave chaotically.
Mathematics Subject Classification ID: 39A11
Mathematics Subject Classification ID: 11B39
zbMATH DE Number: 2056493
zbMATH Keywords: difference equation
zbMATH Keywords: periodic solution
zbMATH Keywords: convergence
zbMATH Keywords: snap-back repeller
zbMATH Keywords: chaotic behavior
zbMATH Keywords: Fibonacci numbers
zbMATH Keywords: cycles
zbMATH Keywords: fixed point
reviewed by: Oleg Anashkin
MaRDI profile type: Publication
full work available at URL: j.jmaa.2003.10.018
OpenAlex ID: W1965245321
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DOI: 10.1016/J.JMAA.2003.10.018
description: scientific article; zbMATH DE number 2056493