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A nonlinear difference equation with two parameters

Published online by Cambridge University Press:  17 February 2009

A. Brown
A. Brown
Affiliation:
Department of Theretical Physice, Research School of Physical Sciences, Australian National University, G.P.O. Box 4, Canberra, A. C. T. 2601.
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Abstract

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The paper is mainly concerned with the difference equation

where k and m are parameters, with k > 0. This equation arises from a method proposed for solving a cubic equation by iteration and represents a standardised form of the general problem. In using the above equation it is essential to know when the iteration process converges and this is discussed by means of the usual stability criterion. Critical values are obtained for the occurrence of solutions with period two and period three and the stability of these solutions is also examined. This was done by considering the changes as k increases, for a give value of m, which makes it effectively a one-parameter problem, and it turns out that the change with k can differ strongly from the usual behaviour for a one-parameter difference equation. For m = 2, for example it appears that the usual picture of stable 2-cycle solutions giving way to stable 4-cycle solutions is valid for smaller values of k but the situation is recersed for larger values of k where stable 4-cycle solutions precede stable 2-cycle solutions. Similar anomalies arise for the 3-cycle solutions.

Type
Research Article
Information
The ANZIAM Journal , Volume 26 , Issue 4 , April 1985 , pp. 430 - 451
Copyright
Copyright © Australian Mathematical Society 1985

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