Version 1
: Received: 6 April 2020 / Approved: 8 April 2020 / Online: 8 April 2020 (03:59:40 CEST)

How to cite:
Hassan, S.S. Dynamics of the Rational Difference Equation $x_{n+1}=px_{n}+\frac{q}{x_{n-1}^2}$. Preprints2020, 2020040113. https://doi.org/10.20944/preprints202004.0113.v1
Hassan, S.S. Dynamics of the Rational Difference Equation $x_{n+1}=px_{n}+\frac{q}{x_{n-1}^2}$. Preprints 2020, 2020040113. https://doi.org/10.20944/preprints202004.0113.v1

Hassan, S.S. Dynamics of the Rational Difference Equation $x_{n+1}=px_{n}+\frac{q}{x_{n-1}^2}$. Preprints2020, 2020040113. https://doi.org/10.20944/preprints202004.0113.v1

APA Style

Hassan, S.S. (2020). Dynamics of the Rational Difference Equation $x_{n+1}=px_{n}+\frac{q}{x_{n-1}^2}$. Preprints. https://doi.org/10.20944/preprints202004.0113.v1

Chicago/Turabian Style

Hassan, S.S. 2020 "Dynamics of the Rational Difference Equation $x_{n+1}=px_{n}+\frac{q}{x_{n-1}^2}$" Preprints. https://doi.org/10.20944/preprints202004.0113.v1

Abstract

A second order rational difference equation $$x_{n+1}=px_{n}+\frac{q}{x_{n-1}^2}$$with the parameters $p$ and $q$ which lies in $(0,1)$, is studied. The dynamics of the equilibrium is characterized through the trichotomy of the parameter $p<\frac{1}{2}$, $p=\frac{1}{2}$ and $p>\frac{1}{2}$. It is found that there is no periodic solution of period $2$ and $3$ but there exists periodic solutions with only periodic solution $5$ and $10$ are achieved computationally.

Keywords

rational difference equation; asymptotic stability; periodic solutions; chaos and fractal

Subject

Computer Science and Mathematics, Computational Mathematics

Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.