Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Dynamics of the Rational Difference Equation $x_{n+1}=px_{n}+\frac{q}{x_{n-1}^2}$

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 $x_{n+1}=px_{n}+\frac{q}{x_{n-1}^2}$. Preprints 2020, 2020040113 (doi: 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 (doi: 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.

## Subject Areas

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

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