Preprint Article Version 1 This version is not peer-reviewed

Asymptotic Dynamics of a Class of Third Order Rational Difference Equations

Version 1 : Received: 6 April 2020 / Approved: 8 April 2020 / Online: 8 April 2020 (04:05:22 CEST)

How to cite: Hassan, S.S.; Mondal, S.; Mandal, S.; Sau, C. Asymptotic Dynamics of a Class of Third Order Rational Difference Equations. Preprints 2020, 2020040114 (doi: 10.20944/preprints202004.0114.v1). Hassan, S.S.; Mondal, S.; Mandal, S.; Sau, C. Asymptotic Dynamics of a Class of Third Order Rational Difference Equations. Preprints 2020, 2020040114 (doi: 10.20944/preprints202004.0114.v1).

Abstract

The asymptotic dynamics of the classes of rational difference equations (RDEs) of third order defined over the positive real-line as $$\displaystyle{x_{n+1}=\frac{x_{n}}{ax_n+bx_{n-1}+cx_{n-2}}}, \displaystyle{x_{n+1}=\frac{x_{n-1}}{ax_n+bx_{n-1}+cx_{n-2}}}, \displaystyle{x_{n+1}=\frac{x_{n-2}}{ax_n+bx_{n-1}+cx_{n-2}}}$$ and $$\displaystyle{x_{n+1}=\frac{ax_n+bx_{n-1}+cx_{n-2}}{x_{n}}}, \displaystyle{x_{n+1}=\frac{ax_n+bx_{n-1}+cx_{n-2}}{x_{n-1}}}, \displaystyle{x_{n+1}=\frac{ax_n+bx_{n-1}+cx_{n-2}}{x_{n-2}}}$$ is investigated computationally with theoretical discussions and examples. It is noted that all the parameters $a, b, c$ and the initial values $x_{-2}, x_{-1}$ and $x_0$ are all positive real numbers such that the denominator is always positive. Several periodic solutions with high periods of the RDEs as well as their inter-intra dynamical behaviours are studied.

Subject Areas

rational difference equations; local asymptotic stability; periodic; Quasi-Periodic and Fractal-like trajectory

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