Qualitative Behavior of Two Rational Difference Equations

Abstract

Obtaining the exact solutions of most rational recursive equations is sophisticated sometimes. Therefore, a considerable number of nonlinear difference equations is often investigated by studying the qualitative behavior of the governing forms of these equations. The prime purpose of this work is to analyse the equilibria, local stability, global stability character, boundedness character and the solution behavior of the following fourth order fractional difference equations: $$ x_{n+1}=\frac{\alpha x_{n}x_{n-3}}{\beta x_{n-3}-\gamma x_{n-2}},\ \ \ \ x_{n+1}=\frac{\alpha x_{n}x_{n-3}}{-\beta x_{n-3}+\gamma x_{n-2}},\ \ \ n=0,1,..., $$ where the constants $\alpha ,\ \beta ,\ \gamma \in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{+}$ and the initial values $x_{-3},\ x_{-2},\ x_{-1}$\ and $x_{0}$ are required to be arbitrary non zero real numbers. Furthermore, some numerical figures will be obviously shown in this paper.

Keywords

• Equilibria,Local stability,Periodicity,Boundedness character,Global behavior,Recursive equations

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