# Qualitative Behavior of a Quadratic Second-Order Rational Difference Equation -by M. B. Almatrafi, E. M. Elsayed and Faris Alzahrani

**Abstract: **This article demonstrates the existence of boundedness, asymptotic behavior

and the periodicity of the following quadratic second-order rational

difference equation:

\begin{equation*}

x_{n+1}=ax_{n}+\dfrac{bx_{n}^{2}+cx_{n}x_{n-1}+dx_{n-1}^{2}}{\alpha

x_{n}^{2}+\beta x_{n}x_{n-1}+\gamma x_{n-1}^{2}},\;\;\;n=0,1,...,

\end{equation*}

where the constants $a,\ b,\ c,\ d,\ \alpha ,\ \beta $

and $\gamma $ are positive real numbers and the initial conditions $x_{-1}$\

and\ $x_{0}\;$are arbitrary non zero real numbers.

**Keywords: **stability, boundedness, periodicity, equilibrium, difference equations.