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.