Abdul Khaliq and SK. Sarif Hassan, Dynamics of a rational difference equation $x_{n+1}=ax_{n}+\dfrac{\alpha +\beta x_{n-k}}{A+Bx_{n-k}}$, Volume 2018, Number 1, Pages 159-179, 2018

Abdul Khaliq and SK. Sarif Hassan, Dynamics of a rational difference equation $x_{n+1}=ax_{n}+\dfrac{\alpha +\beta x_{n-k}}{A+Bx_{n-k}}$, Volume 2018, Number 1, Pages 159-179, 2018

Abstract: A nonlinear rational difference equation $x_{n+1}=ax_{n}+\dfrac{\alpha +\beta x_{n-k}}{A+Bx_{n-k}},\;\;\;n=0,1,...,$ of higher order is considered to apprehend the dynamics viz. the invariant intervals, periodic solutions, the character of semi-cycles and global asymptotic stability. {\Large \noindent }Here all the parameters $a$, $% \alpha$, $\beta$ and $A$, $B$ and the initial conditions $x_{k},\ ...,\ x_{1},\ x_{0}$ are positive real numbers $k=\{1,2,3,...\}$. It is shown that the equilibrium point is globally asymptotically stable under the condition \ $\beta \leq A$, and the unique positive solution is also globally asymptotically stable under the conditions $\beta$ $\leq$ $A$ $\leq$ $% \beta$. Finally, we study the global stability of this equation through numerically solved examples and confirm our theoretical discussion through it. We also have considered parameters as real numbers (positive and negative) just to comprehend additional dynamics.

Keywords: Asymptotic stability, periodic solutions, equilibrium point, chaos, difference equations.

Abdul Khaliq and SK. Sarif Hassan, Dynamics of a rational difference equation $x_{n+1}=ax_{n}+\dfrac{\alpha +\beta x_{n-k}}{A+Bx_{n-k}}$, International Journal of Advances in Mathematics, Volume 2018, Number 1, Pages 159-179, 2018.