International Journal of Advances in Mathematics
International Journal of Advances in Mathematics
2456-6098
Volume 2018
Number 1
2018
January
1
Dynamics of a rational difference equation $
x_{n+1}=ax_{n}+\dfrac{\alpha +\beta x_{n-k}}{A+Bx_{n-k}}$
159
179
EN
Abdul
Khaliq
Department of Mathematics, Riphah Institute of Computing and
Applied Sciences, Riphah International University, Lahore, Pakistan.
khaliqsyed@gmail.com
Sarif
Hassan
Department of Mathematics, Pingla Thana Mahavidyalaya,\\
Paschim Medinipur, 721400, West Bengal, India.
khaliqsyed@gmail.com
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.
Asymptotic stability,
periodic solutions, equilibrium point, chaos, difference equations.
http://adv-math.com/dynamics-rational-difference-equation-x_n1ax_ndfracalpha-beta-x_n-kabx_n-k/
http://adv-math.com/wp-content/uploads/2018/01/final-201737-1.pdf