Dynamics of a Two-Dimensional Cooperative System of Polynomial Difference Equations With Cubic Terms
DOI:
https://doi.org/10.5644/SJM.18.01.09Keywords:
Difference equation, Equilibrium, Period-two solutions, Basin of Attraction, Global dynamicsAbstract
In this paper we present a local dynamics and investigate the global behavior of the following system of difference equations
$x_{n+1}=ax_{n}^{3}+by_{n}^{3}$
$y_{n+1}=Ax_{n}^{3}+By_{n}^{3}$
$n\in\mathbb{N}_0$
with non-negative parameters and initial conditions $x_{0}$ and $y_{0}$ that are real numbers. We establish the relations for local stability of equilibriums and necessary and sufficient conditions for the existence of period-two solution(s). We then use this result to give global behavior results for special ranges of parameters and determine the basins of attraction of all equilibrium points.
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