Abstract
In this paper, we study the equilibrium points, local asymptotic stability of an equilibrium point, instability of equilibrium points, periodicity behavior of positive solutions, and global character of an equilibrium point of a fourth-order system of rational difference equations of the form , where the parameters α, β, γ, , , and initial conditions , , , , , , , are positive real numbers. Some numerical examples are given to verify our theoretical results. MSC:39A10, 40A05.
Highlights
Introduction and preliminariesThe theory of discrete dynamical systems and difference equations developed greatly during the last twenty-five years of the twentieth century
It is very interesting to investigate the behavior of solutions of a system of higher-order rational difference equations and to discuss the local asymptotic stability of their equilibrium points
We investigated some dynamics of an eight-dimensional discrete system
Summary
The following statements are true: (i) A necessary and sufficient condition for all of the roots of ) to have a negative real part is det(Hj) > for j = , , . (ii) A necessary and sufficient condition for the existence of a root of ) with a positive real part is det(Hj) < for some j ∈ { , , . For the equilibrium point P = ( , ) of Equation (i) Let α < β and α < β , the equilibrium point P = ( , ) of the system (ii) If α > β or α > β , the equilibrium point P = ( , ) of the system ) about the equilibrium point ( , ) is given by.
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