Explicit Solutions and Bifurcations for a System of Rational Difference Equations

Bashir Al-Hdaibat,Ramadan Sabra,Saleem Al-Ashhab

doi:10.3390/math7010069

Abstract

In this paper, we consider the explicit solution of the following system of nonlinear rational difference equations: x n + 1 = x n - 1 / x n - 1 + r , y n + 1 = x n - 1 y n / x n - 1 y n + r , with initial conditions x - 1 , x 0 and y 0 , which are arbitrary positive real numbers. By doing this, we encounter the hypergeometric function. We also investigate global dynamics of this system. The global dynamics of this system consists of two kind of bifurcations.

Highlights

Introduction and PreliminariesDifference equations play an important role in many disciplines including biology, ecology, physics, economics, and many more [1,2]
The study of rational difference equations is of crucial importance, since we know so little about such equations
Our goal in this paper is to investigate the explicit solution of the following system of nonlinear rational difference equations systems

Summary

Introduction and Preliminaries

Difference equations play an important role in many disciplines including biology, ecology, physics, economics, and many more [1,2] These equations appear naturally as discrete analogs of differential equations, and as numerical solutions of differential and delay differential equations. Amleh et al [6] considered the nonlinear difference equation xn = α + xn−1 /xn , where the parameter α and the initial condition x0 are arbitrary positive real number. They proved that, if xn is a nontrivial solution of the equation so that there is a n0 such that xn ≥ α + 1 for n ≥ n0 , xn is monotonically convergent to zero. The bifurcations that occur in Equation (1) are numerically studied with a one-parameter bifurcation analysis

The General Solution

Bifurcation Analysis