Analytical Study of a Class of Rational Difference Equations

TL;DR

This paper analytically solves a specific fourth-order rational difference equation, explores the behavior of its solutions including convergence and unboundedness, and illustrates findings with numerical examples.

Contribution

It provides an explicit solution to a class of fourth-order rational difference equations and analyzes their dynamic behavior.

Findings

Solutions can converge, diverge, or become periodic.

Conditions for convergence and unboundedness are identified.

Numerical examples illustrate theoretical results.

Abstract

We obtain the solution of the fourth order difference equation $$ x_{n+1}=\frac{ \alpha x_{n-3}}{A+B x_{n-1}x_{n-3}}$$ with the initial conditions; $x_{-3}=d,$ $x_{-2}=c,$ $x_{-1}=b,$ and $x_{0}=a$ are arbitrary nonzero real numbers, $\alpha$, $A$ and $B$ are arbitrary constants. The result is used to study the convergence of solutions, the existence of unbounded solutions and the convergence to periodic solutions. We illustrate the results by several numerical examples.