Forbidden Set of the Rational Difference Equation $x_{n+1} = x_n x_{n-k}/(ax_{n-k+1} +x_n x_{n-k+1} x_{n-k})$

TL;DR

This paper determines the forbidden set of a specific nonlinear rational difference equation by deriving its closed-form solution, and discusses its limiting behavior, periodicity, and potential generalizations, addressing an open problem in the field.

Contribution

The paper explicitly finds the forbidden set of a nonlinear difference equation by solving it in closed form, providing new insights into its dynamics and open problems.

Findings

Forbidden set explicitly characterized

Solutions exhibit specific limiting and periodic behaviors

Numerical examples illustrate theoretical results

Abstract

This short note aims to answer one of the open problems raised by F. Balibrea and A. Cascales in \cite{bc}. In particular, the forbidden set of the nonlinear difference equation $x_{n+1} = x_n x_{n-k}/(ax_{n-k+1} +x_n x_{n-k+1} x_{n-k})$, where $k$ is a positive integer and $a$ is a positive constant, is found by first computing the closed form solution of the given equation. Additional results regarding the limiting properties and periodicity of its solutions are also discussed. Numerical examples are also provided to illustrate the exhibited results. Lastly, a possible generalization of this present work is offered as an open problem.