Complex Dynamics of the Difference Equation $z_{n+1}=\frac{\alpha}{z_{n}}+ \frac{\beta}{z_{n-1}}$

TL;DR

This paper explores the complex dynamics of a second-order rational difference equation with complex parameters and initial conditions, revealing rich behaviors, stability properties, and open problems in the complex plane.

Contribution

It provides a detailed analysis of stability, periodicity, and complex behaviors of the difference equation, introducing new insights into its chaotic and higher-order solutions.

Findings

Rich asymptotic behaviors in the complex plane

Stability and periodicity conditions analyzed

Identification of open problems and conjectures

Abstract

The dynamics of the second order rational difference equation in the title with complex parameters and arbitrary complex initial conditions is investigated. Two associated difference equations are also studied. The solutions in the complex plane of such equations exhibit many rich and complicated asymptotic behavior. The analysis of the local stability of these three difference equations and periodicity have been carried out. We further exhibit several interesting characteristics of the solutions of this equation, using computations, which does not arise when we consider the same equation with positive real parameters and initial conditions. Many interesting observations led us to pose several open problems and conjectures of paramount importance regarding chaotic and higher order periodic solutions and global asymptotic convergence of such difference equations. It is our hope that these observations of these complex difference equations would certainly be new add-ons to the present art of research in rational difference equations in understanding the behavior in the complex domain.