Complex Dynamics of $\displaystyle{z_{n+1}=\frac{\alpha + \beta z_{n}+ \gamma z_{n-1}}{A + B z_n + C z_{n-1}}}$

TL;DR

This paper explores the complex dynamics of a second order rational difference equation with complex parameters, revealing stability, boundedness, and chaotic behaviors unique to the complex setting, and posing open problems.

Contribution

It introduces the study of chaos and complex solution behaviors in this difference equation, which were not observed in the real parameter case.

Findings

Identification of stability and boundedness conditions in the complex domain

Discovery of chaotic solutions unique to complex parameters

Observation of higher order periodic solutions and open problems

Abstract

The dynamics of the second order rational difference equation $\displaystyle{z_{n+1}=\frac{\alpha + \beta z_{n}+ \gamma z_{n-1}}{A + B z_n + C z_{n-1}}}$ with complex parameters and arbitrary complex initial conditions is investigated. In the complex set up, the local asymptotic stability and boundedness are studied vividly for this difference equation. 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 are shown. The chaotic solutions of the difference equation is absolutely new feature in the complex set up which is also shown in this article. Some of the interesting observations led us to pose some open interesting problems regarding chaotic and higher order periodic solutions and global asymptotic convergence of this equation.