Dynamics of $\displaystyle{z_{n+1}=\frac{\alpha + \alpha z_{n}+\beta z_{n-1}}{1+z_{n}}}$ in Complex Plane

TL;DR

This paper explores the complex dynamics of a second-order rational difference equation, revealing parameter trichotomy and chaotic solutions in the complex plane, extending known real-case results.

Contribution

It extends the analysis of a known difference equation to complex parameters and initial conditions, uncovering chaos and confirming parameter trichotomy in the complex domain.

Findings

Parameter trichotomy in complex plane confirmed

Chaotic solutions identified in complex setting

Extension of real-case dynamics to complex parameters

Abstract

The dynamics of the second order rational difference equation $\displaystyle{z_{n+1}=\frac{\alpha + \alpha z_{n}+\beta z_{n-1}}{1+z_{n}}}$ with complex parameters $\alpha$, $\beta$ and arbitrary complex initial conditions is investigated. The same difference equation is well studied with positive real parameters and initial values and one of the main results on trichotomy of parameters is revisited in the complex set-up and the similar result is found to be true. In addition, the chaotic solutions of the equation is fetched out in the complex set-up which was absent in the real scenario.