Dynamics of Delay Logistic Difference Equation in the Complex Plane

TL;DR

This paper explores the complex delay logistic difference equation, analyzing its stability, solution behaviors, and conjecturing about chaos and periodicity, thus extending understanding of rational difference equations in the complex plane.

Contribution

It introduces a detailed analysis of the delay logistic equation with complex parameters, revealing new behaviors and posing open problems in the complex domain.

Findings

Identification of local stability conditions

Discovery of unique solution characteristics in the complex plane

Open problems on chaos and periodic solutions

Abstract

The dynamics of the delay logistic equation with complex parameters and arbitrary complex initial conditions is investigated. The analysis of the local stability of this difference equation has 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. Some of the interesting observations led us to pose some open problems and conjectures regarding chaotic and higher order periodic solutions and global asymptotic convergence of the delay logistic equation. It is our hope that these observations of this complex difference equation would certainly be an interesting addition to the present art of research in rational difference equations in understanding the behaviour in the complex domain.