Dynamics of the complex rational delay recursive sequence $\displaystyle{z_{n+1}=\frac{\alpha+\beta z_{n-k}}{\gamma - z_{n}}}$

TL;DR

This paper investigates the complex dynamics of a rational delay difference equation, revealing the existence of periodic and chaotic solutions, which differ significantly from the real parameter case.

Contribution

It introduces the analysis of complex parameters in the delay rational difference equation, demonstrating new dynamical behaviors including chaos.

Findings

Existence of prime period two solutions

Higher order periodic solutions

Chaotic solutions confirmed computationally

Abstract

Dynamics of the delay rational difference equation $\displaystyle{z_{n+1}=\frac{\alpha+\beta z_{n-k}}{\gamma - z_{n}}}$ with complex parameters $\alpha$, $\beta$, $\gamma$ and arbitrary complex initial conditions is investigated. Existence of prime period two solutions and higher order periods are ensured in the complex parameters unlike in the case of real parameters of the same rational difference equation. In addition, a new dynamical behavior, chaotic solutions of the difference equation are ensured computationally.