Computational Complex Dynamics of $\displaystyle{f_{\alpha, \beta, \gamma, \delta}(z)=\frac{\alpha z + \beta}{\gamma z^2 +\delta z}}$

TL;DR

This paper computationally explores the complex dynamics of a family of rational maps, revealing higher-period solutions, chaos, and fractal structures absent in real line cases.

Contribution

It introduces a detailed computational analysis of complex rational maps, uncovering new periodic and chaotic behaviors not seen in real dynamics.

Findings

Existence of higher-period solutions in complex plane

Presence of chaotic fractal and non-fractal solutions

Special parameter cases exhibit unique dynamics

Abstract

The dynamics of the family of maps $\displaystyle{f_{\alpha, \beta, \gamma, \delta}(z)=\frac{\alpha z + \beta}{\gamma z^2 +\delta z}}$ in complex plane is investigated computationally. This dynamical system $z_{n+1}=f_{\alpha, \beta, \gamma, \delta}(z_n)=\frac{\alpha z_n + \beta}{\gamma z_n^2 +\delta z_n}$ has periodic solutions with higher periods which was absent in the real line scenario. It is also found that there are chaotic fractal and non-fractal like solutions of the dynamical systems. A few special cases of parameters are also have been taken care.