Stability range of parameters at fixed points for a class of complex dynamics

TL;DR

This paper investigates the stability ranges of parameters at fixed points for a class of complex rational functions, deriving explicit relations and numerically analyzing stability conditions.

Contribution

It provides explicit formulas for parameters at fixed points and explores their stability ranges in complex dynamics.

Findings

Explicit expressions for parameters a and c in terms of the derivative at fixed points.

Numerical stability regimes for different parameter cases.

Relationship between parameters a and c at fixed points.

Abstract

We study the parameters range for the fixed point of a class of complex dynamics with the rational fractional function as $R_{n,a,c}(z)=z^n+\frac{a}{z^n}+c$, where $n=1,2,3,4$ is specified, $a$ and $c$ are two complex parameters. The relationship between two parameters, $a$ and $c$, is obtained at the fixed point. Moreover the explicit expression of the parameter $a$ and $c$ in terms of $\lambda$ is derived, where $\lambda$ is the derivative function at fixed point. The parameter regimes for the stability of the fixed point are presented numerically for some typical different cases.