Chaos in Dynamics of a Family of Transcendental Meromorphic Functions

TL;DR

This paper investigates the chaotic dynamics and Julia sets of a family of transcendental meromorphic functions, analyzing bifurcations, computing Lyapunov exponents, and characterizing the Julia sets through computational simulations.

Contribution

It provides a detailed analysis of bifurcations, chaos, and Julia set characterization for the family ta_mbda(z), including computational visualization and comparison with related functions.

Findings

Chaotic behavior occurs when ta_mbda(z) crosses certain parameter thresholds.

Lyapunov exponents quantify the degree of chaos for specific parameter values.

Julia sets are characterized as complements of basins of attraction and contain complements of attracting periodic orbits for large mbda.

Abstract

The characterization and properties of Julia sets of one parameter family of transcendental meromorphic functions $\zeta_\lambda(z)=\lambda \frac{z}{z+1} e^{-z}$, $\lambda >0$, $z\in \mathbb{C}$ is investigated in the present paper. It is found that bifurcations in the dynamics of $\zeta_\lambda(x)$, $x\in {\mathbb{R}}\setminus \{-1\}$, occur at several parameter values and the dynamics of the family becomes chaotic when the parameter $\lambda$ crosses certain values. The Lyapunov exponent of $\zeta_\lambda(x)$ for certain values of the parameter $\lambda$ is computed for quantifying the chaos in its dynamics. The characterization of the Julia set of the function $\zeta_\lambda(z)$ as complement of the basin of attraction of an attracting real fixed point of $\zeta_\lambda(z)$ is found here and is applied to computationally simulate the images of the Julia sets of $\zeta_\lambda(z)$. Further, it is established that the Julia set of $\zeta_\lambda(z)$ for $\lambda>(\sqrt{2}+1) e^{\sqrt{2}}$ contains the complement of attracting periodic orbits of $\zeta_\lambda(x)$. Finally, the results on the dynamics of functions $\lambda \tan z$, $\lambda \in {\mathbb{\hat{C}}}\setminus\{0\}$, $E_{\lambda}(z) = \lambda \frac{e^{z} -1}{z}$, $\lambda > 0$ and $f_{\lambda}=\lambda f(z)$, $\lambda>0$, where $f(z)$ has certain properties, are compared with the results found in the present paper.