Paper detail

Chaos in Dynamics of a Family of Transcendental Meromorphic Functions

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