Chaos in Dynamics of a Family of Transcendental Meromorphic Functions

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 $\lambda \tan z$, $\lambda \in \hat{\mathbb{C}}\setminus \{0\}$, $E_{\lambda }(z)=\lambda \frac{e^z-1}{z}$, $\lambda >0$ and $f_{\lambda }=\lambda f(z)$, $\lambda >0$, where $f(z)$ has certain properties, are compared with the results found in the present paper.

18 pages