Iterations of Meromorphic Functions involving Sine

In this article, the dynamics of a one-parameter family of functions $f_λ(z) = \frac{\sin{z}}{z^2 + λ},$ $λ>0$, are studied. It shows the existence of parameters $0< λ_{1}< λ_{2}$ such that bifurcations occur at $λ_1$ and $λ_2$ for $f_λ$. It is proved that the Fatou set $\mathcal{F}(f_λ)$ is the union of basins of attraction in the complex plane for $λ\in (λ_1, λ_2) \cup (λ_2, \infty)$. Further, every Fatou component of $f_λ$ is simply connected for $λ\geq λ_1$. The boundary of the Fatou set $\mathcal{F}(f_λ)$ is the Julia set $\mathcal{J}(f_λ)$ in the extended complex plane for $λ> 1$. Interestingly, it is found that $f_λ$ has only one completely invariant Fatou component, say $U_λ$ such that $\mathcal{F}(f_λ) = U_λ$ for $λ>λ_2$. Moreover, the characterization of the Julia set of $f_λ$ is seen for $λ\in (λ_1, \infty)\setminus \{λ_2\}$.