Chaotic burst in the dynamics of $f_{\lambda}(z) = \lambda \frac{\sinh(z)}{z}$

In this paper, a one-parameter family of non-critically finite entire functions $\mathscr{F} \equiv {f_{\lambda}(z) = \lambda f(z) : \lambda \in \mathbb{R} \backslash {0}}$ with $f (z) = \lambda \frac{\sinh(z)}{z}$ is considered and the dynamics of the entire transcendental functions $f_{\lambda} \in \mathscr{F}$ is studied in detail. It is shown that there exists a parameter value $\lambda^{*} > 0$ such that the Julia set of $f_{\lambda}(z)$ is nowhere dense subset for $0<|\lambda| \leqslant \lambda^{*} (\approx 1.104)$. For $|\lambda| > \lambda^{*}$ the set explodes and becomes equal to the extended complex plane. This phenomenon is referred to as a chaotic burst in the dynamics of the functions $f_{\lambda}$ in the one-parameter family $\mathscr{F}$.