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 $\mathscr{F} \equiv \{f_\lambda(z)=\lambda f(z): \lambda \in \mathbb{R} \setminus \{0\}\}$ with $f(z) = \frac{\sinh z}{z}$ 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 $\mathscr{F}$.