Dynamics of two families of meromorphic functions involving hyperbolic cosine function

Abstract

In this paper, one-parameter families ${\mathcal {F}}\equiv \left\{ f_{\lambda }(z)=\lambda \left( \cosh z+\frac{1}{\cosh z}\right) \;\text{ for }\; z\in {\mathbb {C}}: \lambda >0\right\}$ and ${\mathcal {G}}\equiv \left\{ g_{\lambda }(z)=\lambda \left( \cosh z-\frac{1}{\cosh z}\right) \;\text{ for }\; z\in {\mathbb {C}}: \lambda >0\right\}$ are considered and the dynamics of functions $f_{\lambda }\in {\mathcal {F}}$ and $g_{\lambda }\in {\mathcal {G}}$ are investigated. It is shown that both the functions $f_{\lambda }$ and $g_{\lambda }$ have finite number of singular values and the origin is always an attracting fixed point of $g_{\lambda }(z)$. The dynamics of $f_{\lambda }(z)$ and $g_{\lambda }(z)$ on the extended complex plane are studied by investigating the nature of the real fixed points and the singular values of $f_{\lambda }$ and $g_{\lambda }$. It is shown that a bifurcation and chaotic burst occur at a certain parameter value of $\lambda$ for the functions $f_{\lambda }$ in the family ${\mathcal {F}}$ but there is no bifurcation in the family ${\mathcal {G}}$.