Dynamics of two families of meromorphic functions involving hyperbolic cosine function

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