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 \({\mathcal {G}}\).

在本文中，单参数族\({\mathcal {F}}\equiv \left\{ f_{\lambda }(z)=\lambda \left( \cosh z+\frac{1}{\cosh z} \right) \;\text{ for }\; z\in {\mathbb {C}}: \lambda >0\right\} \)和\({\mathcal {G}}\equiv \left\{ g_ {\lambda }(z)=\lambda \left( \cosh z-\frac{1}{\cosh z}\right) \;\text{ for }\; z\in {\mathbb {C}}: \lambda >0\right\} \)被考虑，函数\(f_{\lambda }\in {\mathcal {F}}\)和\(g_{\lambda }\in {\mathcal {G }}\)进行调查。结果表明，函数\(f_{\lambda }\)和\(g_{\lambda }\)都具有有限个奇异值并且原点始终是\(g_{\lambda }( z)\). 通过研究实不动点的性质和\的奇异值，研究了\(f_{\lambda }(z)\)和\(g_{\lambda }(z)\)在扩展复平面上的动力学(f_{\lambda }\)和\(g_{\lambda }\)。结果表明，对于族\({\mathcal {F}}\) 中的函数\(f_{\lambda }\)，在\(\lambda \)的某个参数值处发生分叉和混沌爆发，但有在家庭\({\mathcal {G}}\) 中没有分叉。