Dynamics of two families of meromorphic functions involving hyperbolic cosine function

In this paper, one-parameter families F equivalent to {f(lambda)(z) =lambda (cosh z + 1/cosh z) for z is an element of C : lambda > 0} and G equivalent to {g(lambda)(z) = lambda(cosh z - 1/cosh z) for z is an element of C : lambda > 0} are considered and the dynamics of functions f(lambda) is an element of F and g(lambda) is an element of 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. for the functions f. in the family F but there is no bifurcation in the family G.

Automated summary

This paper investigates the dynamics of functions in two families, F and G, showing that bifurcation and chaotic bursts occur at a certain parameter value for functions in family F, but not in family G.