Chaotic behavior in real dynamics and singular values of family of generalized generating function of Apostol-Genocchi numbers

Abstract

Chaotic behavior in the real dynamics and singular values of a two-parameter family of generalized generating function of Apostol-Genocchi numbers, f λ,a (z) = λ 2z e az +1 , λ, a ∈ R\{0}, are investigated. The real fixed points of f λ,a (z) and their nature are studied. It is seen that bifurcation and chaos occur in the real dynamics of f λ,a (z). It is also found that the function f λ,a (z) has infinitely many singular values for a > 0 and a < 0. The critical values of f λ,a (z) lie inside the open disk, the annulus and exterior of the open disk at center origin for a > 0 and a < 0.

Chaotic Behaviour and Bifurcation in Real Dynamics of Two-Parameter Family of Functions including Logarithmic Map

“…The real dynamics of ðb x − 1Þ/x associated with one parameter is explained in [42]. For two-parameter families of special type of generating functions, the bifurcation as well as chaos in the real dynamics is described by author [43,44]. The description of the real dynamics of transcendental functions is essential in many phenomena.…”