Bifurcation and Chaos in Real Dynamics of a Two-Parameter Family Arising from Generating Function of Generalized Apostol-Type Polynomials

Abstract

The aim of this paper is to investigate the bifurcation and chaotic behaviour in the two-parameter family of transcendental functions f λ , n ( x ) = λ x ( e x + 1 ) n , λ > 0 , x ∈ R , n ∈ N \ { 1 } which arises from the generating function of the generalized Apostol-type polynomials. The existence of the real fixed points of f λ , n ( x ) and their stability are studied analytically and the periodic points of f λ , n ( x ) are computed numerically. The bifurcation diagrams and Lyapunov exponents are simulated; these demonstrate chaotic behaviour in the dynamical system of the function f λ , n ( x ) for certain ranges of parameter λ .