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

The aim of this paper is to investigate the bifurcation and chaotic behaviour in the two-parameter family of transcendental functions f(lambda,n) (x) = lambda x/(e(x)+1)(n), lambda > 0, x is an element of R, n is an element of N\{1} which arises from the generating function of the generalized Apostol-type polynomials. The existence of the real fixed points of f(lambda,n) (x) and their stability are studied analytically and the periodic points of f(lambda)(,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(lambda,n) (x) for certain ranges of parameter lambda.