Chaotic Behaviour in Two-parameter Family of Transcendental Functions Associated with Exponential Map

Abstract

This article is devoted to the study of chaos and bifurcation in the real dynamics of a newly proposed two-parameter family of transcendental functions. We assume that one parameter is continuous and other parameter is discrete. For certain parameters, the theoretical computations of the real fixed points of a family of functions are given. The numerical simulations of the real periodic points of functions are described. The bifurcation diagrams of the real dynamics of these functions for some selected parameter values are provided. In these bifurcation diagrams, the period-doubling occurs which proceeds to a pathway toward chaos in the dynamics of functions. Further, the periodic-three window is visible in the bifurcation diagrams which implies chaos. Lastly, chaos is quantified in the dynamics of functions by calculating Lyapunov exponents.