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

Abstract

The focus of this research work is to obtain the chaotic behaviour and bifurcation in the real dynamics of a newly proposed family of functions $f_{\lambda ,a}(x)=x+(1–\lambda x)$ ln , depending on two parameters in one dimension, where assume that $\lambda$ is a continuous positive real parameter and $a$ is a discrete positive real parameter. This proposed family of functions is different from the existing families of functions in previous works which exhibits chaotic behaviour. Further, the dynamical properties of this family are analyzed theoretically and numerically as well as graphically. The real fixed points of functions $f_{\lambda ,a}(x)$ are theoretically simulated, and the real periodic points are numerically computed. The stability of these fixed points and periodic points is discussed. By varying parameter values, the plots of bifurcation diagrams for the real dynamics of $f_{\lambda ,a}(x)$ are shown. The existence of chaos in the dynamics of $f_{\lambda ,a}(x)$ is explored by looking period-doubling in the bifurcation diagram, and chaos is to be quantified by determining positive Lyapunov exponents.