Chaotification and chaos control of <i>q</i>-homographic map

This paper concerns the dynamical study of the q -deformed homographic map, namely, the q -homographic map, where q -deformation is introduced by Jagannathan and Sinha with the inspiration from Tsalli’s q -exponential function. We analyze the q -homographic map by computing its basic nonlinear dynamics, bifurcation analysis, and topological entropy. We use the notion of a false derivative and the generalized Lambert W function of the rational type to estimate the upper bound on the number of fixed points of the q -homographic map. Furthermore, we discuss chaotification of the q -deformed map to enhance its complexity, which consists of adding the remainder of multiple scaling of the map’s value for the next generation using the multiple remainder operator. The chaotified q -homographic map shows high complexity and the presence of robust chaos, which have been theoretically and graphically analyzed using various dynamical techniques. Moreover, to control the period-doubling bifurcations and chaos in the q -homographic map, we use the feedback control technique. The theoretical discussion of chaos control is illustrated by numerical simulations.