Chaotification and chaos control of q-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.