Difference map and its electronic circuit realization

Abstract

In this paper we study the dynamical behav- ior of the one-dimensional discrete-time system, the so-called iterated map. Namely, a bimodal quadratic map is introduced which is obtained as an amplifica- tion of the difference between well-known logistic and tent maps. Thus, it is denoted as the so-called differ- ence map. The difference map exhibits a variety of be- haviors according to the selection of the bifurcation parameter. The corresponding bifurcations are studied by numerical simulations and experimentally. The sta- bility of the difference map is studied by means of Lyapunov exponent and is proved to be chaotic ac- cording to Devaney's definition of chaos. Later on, a design of the electronic implementation of the differ- ence map is presented. The difference map electronic circuit is built using operational amplifiers, resistors and an analog multiplier. It turns out that this elec- tronic circuit presents fixed points, periodicity, chaos and intermittency that match with high accuracy to the corresponding values predicted theoretically.