Study of the dynamical behavior of an Ikeda-based map with a discrete memristor

Abstract

In 1971, Chua suggested that there should be a fourth electrical component in addition to resistor, capacitor and inductor. This new component was proposed to be called “memristor”. Due to lack of commercially available memristors many emulators are mainly used in nonlinear circuits. Also, the memristance functions are used in many dynamical systems in order to enhance their complexity. Furthermore, the use of memristors in discrete chaotic maps is an interesting research topic, due to the applications of such systems. In this work, a memristor-based Ikeda mapping model is presented by coupling a discrete memristance function with Ikeda map. To investigate system’s dynamical behavior a host of nonlinear tools has been used, such as bifurcation and continuation diagrams, maximal Lyapunov exponent diagrams, and Kaplan–Yorke conjecture. Interesting phenomena related to chaos has been observed. More specifically, regular (periodic and quasiperiodic) and chaotic orbits, route to chaos through the mechanism of period doubling and crisis phenomena, have been found. Moreover, higher value of the internal state of the memristor, revealed chaotic behavior in a bigger area. Also, from the comparison of the bifurcation diagrams with the respective continuation diagrams coexisting attractors have been found. Finally, two-parameter bifurcation-like diagrams revealed the system’s rich dynamical behavior.