Design and energy function determination of a memristor-coupled circuit representing a two-dimensional chaotic map

The modeling and dynamical analysis of discrete chaotic systems is a vital research field, and various chaotic maps have been developed using mathematical and control-theoretic approaches. However, physical circuit design of mathematically defined discrete chaotic systems and the computation of their energy functions remain challenging and open problems. In this study, a two-dimensional (2D) chaotic map is constructed using an open-loop modulation coupling method, and its dynamical characteristics are analyzed using bifurcation diagrams. Lyapunov exponents (LEs) and spectral entropy (SE) complexity are also inspected under different parameter configurations. Furthermore, the proposed chaotic map is expressed using two distinct physical memristive circuits: one is composed of a magnetic flux-controlled memristor, a nonlinear resistor, and a capacitor; the other utilizes a charge-controlled memristor, a nonlinear resistor, and an inductor. Moreover, two energy functions are derived from the two memristor-coupled circuits for the proposed chaotic map. The results demonstrate that the mathematical model of the discrete chaotic system can be effectively expressed through these two nonlinear circuits. Our study offers a theoretical foundation and viable methodology for the physical circuit representation of discrete chaotic systems and determination of their energy functions.