Design chaotic maps with elegance of mathematical equations and strengthened by the discrete memristor.

Abstract

Despite the plethora of classical chaotic maps that have been proposed, the endeavor to discover novel chaotic maps exhibiting various forms of nonlinearity remains a formidable challenge. Inspired by the esthetic allure of mathematical curves, such as the cardioid and rose curves, a series of chaotic maps have been proposed. The results demonstrate that the resultant phase diagrams reflect the contours of these sophisticated curves. The presence of chaos is substantiated through the estimation of Lyapunov exponents and the application of the 0-1 test algorithm. Upon incorporating the discrete memristor into chaotic maps in two distinct manners, it is revealed that the resultant memristive chaotic maps exhibit heightened complexity. Given that the discrete memristor augments the dimensionality of the chaotic maps, hyperchaos phenomena are observed. Finally, analog circuits of two chaotic maps, namely, a cardioid chaotic map and its counterpart with a discrete memristor, are designed to show the physical realizability. This approach offers an alternative method for the design of chaotic maps and underscores the efficacy of discrete memristors in the enhancement of chaotic behaviors.