A fractal–fractional perspective on chaotic behavior in 4D memristor-nonlinear system

Abstract

The use of fractal–fractional derivatives has attracted considerable interest in the analysis of chaotic and nonlinear systems as they provide a unique capability to represent complex dynamics that cannot be fully described by integer-order derivatives. The fractal–fractional derivative with a power law kernel is used in this paper as an analytical tool to analyze the dynamics of a chaotic integrated circuit. Using coupled ordinary differential equations of classical order, the complexity of an integrated circuit is modeled. The classical order model is generalized via fractal–fractional derivatives of the power law kernel. Moreover, this paper is concerned with investigating the Ulam stability of the model and conducting theoretical studies in order to analyze equilibrium points, identify unique solutions, and verify the existence of such solutions. By examining the complex dynamics that result in chaotic behavior, these investigations shed light on the fundamental properties of integrated circuits. For the purpose of exploring the non-linear fractal–fractional order system, a numerical algorithm has been developed to facilitate our analysis. MATLAB software has been used to implement this algorithm, making it possible to carry out detailed simulations. Simulating solutions are accomplished using 2D and 3D portraits, which provide visual and graphical representations of the results. Throughout the simulation phase, particular attention is given to the impact of fractional order parameter and fractal dimension. As a result of this study, we have gained a comprehensive understanding of the behavior of the system and its response to variations in values.