Electronic Implementation Replicates Numerical Error in the Chaotic Attractor Formation in a Memristive Circuit

Abstract

Electronic circuit implementation has been widely used to corroborate the numerically detected occurrence of chaotic dynamics in nonlinear dynamical systems. In this paper, we demonstrate that careful consideration is required when dealing with such behaviors, both in terms of numerical methods and the corresponding electronic circuit designs. Specifically, we investigate the dynamics of a circuit consisting of three elements: a passive linear inductor, a passive linear capacitor, and a locally active generic memristor, which is modeled by a three-dimensional four-parameter differential system, having the plane x = 0 invariant with respect to its flow in the phase space. For a particular set of parameter values, the system exhibits a chaotic attractor, numerically obtained. However, depending on the numerical method used, solutions traverse the invariant plane x = 0, which contradicts the analytical properties of the system. Furthermore, an electronic circuit implementation of the model surprisingly mimics such a numerical error, producing the same erroneous crossing behavior. This peculiarity made us consider other numerical methods, seeking to identify the reasons for the behavior, either via stiffness, or by constant step methods, etc. Going beyond the usual numerical difficulties, the search for a more suitable method was intense and culminated in an implementation specifically made to be used in problems with sharp growth, known as the Cash-Karp method, which produced reasonable solutions. In this context, we provide results by different numerical methods, highlighting potential pitfalls in both numerical simulations and physical implementation in the study of chaotic systems.