Local-Activity and Simultaneous Zero-Hopf Bifurcations Leading to Multistability in a Memristive Circuit

In this paper, we consider a memristive circuit consisting of three elements: a passive linear inductor, a passive linear capacitor and an active memristive device. The circuit is described by a four-parameter system of ordinary differential equations. We study in detail the role of parameters in the dynamics of the system. Using the existence of first integrals, we show that the circuit may present a continuum of stable periodic orbits, which arise due to the occurrence of infinitely many simultaneous zero-Hopf bifurcations on a line of equilibria located in the region where the memristance is negative and, consequently, the memristive device is locally-active. These bifurcations lead to multistability, which is a difficult and interesting problem in applied models, since the final state of a solution depends crucially on its initial condition. We also study the control of multistability by varying a parameter related to the state variable of the memristive device. All analytical results obtained were corroborated by numerical simulations.

Dynamic Behaviors Analysis of a Novel Fractional-Order Chua’s Memristive Circuit

This paper proposed a novel fractional-order Chua’s memristive circuit. Firstly, a fractional-order mathematical model of a diode bridge generalized memristor with RLC filter cascade is established, and simulations verify that the fractional-order generalized memristor satisfies the basic characteristics of a memristor. Secondly, the capacitor and inductor in Chua’s chaotic circuit are extended to the fractional order, and the fractional-order generalized memristor is used instead of Chua’s diode to establish the fractional-order mathematical model of chaotic circuit based on RLC generalized memristor. By studying the stability analysis of the equilibrium point and the influence of the circuit parameters on the system dynamics, the dynamic characteristics of the proposed chaotic circuit are theoretically analyzed and numerically simulated. The results show that the proposed fractional-order memristive chaotic circuit has gone through three states: period, bifurcation, and chaos, and a narrow period window appears in the chaotic region. Finally, the equivalent circuit method is adopted in PSpice to realize the construction of the fractional-order capacitance and inductance, and the simulation of the fractional-order memristive chaotic circuit is completed. The results further verify the correctness of the theoretical analysis.

Recent Advances in Dimensionality Reduction Modeling and Multistability Reconstitution of Memristive Circuit

Due to the introduction of memristors, the memristor-based nonlinear oscillator circuits readily present the state initial-dependent multistability (or extreme multistability), i.e., coexisting multiple attractors (or coexisting infinitely many attractors). The dimensionality reduction modeling for a memristive circuit is carried out to realize accurate prediction, quantitative analysis, and physical control of its multistability, which has become one of the hottest research topics in the field of information science. Based on these considerations, this paper briefly reviews the specific multistability phenomenon generating from the memristive circuit in the voltage-current domain and expounds the multistability control strategy. Then, this paper introduces the accurate flux-charge constitutive relation of memristors. Afterwards, the dimensionality reduction modeling method of the memristive circuits, i.e., the incremental flux-charge analysis method, is emphatically introduced, whose core idea is to implement the explicit expressions of the initial conditions in the flux-charge model and to discuss the feasibility and effectiveness of the multistability reconstitution of the memristive circuits using their flux-charge models. Furthermore, the incremental integral transformation method for modeling of the memristive system is reviewed by following the idea of the incremental flux-charge analysis method. The theory and application promotion of the dimensionality reduction modeling and multistability reconstitution are proceeded, and the application prospect is prospected by taking the synchronization application of the memristor-coupled system as an example.