Complexity

Volume 2018, Article ID 5935637, 13 pages

https://doi.org/10.1155/2018/5935637

## Memristor-Based Canonical Chua’s Circuit: Extreme Multistability in Voltage-Current Domain and Its Controllability in Flux-Charge Domain

School of Information Science and Engineering, Changzhou University, Changzhou 213164, China

Correspondence should be addressed to Bocheng Bao; moc.621@oabnivrem

Received 16 December 2017; Accepted 17 January 2018; Published 25 March 2018

Academic Editor: Viet-Thanh Pham

Copyright © 2018 Han Bao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper investigates extreme multistability and its controllability for an ideal voltage-controlled memristor emulator-based canonical Chua’s circuit. With the voltage-current model, the initial condition-dependent extreme multistability is explored through analyzing the stability distribution of line equilibrium point and then the coexisting infinitely many attractors are numerically uncovered in such a memristive circuit by the attraction basin and phase portraits. Furthermore, based on the accurate constitutive relation of the memristor emulator, a set of incremental flux-charge describing equations for the memristor-based canonical Chua’s circuit are formulated and a dimensionality reduction model is thus established. As a result, the initial condition-dependent dynamics in the voltage-current domain is converted into the system parameter-associated dynamics in the flux-charge domain, which is confirmed by numerical simulations and circuit simulations. Therefore, a controllable strategy for extreme multistability can be expediently implemented, which is greatly significant for seeking chaos-based engineering applications of multistable memristive circuits.

#### 1. Introduction

Initial condition-dependent extreme multistability, first encountered in several coupled nonlinear dynamical systems [1–3], is a coexisting phenomenon of infinitely many attractors for a given set of system parameters. More recently, due to the existence of infinitely many equilibrium points, for example, line equilibrium point or plane equilibrium point, this special dynamical phenomenon of extreme multistability is naturally exhibited in a class of ideal flux/voltage-controlled memristor-based chaotic circuits/systems [4–9], thereby leading to the emergence of infinitely many disconnected attractors.

Extreme multistability is a fantastic kind of multistability, which makes a nonlinear dynamical circuit or system supply great flexibility for its potential uses in chaos-based engineering applications [10–12], but also raises new challenges for its control of the existing multiple stable states [11–14]. Generally, multistability is confirmed in hardware experiments by randomly switching on and off experimental circuit supplies [9, 15–21] or by MATLAB numerical or PSPICE/PSIM circuit simulations [4–8, 22–28]. Consequently, to direct the nonlinear dynamical circuit or system to a desired oscillating mode, an effective control approach should be proposed [12]. To this end, this paper takes an ideal voltage-controlled memristor emulator-based canonical Chua’s circuit as an example; a controllable strategy for extreme multistability is achieved through converting the initial condition-dependent dynamics in the voltage-current domain into the system parameter-associated dynamics in the flux-charge domain [29, 30].

Besides, for a memristor-based circuit or system with line equilibrium point or plane equilibrium point, its stability at the equilibrium point is very difficult to be determined due to the existence of one or two zero eigenvalues [5–9], which results in the fact that the coexisting infinitely many attractors’ behaviors can not be precisely interpreted from the stabilities of the nonzero eigenvalues. As a matter of fact, the memristor initial condition and other initial conditions all have dynamical effects on the memristor-based circuit or system [8, 9]. However, the dynamical effects are implied, which can not be explicitly expressed in the voltage-current domain. How about the memristor-based circuit or system in the flux-charge domain?

Flux-charge analysis method was first postulated as a tool of dimensionality reduction [31–36], in which the initial conditions of the memristor-based circuit or system are not precisely formulated, thereby leading to the absence of the initial condition-dependent dynamical behaviors [34–36]. In the last two years, a new flux-charge analysis method is reported in [29, 30], which judiciously utilizes the incremental flux and charge to substitute the conventional flux and charge and efficaciously solves the issue of the original flux-charge analysis method. Accordingly, based on the voltage-current relation, an accurate flux-charge relation of the ideal voltage-controlled memristor emulator is established. With the accurate constitutive relation, an incremental flux-charge model for the memristor-based canonical Chua’s circuit is constructed, upon which all the initial conditions in the voltage-current model can be explicitly formulated by the system parameters in the flux-charge model and the multiple stable states can be consequently controlled by changing the initial condition-related system parameters.

The rest of the paper is structured as follows. In Section 2, an ideal voltage-controlled memristor emulator-based canonical Chua’s circuit is presented. With the voltage-current model, the initial condition-dependent extreme multistability is explored and then the coexisting infinitely many attractors are numerically uncovered. In Section 3, based on the accurate constitutive relation of the memristor emulator, a set of incremental flux-charge equations for the memristor-based canonical Chua’s circuit are formulated and a dimensionality reduction model is thus established, upon which the feasibility of the flux-charge analysis method is verified by MATLAB numerical simulations. In Section 4, an equivalent circuit of the incremental flux-charge model is designed and circuit simulations for the initial condition-dependent behaviors are executed, from which the controllability of extreme multistability is physically confirmed. The conclusions are drawn in the last section.

#### 2. Extreme Multistability in the Voltage-Current Domain

Based on a canonical Chua’s circuit and an ideal voltage-controlled memristor emulator, a new memristor-based canonical Chua’s circuit is constructed, as shown in Figure 1(a), which is simple and physically realizable. The ideal voltage-controlled memristor emulator is equivalently implemented with an electronic circuit via op-amp integrators and analog multipliers [5, 6, 34], as shown in Figure 1(b). In our next work, the considered circuit parameters remained unchanged and are listed in Table 1, where is the total gain of two multipliers and .