Complexity

Volume 2019, Article ID 3870327, 17 pages

https://doi.org/10.1155/2019/3870327

## A Novel Memductor-Based Chaotic System and Its Applications in Circuit Design and Experimental Validation

^{1}School of Mechanical and Precision Instrument Engineering, Xi’an University of Technology, Xi’an 710048, China^{2}School of Physics and Electromechanical Engineering, Hexi University, Zhangye 734000, China^{3}State Key Laboratory of ASIC & System, Fudan University, Shanghai 200433, China^{4}School of Printing, Packaging Engineering and Digital Media Technology, Xi’an University of Technology, Xi’an 710048, China^{5}School of Information Science and Engineering, Lanzhou University, Lanzhou 730000, China

Correspondence should be addressed to Yanjun Lu; nc.ude.tuax@ulnujnay

Received 16 April 2018; Revised 16 July 2018; Accepted 5 August 2018; Published 3 January 2019

Academic Editor: Viet-Thanh Pham

Copyright © 2019 Li Xiong 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 is expected to introduce a novel memductor-based chaotic system. The local dynamical entities, such as the basic dynamical behavior, the divergence, the stability of equilibrium set, and the Lyapunov exponent, are all investigated analytically and numerically to reveal the dynamic characteristics of the new memductor-based chaotic system as the system parameters and the initial state of memristor change. Subsequently, an active control method is derived to study the synchronous stability of the novel memductor-based chaotic system through making the synchronization error system asymptotically stable at the origin. Further to these, a memductor-based chaotic circuit is designed, realized, and applied to construct a new memductor-based secure communication circuit by employing the basic electronic components and memristor. Furthermore, the design principle of the memductor-based chaotic circuit is thoroughly analyzed and the concept of “the memductor-based chaotic circuit defect quantification index” is proposed for the first time to verify whether the chaotic output is consistent with the mathematical model. A good qualitative agreement is shown between the simulations and the experimental validation results.

#### 1. Introduction

With the deep study of the chaotic systems and chaotic circuits, the concept of memristor was first put forwarded by Chua in 1971 [1]. Memristor is the fourth circuit component after capacitor, resistor, and inductor were coined, which is actually a nonlinear resistor with natural memory function. Nevertheless, we did not see significant progress on relevant research at that time on account of insufficient attention was paid to the memristor. The immature nanomanufacturing technology and difficult manufacturing of memristor with real materials all contributed to the slow progress on memristor [2]. It was not until 2008 that the HP Laboratories confirmed the existence of memristor and simultaneously a memristor-based real device was coined with its results published in *Nature* [3, 4]. Since then, memristor has become a hot research spot of chaos and it drew much more eyes from researchers engaged in various areas of science and engineering [5–10]. It is well known that memristor has two models, namely, charge control and chain control. Among them, charge control exports memristor, while chain control exports memductor. If the memristor is a constant, it becomes the same concept as resistor. Correspondingly, the physical meaning of memductor is equivalent to conductance. Because the design of memductor is more convenient than the design of memristor in the design of chaotic circuits, the model of memductor is studied in this paper.

As a tunable nonlinear device with small size and low power consumption, memristor is quite suitable for the applications of high-frequency chaotic circuit, image encryption, and chaotic secure communication. It is no wonder that, in recent years, utilizing memristor to construct chaotic circuits has attracted close attention of quite a number of researchers [11–15]. Among them, Itoh and Chua adopted the memristor with a characteristic curve for the monotone rise and piecewise linear to replace the diode in Chua’s circuit and followed by the chaotic oscillation circuit based on memristor was derived [6]. Similarly, Muthuswamy and Kokate replaced the memristor with piecewise linear model instead of Chua’s diode and analyzed the dynamic characteristics of the system after replacement. The results indicated that the chaotic characteristics of the system were more complex than that of the classical Chua’s [7]. In 2010, Muthuswamy and Chua proposed the most simple third-order memristor chaotic circuit so far and in [8, 9] showed the experimental results of the corresponding hardware circuit, whose greatest feature was the simple structure. It was connected in series simply by a linear inductor, a linear capacitor, and a nonlinear memristor. In addition, Bao et al. carried on the research on the memristor chaotic circuit and realized a series of new Chua’s memristor chaotic circuits by using the smooth model magnetic controlled memristor [10–12]. At present, the proposed memristor chaotic oscillation circuits of different structure and types [13–23] include the chaotic circuits with two memristors [16], integer-order memristor chaotic circuit [18], fractional-order memristor chaotic circuit [19], and memristor-based circuit for neural networks [23], whereas most of the researchers focus on theoretical analysis and numerical simulation for the memristive chaotic system and the experimental validation of the hardware circuit is rarely seen because those memristive chaotic circuits are theoretically established and their feasibility to be implemented by using hardware circuit is still not known. In particular, it is more difficult to design and implement a practical circuit for certain more complicated memductor chaotic systems. Therefore, we construct a novel memductor-based chaotic circuit and implement the experimental validation of the hardware circuit for above reason.

Moreover, in order to meet the security requirements of chaotic secure communication, researchers proposed a method to improve the predictability and complexity of the system by constructing hyperchaotic systems [24–26] and memristor-based chaotic systems, since memristor is a nonlinear component, whose memory ability [27–31] of the current by convection is not available in conventional chaotic circuit elements. In this way, it is especially suitable for the chaotic secure communication field [32–36]. Although the application research of memristor is just the beginning in the field of chaotic secure communication, it has great potentials and advantages in improving the confidentiality and security of chaotic secure communication system. So far, there is no literature to implement the memductor-based chaotic secure communication in chaotic modulation way. In this paper, chaotic modulation is adopted to implement the memductor-based secure communication based on the novel memductor-based chaotic circuit.

The contribution of this paper is that a new method for constructing ordinary chaotic system into memductor-based chaotic system is proposed by using memristor as nonlinear term. Then, we perform a detailed analysis, active control, synchronous stability analysis [37–40], and secure communication of the novel memductor-based chaotic system. The active control is implemented, and the synchronization stability results are determined by using Lyapunov stability theory. The corresponding physical circuit implementation is also proposed to show the accuracy and efficiency of the memductor-based chaotic circuit. The analog circuit implementation results match with the Multisim and MATLAB simulation results. In addition, the concept of “the memductor-based chaotic circuit defect quantification index” is first proposed to verify whether the chaotic output is consistent with the mathematical model through deep analysis on the design principle of memductor-based chaotic circuit. Our research provides important theoretical and technical basis for the realization of the large-scale integrated circuit with memductor. This paper is expected to serve as a further step to apply memductor into real-world secure communication.

This paper falls into 6 parts. In Section 2, a novel 4D memductor-based chaotic system is constructed. In following Section 3, several qualitative issues about the novel memductor-based chaotic system, such as the basic dynamical behavior, divergence, stability of the equilibrium set, bifurcation, Poincaré map, and synchronous stability, are investigated analytically and numerically. In Section 4, the proposed memductor-based chaotic circuit is implemented in an analog electronic circuit. After that, a new memductor-based chaotic secure communication circuit is proposed based on the novel memductor-based chaotic circuit in Section 5. Finally, some conclusions and discussions are drawn in Section 6.

#### 2. The Construction of a Novel Memductor-Based Chaotic System

##### 2.1. A Specific Memductor Model

Apart from the three basic circuit components, including capacitor, resistor, and inductor, the fourth circuit component is memristor, which derives from the magnetic flux and charge in the circuit. And the resistance value of the memristor varies with the current flowing through the circuit. When the circuit is powered down, the resistance value of the memristor still remains valid before the power is broken. Therefore, memristor is actually a nonlinear resistor with natural memory function.

The memristor is defined as the relation between the magnetic flux and the charge quantity, that is,

Memristor can be divided into accumulation charge memristor and magnetic flux-controlled memristor. For a charge-controlled memristor, is easily obtained by

For (2), differentiation can be easily obtained as follows:

Thus, can be obtained as follows:

According to Ohm’s law, is obtained as follows:

Thus, a memristance value is obtained as follows: where is the memristance, and its unit is Ohm (Ω). If the memristance value is a constant, then it becomes the same concept as resistance. It can also be obtained by a linear relationship between the current and the voltage.

For the magnetic flux-controlled memristor, is easily obtained by

From , we can get where is the memductance. In the chaotic circuits, the use of memductor is more extensive. This is because the design of memductor in chaotic circuits is more convenient than memristor design.

Here, a magnetically controlled memristor is defined with a smooth cubic monotonic rise nonlinear characteristic curve. The model is a nonlinear memductor, and the nonlinearity is modeled by using a cubic curve model. The formula is described as follows:

Act on the equation ends of the sign with , that is,

In consideration of , , and , we can obtain and

Equation (11) is the VAR (volt ampere relation) expression of the memductor. It makes the physical concept of memductor more distinct; thus, we can clearly see that the dimension of is conductance. Equation (12) seems useless, but it is very important for engineering design. The specific circuit of memristor can be directly designed by (12). Even when the model represented by (10) changes, we can also design corresponding memductor-based or memristor-based circuits according to this method.

##### 2.2. Realization Circuit of the Specific Memductor Element

According to (12), the specific circuit of memristor can be designed directly. An equivalent memductor-based circuit consisting of operational amplifier, analog multiplier, resistor, and capacitor is shown in Figure 1.