Complexity

Volume 2019, Article ID 4647608, 11 pages

https://doi.org/10.1155/2019/4647608

## Dynamics and Synchronization of a Memristor-Based Chaotic System with No Equilibrium

College of Physics and Electronics, Hunan Institute of Science and Technology, Yueyang 414006, China

Correspondence should be addressed to Chun-Lai Li; moc.361@lhciltsinh

Received 26 July 2019; Accepted 17 September 2019; Published 28 October 2019

Guest Editor: Viet-Thanh Pham

Copyright © 2019 Hong-Min Li 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

The topics of memristive system and synchronization are two hot fields of research in nonlinear dynamics. In this paper, we introduce a memristor-based chaotic system with no equilibrium. It is found that the memristor-based system under investigation exhibits fruitful dynamic behaviors such as coexisting bifurcation, multistability, transient chaos, and transient quasiperiod. Thus, it is difficult to reproduce the accurate dynamics of the system, which is highly advantageous in encryption and communication. Then, a simple intermittent control scheme with adaptive mechanism is developed to achieve complete synchronization for the introduced system. Because the output signal is transmitted intermittently to the receiver system, more channel capacity can be saved and the security performance can be improved naturally in practical communication.

#### 1. Introduction

As the fourth basic circuital element along with resistor, inductor, and capacitor, the memristor was postulated by Chua in 1971 [1], and it was then successfully fabricated by the HP laboratories in 2008 [2]. Since then, the memristor was recognized to perfect the symmetry of the four fundamental circuital variables and has aroused wide interest in academia [3–5]. The memristor is commonly defined as a two-terminal nonlinear component with controllable resistance called memristance that varies according to the amount of charge or flux flowing through it [6]. The fingerprint of a memristor is composed of a current-voltage characteristic curve, which shows a pinched hysteresis loop whose shape varies with frequency and converges to a straight line with the increase of frequency [7].

The memristor is currently used to design flash memory, improve neural networks, and construct chaotic circuits, for the intrinsic characteristics of memory, nanoscale device, and inherent nonlinearity. Itoh and Chua constructed the memristive chaotic oscillator in 2008, by replacing Chua’s diodes in Chua’s circuit with the piecewise linear memristor [8]. Afterwards, many memristor-based chaotic oscillators were constructed. For example, by replacing the single diode with a memristor in the original circuit, Pelap postulated an emendatory Tamasevicius oscillator [9]. Bi-Rong designed a simple chaotic circuit consisting of an inductor, a capacitor, and a voltage-controlled memristor [10]. Zhao et al. proposed a memristor-based chaotic system by replacing the nonlinear diode in the Chua circuit with an active flux-controlled memristor [11]. In order to increase the complexity of memristor-based system, Teng et al. used a fourth-degree polynomial memristance to produce a multiscroll chaotic attractor [12]. By replacing Chua’s diode with a physical SBT memristor and a negative conductance in the canonical Chua’s circuit, a new memristor-based modified Chua’s circuit is constructed [13]. There usually emerges special dynamics in this kind of memristive systems, such as initial sensitivity, coexisting bifurcation, coexistence attractors, and transient dynamics. Therefore, the memristive chaotic system will provide more complex dynamics and facilitates the engineering applications of information encryption, secure communication, and signal processing [14–17].

Meanwhile, close attention was paid to chaotic system without equilibrium [18–20]. From the computer-processing perspective, it is challenging to numerically localize the attractor in such system since there is no transient process leading from the vicinity of unstable equilibrium point. In other words, the attracting basin of such system does not intersect with any small neighborhood of its equilibrium point, or the attractor is “hidden” [21–24]. Up to now, little information is known about the dynamical behavior in such system, and what is worse is that the Shilnikov criteria cannot be employed to prove the chaos for the absence of heteroclinic or homoclinic orbit [25, 26].

Because of its application in secure communication, digital signal, neural network, and other fields, the synchronization of chaotic system is a fashionable subject in nonlinear science. Since the first scheme was carried out by Pecora and Carroll for the synchronization of two identical chaotic systems [27], a great diversity of methods have been proposed to synchronize chaotic systems, such as active control, adaptive control, impulsive control, sliding mode control, intermittent control, pinning control, and hybrid control [28–31]. Generally, a chaotic communication system can be constructed based on master-slave synchronization, where the message is modulated by the transmitting system and is then sent to the receiving system. Also, in the receiver, the designed synchronization scheme is used to demodulate the received signal and extract the message [32]. The intermittent synchronization implies that the slave system receives the demodulated information from the master system intermittently. Therefore, the intermittent synchronization scheme will decrease the amount of conveyed information and the communication channel capacity will be reserved for more message transmission. Also, accordingly, the security of the chaotic communication system will be improved since the redundancy of the synchronization information in the channel is reduced. Therefore, the intermittent synchronization scheme is especially fit for the design of practical chaos-based communication system.

In this paper, we introduce a memristor-based chaotic system with no equilibrium. The dynamical evolution of the memristive system is studied by using phase diagram, time-domain trajectory, bifurcation diagram, and Lyapunov exponent. It is found that by changing system parameters or initial condition, the reported system exhibits different topological structures of coexisting bifurcation, multistability, transient dynamics. The coexisting hidden attractors signify that the system has fruitful and complex dynamic behaviors, which is highly advantageous in encryption and communication for the difficulty of reproducing the accurate dynamics of the system. Then, a simple intermittent control scheme with adaptive mechanism is developed to achieve complete synchronization for the introduced memristive system. Since the output signal is transmitted intermittently to the receiver system, more channel capacity can be saved and the security performance of the communication system can be improved naturally in practical communication. Theoretical analysis and illustrative examples are executed to verify the effectiveness of the proposed synchronization scheme.

#### 2. Memristor-Based Chaotic System with No Equilibrium

##### 2.1. Model Description

Based on Sprott A system, the constructed memristive chaotic system can be described by the following differential equations:where *x*, *y*, *z* are state variables; the function *W* represents the model of a flux-controlled memristor, depicted as ; and *a*, *b*, *c* are positive parameters.

The dissipativity is decided by ; thus, system (1) is non-Hamiltonian conservative of phase volume [33]. It is palpable that there exists no equilibrium in system (1). Therefore, the strange attractor is “hidden” in the sense of classification method described by Leonov et al. [21], and the Shilnikov method cannot be employed to verify the emergence of chaos since there is no heteroclinic or homoclinic orbit in this system. It is easy to know that the system is symmetric with respect to the *y*-axis in the sense of coordinate transformation (*x*, *y*, *z*, *t*) ⟶ (−*x*, *y*, −*z*, −*t*).

When choosing the parameters *a* = 3, *b* = −1, and *c* = 1 and initial condition (0.2, 0.15, 0), system (1) appears a chaotic state with the Lyapunov exponents 0.1062, 0, −0.1062, as illustrated by the phase portrait in Figure 1.