Complexity

Volume 2018, Article ID 6069401, 12 pages

https://doi.org/10.1155/2018/6069401

## A New Memristor-Based 5D Chaotic System and Circuit Implementation

^{1}Tianjin Key Laboratory for Civil Aircraft Airworthiness and Maintenance, Civil Aviation University of China, Tianjin 300300, China^{2}College of Information Engineering and Automation, Civil Aviation University of China, Tianjin 300300, China^{3}Engineering Technology Department, Sichuan Airlines, Chengdu 610202, China^{4}Department of Engineering Design and Mathematics, University of the West of England, Frenchay Campus, Coldharbour Lane, Bristol, BS 161QY, UK

Correspondence should be addressed to Rui Wang; nc.ude.cuac@gnawiur and Hui Sun; nc.ude.cuac@nus-h

Received 30 June 2018; Revised 5 October 2018; Accepted 29 October 2018; Published 2 December 2018

Guest Editor: Viet-Thanh Pham

Copyright © 2018 Rui Wang 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 proposes a new 5D chaotic system with the flux-controlled memristor. The dynamics analysis of the new system can also demonstrate the hyperchaotic characteristics. The design and analysis of adaptive synchronization for the new memristor-based chaotic system and its slave system are carried out. Furthermore, the modularized circuit designs method is used in the new chaotic system circuit implementation. The Multisim simulation and the physical experiments are conducted, compared, and matched with each other which can demonstrate the existence of the attractor for the new system.

#### 1. Introduction

Memristors are the fourth kind of circuit elements except for resistors, capacitors, and conductors and are conceived by Chua in 1971 through the basic symmetric principle [1]. Furthermore, the corresponding theory was applied to memristive devices in 1976 [2]. It took a long time to develop the hardware memristor model. Until 2008 HP labs first realized the memristor of nanoscale in the form of crossbar array [3]. Since memristors have the potential applications in the wide range of fields, memristor study becomes hotter, and a huge amount of researchers have paid immense attention on memristor studies from industry and academics, respectively [4, 5]. The typical examples include nonvolatile memories of nanoscale [6], memristor-based synapse in neuromorphic systems [7, 8], logic operations through material implication [9–12], and nonlinear dynamics in chaotic system [13–17].

With rapid development of memristor models, some studies combine the memristor and chaotic systems including dynamics analysis, image encryption applications and circuit implementations which have grown up quickly in recent years [18–22]. One of the typical early memristor-based chaotic systems was developed by Itoh and Chua in 2008 [19]. This paper developed some nonlinear oscillators by using memristors based on Chua’s oscillators. Petráš derived and investigated a fractional-order memristor-based Chua’s circuit in [20]. Chua and Muthuswamy also discussed circuit topology and developed the simplest memristor-based circuits [21]. These papers demonstrate that the memristor oscillators own the special nonlinear dynamics due to memristors’ extinguished characteristics. One of these significant characteristics is that the behaviors are dependent on initial states and circuit parameters. Li et al. proposed a scroll chaotic system circuit implementation by using HP memristor [22]. Ma et al. developed a four-wing hyperchaotic system by using a memristor adding over a three-dimensional chaotic system [16]. Dimitrios et al. found a new 4-D memristive chaotic system and investigated the behavior with hidden attractors of the system through numerical simulations [23]. Wang et al. proposed a flux-controlled memristor model and established a 4-D chaotic system with this model. The numerical analysis and circuit implementation simulation verification were conducted [24]. Mou et al. discussed the characteristics of dynamical behaviors of a fractional-order 4D hyperchaotic memristive system and circuit simulation verification [25]. Other memristor-based hyperchaotic systems were also investigated such as numerical analysis about a four-dimensional hyperchaotic system with memristor and conducted circuit simulation verifications [26–29]. There are also some other chaotic systems developed by memristive models and its applications [30–32].

The above memristor-based chaotic system literatures focus on four or lower-dimensional chaotic systems and investigate the detailed numerical analysis and the corresponding numerical and circuit implementation simulation verifications. However, the higher-dimensional (5-D or above) memristor-based hyperchaotic systems and the corresponding physical hardware experiments are not found. Therefore, the paper analyzes a new memristor-based hyperchaotic system and develops a circuit physical implementation method by using the modularized design method. This method is used to design the circuit without dimensions for chaotic circuit designs and is easy to be implemented in the circuit by using less circuit parts [33–38].

The novelty of this paper is to develop a new memristor-based 5D hyperchaotic system, design and analyze the adaptive synchronization of this new system, implement the physical experiment circuit hardware, and verify the existence of system attractors. The improved modularized design method is used to implement the circuit of the system to verify the existence of attractors.

The rest of the paper is organized as follows. Section 2 analyzes the fundamental characteristics of the new memristor-based chaotic system. Section 3 investigates the adaptive synchronization of the new memristor-based hyperchaotic system. Section 4 discusses the circuit implementation of the new system and verifies the existence of attractors. Conclusions are presented in Section 5.

#### 2. Analysis of a New Memristor-Based Hyperchaotic System

In this section, numerical analyses are conducted for a flux-controlled memristor-based new 5D hyperchaotic system derived from Wang’s 4D hyperchaotic system.

As illustrated in [16], memristor model is based on the fundamental characteristics of a flux-controlled memristor described below.where and are the current and the voltage of the device terminal, respectively. is the incremental memductance defined as

This demonstrates that the characteristics of a memristor are a nonlinear function reflecting the relationship among the charge and flux across and through the device.

Furthermore, this paper consistently uses the smooth cubic monotone-increasing continuous nonlinearity described as follows [20–22, 33].where .

Then the memductance is shown below

This paper develops a 5D memristor-based chaotic system which is derived from four-wing autonomous chaotic dynamics systems reported by Wang et al. [34, 37]This system has four state variables , , , and , and .

Substitute (4) into (5), a 5D memristor-based system is obtained.where and , , , are positive parameters.

##### 2.1. Equilibria and Stability

The equilibrium points of System (6) can be calculated by solving the equations as shown belowwhere , , and .

System (6) has only one real equilibrium point with (0, 0, 0, 0, 0) and has typical characteristics with a line equilibrium in (0, 0, 0, 0,* δ*), given a real constant.

First, analyze the zero equilibrium point (0, 0, 0, 0, 0). The Jacobian matrix of System (6) on the zero equilibrium point is and the corresponding polynomial isandIt is obvious that 0 and are eigenvalues of System(6) for the (0, 0, 0, 0, 0) equilibrium point. According to Routh-Hurwitz condition, if and only if , , , and coexist, has the negative real number. However, the above four inequalities are not able to be realized simultaneously. Then not all real parts of the eigenvalues are negative. Therefore, it is not a stable equilibrium point.

Second, analyze the eigenvalues of Jacobian matrix of System (6) on the line equilibrium in (0, 0, 0, 0,* δ*).Typically, when and , is calculated as shown belowTwo of the five eigenvalues of are complex conjugates; therefore, it is difficult to determine the stability of the line equilibria.

##### 2.2. Symmetry

System (6) is symmetric with respect to axis since it is invariant when applying the coordinate transformations.

##### 2.3. Dissipativity

Furthermore, dissipative characteristics analysis of System (6) is shown below. The system divergence is given bywhen , , , , System (6) is dissipative. The paper selects , for equation (7), and for System (6).

##### 2.4. Lyapunov Spectrum and Bifurcation Diagram

Fix parameters , and vary the parameter , and the graphs about the Lyapunov exponents versus , bifurcation diagram, and phase portraits are shown in Figure 1.