Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 2313768, 13 pages

https://doi.org/10.1155/2017/2313768

## Characteristic Analysis of Fractional-Order 4D Hyperchaotic Memristive Circuit

^{1}School of Physics and Electronics, Central South University, Changsha 410083, China^{2}School of Information Science and Engineering, Dalian Polytechnic University, Dalian 116034, China

Correspondence should be addressed to Kehui Sun

Received 6 March 2017; Accepted 28 May 2017; Published 10 July 2017

Academic Editor: Alessandro Lo Schiavo

Copyright © 2017 Jun Mou 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

Dynamical behaviors of the 4D hyperchaotic memristive circuit are analyzed with the system parameter. Based on the definitions of fractional-order differential and Adomian decomposition algorithm, the numerical solution of fractional-order 4D hyperchaotic memristive circuit is investigated. The distribution of stable and unstable regions of the fractional-order 4D hyperchaotic memristive circuit is determined, and dynamical characteristics are studied by phase portraits, Lyapunov exponents spectrum, and bifurcation diagram. Complexities are calculated by employing the spectral entropy (SE) algorithm and* C*_{0} algorithm. Complexity results are consistent with that of the bifurcation diagrams, and this means that complexity can also reflect the dynamic characteristics of a chaotic system. Results of this paper provide a theoretical and experimental basis for the application of fractional-order 4D hyperchaotic memristive circuit in the field of encryption and secure communication.

#### 1. Introduction

According to the principle of completeness with variable combination, Professor Chua predicted the existence of memristor in 1971 [1]. In 1976, he expounded the character of memristor, composition principle, and applications [2]. For a long time, the existence of element which satisfied the character of memristor was not discovered, so the study of memristor did not rise to the attention of scientific community and engineering circles. In 2008, the HP laboratory reported the realization of memristor firstly [3, 4], and, since then, the memristor has attracted much attention all over the world. Memristors are often divided into charge-controlled memristor and flux-controlled memristor. Both of them are typical nonlinear elements. It is expected to be an effective memory storage device in computers. It means lower power consumption and less thermal design to deal with, and it is easy to generate a chaotic vibration signal by employing this element. On one hand, we need to prevent the harm of chaos phenomenon in the application. On the other hand, the chaotic vibration signal by employing the memristor can be applied to many fields such as secure communication and aerospace industry. So researchers began to focus on the design and realization of memristive chaotic circuit [5–20]. In these literatures, only one memristor was applied in an independent circuit, and the dynamic characteristics of memristive chaotic system are related to the initial state of memristor, including unique nonlinear physics phenomenon. Zhang and Deng studied double-compound synchronization of six memristor-based Lorenz systems [21]. However, this article focuses on the synchronization method of a net with four Lorenz systems, rather than dynamic behaviors of a memristive circuit. When Bao et al. and Mou et al. applied more than one memristor in a single circuit [22, 23], they found that memristors would affect each other, and the dynamical behaviors of the circuit with more than one memristor become more complex. Compared with ordinary memristive chaotic systems, the memristive hyperchaotic systems have excellent security in communication because the memristive hyperchaotic sequence has better complexity and randomicity [24–26].

Fractional calculus is more than 300-year-old topic. It can describe physical phenomena more accurately than that of the integer-order calculus, so fractional calculus has attracted more attention in various areas of applications such as physics, chemistry, bioengineering, signal processing, and control system [27–29]. A very important area of applications is the chaos theory [30–33]. At present, there are three typical methods to solve the fractional-order nonlinear system, such as frequency domain approximation [30], predictor-corrector method (PCM) [34], and Adomian decomposition method (ADM) [35]. The calculation precision of frequency domain approximations is limited. Whether this method accurately reflects the chaos characteristics of a fractional-order nonlinear system was questioned [36, 37]. For PCM, one can obtain more accurate results, but the calculation speed is too slow, and it consumes too much computer resources. Thus, it is unsuitable for engineering practice. Compared with the two solution approaches, ADM is capable of dealing with linear and nonlinear problems in time domain [38, 39]. He et al. concluded the characteristics of this method, such as high accuracy, fast convergence, and less computer resources consumption [40]. Recently, Li et al. proposed a new 4D hyperchaotic memristive circuit [41], which possesses abundant complex dynamics. The main feature of this system is having uncountable infinite number of stable equilibria, which is significantly different from other reported chaotic systems before. However, the article mainly analyzed the equilibrium states, and the analysis of dynamics characteristics of the 4D memristive hyperchaotic system is incomplete. Its fractional-order form has not been studied by now. Thus, it makes a great sense to study the dynamics characteristics of the system and the solution and dynamics of its corresponding fractional-order case. To our knowledge, no one has studied the fractional-order memristive hyperchaotic system. So we will employ ADM algorithm to solve the fractional-order 4D hyperchaotic system.

In this paper, we focus on dynamical characteristic of the fractional-order 4D hyperchaotic memristive circuit. It is organized as follows. The dynamical characteristics of integer-order 4D hyperchaotic memristive circuit are investigated in Section 2. In Section 3, ADM is introduced briefly, and the iterative algorithm of the fractional-order 4D hyperchaotic memristive circuit is deduced. In Section 4, the distribution of stable and unstable regions of the fractional-order 4D hyperchaotic memristive circuit is determined, and the dynamical characteristics of this system are analyzed. Finally, we summarize the results and indicate future directions.

#### 2. 4D Hyperchaotic Memristive Circuit

##### 2.1. Model of the 4D Hyperchaotic Memristive Circuit

Figure 1 shows the 4D hyperchaotic memristive circuit model, which consists of standard integrators, standard multipliers, linear resistors, linear capacitors, and a nonlinear active memristor. We can use , , and to indicate states of voltages. For the memristive circuit, there is an input from to by a flux-controlled memristor, and it is illustrated by in Figure 1. According to Chua’s definition, a memristor is a passive two-terminal circuit element described by a nonlinear* i*-*v* characteristic as follows: or , where *, **, *, and* φ*are the voltage, current, charge, and flux associated with the device. is the memristance defined as follows: . is the memductance defined as follows: . Here, we focus on the flux-controlled memristive system described by the following circuit equation: , where and are two positive constant parameters. According to the volt-ampere characteristics of each element and Kirchhoff’s current and voltage law, the differential equation (1) is obtained:where is a reference resistor and is a reference capacitor; then is the physical time, where is the dimensionless time. The parameters can be taken as follows: = = =

*C*, = =

*R*, , , and . By employing the normalized operation, (1) becomeswhere is a positive parameter indicating the strength of the memristor. We should note here that comes from the memristor and has the same function as the physical memductance mentioned above, but it is dimensionless, which would be convenient for the following discussion.