Complexity

Volume 2018, Article ID 8639470, 9 pages

https://doi.org/10.1155/2018/8639470

## Complexity Analysis of a Mixed Memristive Chaotic Circuit

School of Information Science and Engineering, Dalian Polytechnic University, Dalian 116034, China

Correspondence should be addressed to Jun Mou; nc.ude.usc@nujuom

Received 9 June 2018; Revised 20 August 2018; Accepted 27 August 2018; Published 19 November 2018

Academic Editor: Chittaranjan Hens

Copyright © 2018 Xiaolin Ye 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

In this paper, we design a chaotic circuit with memristors, which consists of two flux-controlled memristors and a charge-controlled memristor, and the dimensionless mathematical model of the circuit was established. Using the conventional dynamic analysis methods, the equilibrium point set and stability of the chaotic system were analyzed, and the distribution of stable and unstable regions corresponding to the memristor initial states was determined. Then, we analyze the dynamical behaviors with the initial states of the memristors and the circuit parameter of the circuit system, respectively. By using spectral entropy (SE) and C_{0} complexity algorithms, the dynamic characteristics of the system were analyzed. In particular, the 2D and 3D complexity characteristics with multiple varying parameters were analyzed. Some peculiar physical phenomenon such as coexisting attractors was observed. Theoretical analysis and simulation results show that the chaotic circuit has rich dynamical behaviors. The complicated physical phenomenon in the new chaotic circuit enriches the related content of chaotic circuit with memristors.

#### 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 characteristic of memristor, composition principle and applications [2]. For a long time, the element which satisfied the characteristic 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 for the first time [3, 4], and since then, 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, and it is easy to generate a chaotic vibration signal by employing this element. So researchers pay more attention to the design and realization of memristive chaotic circuit [5–13]. Corinto et al. analyzed the dynamical behaviors of a memristive oscillator by using the conventional dynamic analysis methods [5]. Teng et al. designed a fractional-order memristor based on the simplest chaotic circuit [6]. Kim et al. constructed a memristor emulator by using the analog multiplier [7]. Itoh and Chua used a memristor with a piecewise linearity to replace Chua’s diode [14] in Chua’s oscillator and got a chaotic oscillation circuit [8]. Muthuswamy and Chua used a memristive circuit with a source and a discontinuous piecewise linearity flux-memristive character to replace Chua’s diode, and they derived some new chaotic circuits [9]. The memristors used in the literature [5–9] are nonsmooth discontinuous piecewise memristors, but it is difficult to realize in practice. After that, the relatively simple mathematical model and the circuit model of physically realizable became a research hotspot in the academia. Muthuswamy put forward a flux-controlled memristor, which has a cubic nonlinear characteristic curve, and used the available devices to build the equivalent circuit [10, 11]. Bao et al. applied the smooth memristor to construct some new circuits and found some special phenomena in the new circuits [12, 13, 15, 16]. In particular, the memristive chaotic systems are also widely applied to related chaotic fields. Zhang and Deng studied the double-compound synchronization of six memristor-based Lorenz systems [17]. In the literature [5–13, 15, 16], only one memristor was applied in an independent circuit. So, it is worth studying whether multiple memristors can coexist in one circuit. When Bao et al. applied two flux-controlled memristors in a single circuit [18], they found that memristors would affect each other, and the dynamical behaviors of the circuit with more than one memristor become more complex. In this paper, based on the circuit model in [18], we design a new mixed memristive chaotic circuit with three memristors. Relative to [18], the higher dimensional circuit system was set up, and the ranges of steady state were different; the new phenomena such as coexisting attractors was observed. In particular, the complexity characteristics with the varying chaotic sequences were analyzed.

In this paper, we focus on the mixed memristive chaotic circuit which has two smooth passive flux-controlled memristors and a charge-controlled memristor. It is organized as follows. The circuit model was designed and its dimensionless mathematical equation was deduced in Section 2. In Section 3, we analyzed the dynamical behaviors of this circuit, including the stability of the equilibrium set, influence of the system initial states, the relationship between the circuit parameter, and the system dynamical behaviors. Particularly, some special dynamic phenomena including coexisting attractors were found; the 2D and 3D complexity characteristics were analyzed. Finally, we summarize the results and indicate future directions.

#### 2. Mixed Chaotic Circuit Based on Memristors

##### 2.1. Model of the Memristors

According to the mathematical model of flux-controlled memristor which was put forward in [19, 20], we can obtain that memristor can be defined as a two-terminal element. For a smooth passive flux-controlled memristor, the magnetic flux between the terminals is a function of the electric charge that passes through the device, and its state variable are where is the memductance. and are real constant, in this article, setting the parameters and .

The volt-ampere characteristic curve of the smooth passive flux-controlled memristor driven by a 2.828 V and 1 Hz sinusoidal signal is shown in Figure 1.