Complexity

Volume 2019, Article ID 6083853, 15 pages

https://doi.org/10.1155/2019/6083853

## Dynamic Behaviors Analysis of a Chaotic Circuit Based on a Novel Fractional-Order Generalized Memristor

^{1}State Key Laboratory Base of Eco-Hydraulic Engineering in Arid Area, Xi'an University of Technology, Xi'an 710048, China^{2}Institute of Water Resources and Hydroelectric Engineering, Xi'an University of Technology, Xi'an 710048, China^{3}College of Electronics and Information, Xi'an Polytechnic University, Xi'an 710048, China^{4}School of Electrical Engineering, Xi’an Jiaotong University, Xi’an 710049, China

Correspondence should be addressed to Shucan Cheng; nc.ude.tuax.uts@cnacuhs and Chaojun Wu; nc.ude.utjx.uts@uw.nujoahc

Received 29 January 2019; Revised 31 March 2019; Accepted 5 May 2019; Published 21 May 2019

Academic Editor: Honglei Xu

Copyright © 2019 Ningning Yang 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, a fractional-order chaotic circuit based on a novel fractional-order generalized memristor is proposed. It is proved that the circuit based on the diode bridge cascaded with fractional-order inductor has volt-ampere characteristics of pinched hysteresis loop. Then the mathematical model of the fractional-order memristor chaotic circuit is obtained. The impact of the order and system parameters on the dynamic behaviors of the chaotic circuit is studied by phase trajectory, Poincaré Section, and bifurcation diagram method. The order, as an important parameter, can increase the degree of freedom of the system. With the change of the order and parameters, the circuit will exhibit abundant dynamic behaviors such as coexisting upper and lower limit cycle, single scroll chaotic attractors, and double scroll chaotic attractors under different initial conditions. And the system exhibits antimonotonic behavior of antiperiodic bifurcation with the change of system parameters. The equivalent circuit simulations are designed to verify the results of the theoretical analysis and numerical simulation.

#### 1. Introduction

The memristor, which is considered as the fourth basic circuit element, was first proposed theoretically by Professor Leon Chua in 1971[1]. Since the development of the practical memristor in HP Laboratory in 2008 [2], the practical application of memristors has attracted wide attention. At present, the research on the memristor mainly focuses on its physical realization such as memristor equivalent circuits [3–5], the dynamical behaviors of chaotic circuits based on the memristor [6–9], and memristive neural networks [10, 11]. As a basic circuit element, memristor is mostly used in various fields in the form of circuit at present, so the application circuit of memristor is rich and diverse. Because memristor has natural nonlinearity and plasticity, it is easy to construct chaotic oscillation circuit based on memristor by organic combination with other circuit elements.

If the volt-ampere characteristics of the circuit ports have three characteristics fingerprints as described in [12], it can be defined as a memristor. Reference [3] presents a generalized memristor diode bridge circuit of cascaded RLC filters consisting of full-wave rectifiers and second-order RLC filters. Reference [4] presents a generalized memristor simulator for first-order parallel RC filters cascaded by a diode bridge. In [5], a first-order generalized memristor simulator based on diode bridge and series RL filter is proposed. In [13], a fractional-order generalized memristor consisting of a diode bridge and a parallel circuit with an equivalent unit circuit and a linear resistance is proposed. If only an inductor cascades a diode bridge, the circuit has simpler topology while it satisfies the memristor characteristics. All the generalized memristor can be simplified as a nonlinear basic element.

The concept of fractional calculus is a development in the field of mathematics, which can be applied to describe memristor characteristics. Fractional calculus plays an important role in signal and image processing [14, 15], control theory [16], and nonlinear dynamical systems [17–21]. A large number of studies have shown that the introduction of fractional-order parameters as adjustable parameters in the model improves the degree of freedom of the model and can more accurately describe the characteristics of actual systems. Because magnetic flux or charge is mathematically a time integral of voltage or current, it can show itself a memory feature [22]. Fractional calculus is especially suitable for describing the memory and hereditary characteristics of the system [23, 24]. It has essentially the same mathematical principles as the memory characteristics of memory circuit elements. It is feasible to introduce fractional calculus theory into the dynamic behavior analysis and application of memory circuit elements [25, 26]. The relationship between fractional calculus and the behavior of memory system is proposed in [17], and it is pointed out that the memristor can be extended to fractional-order one. Reference [18] describes a classical Chua's oscillator based on fractional-order memristor, in which the memristor is a flux-controlled memristor. A novel fractional-order system consisting of memristor, capacitor, and inductor is studied in [19]. In [27], a memory resistor with noncontact polynomial memory resistance function is used in the simplest chaotic circuit. In addition, a Lorenz system based on fractional-order memristors is proposed in [21], its flux-controlled memristors have different permeance functions, and their dynamic characteristics are analyzed, respectively. However few papers mentioned that the chaotic circuit based on fractional-order memristor has coexisting period and chaotic states under different initial values and has antimonotonic behavior of antiperiodic bifurcation.

The rest of this paper mainly includes the following five sections. In Section 2, the generalized fractional-order memristor (FOM) based on single fractional-order inductor and diode bridge is proposed and its memristive characteristics are verified. Then a FOM-based fractional-order chaotic circuit is proposed consisting of a fractional-order capacitor, a fractional-order inductor, and a negative resistor. The equilibrium and stability of the circuit are analyzed. In Section 3, the dynamic behavior of the fractional-order system is analyzed. Firstly, the bifurcation diagram of the system with the order parameter is studied, and the changing process of the dynamic behavior is analyzed combined with the system trajectory under different initial conditions. Then the bifurcation diagrams of the system varying with one of the system parameters at different orders are studied. At the same time, it is found that the fractional-order system has antimonotonic behavior of coexistence and antiperiodicity under different initial conditions. In Section 4 the chaotic circuit composed of equivalent fractional-order inductor and capacitor is simulated to verify the dynamic behavior of the fractional-order chaotic circuit. The last part is the conclusion.

#### 2. Fractional-Order Chaotic Circuit Based on Generalized Memristor

Fractional calculus can be regarded as an extension of classical integer-order calculus, but it has its own unique logic and grammar rules. There are several different definitions involving the fractional versions of the integral and derivative operators: Riemann-Liouville (RL) definition, Grunwald-Letnikov (GL) definition, and Caputo definition [23].

Because the definition of Caputo allows integer-order initial conditions to be used to solve fractional-order differential equations, it is widely used in the modeling of practical problems.

The definition of Caputo iswhere is the first integer which is not less than . is a fractional-order derivative of a continuous function of time, which is given in terms of a time integral.

Laplace transform is a common tool for describing fractional-order systems. The integer-order Laplace transform can be generalized to the fractional-order form.where is an integer value; when the initial value of the system is zero, formula (2) can be simplified asWhen , for single-input single-output (SISO) systems, the relationship between the input and output signals and iswhere .

##### 2.1. Fractional-Order Memristor Based on Diode Bridge Cascaded with a Single Fractional-Order Inductor

A generalized memristor consisting of a single ideal inductor and diode bridge is proposed in [6]. Lots of researches have shown that the real capacitor and inductor can be extended to fractional-order forms to describe their real electricity characteristics. In this paper, the integer-order inductor in the generalized memristor is extended to the fractional-order form. Then the model of integer-order memristor can be extended to the fractional-order one as shown in Figure 1.