Complexity

Volume 2019, Article ID 2051053, 13 pages

https://doi.org/10.1155/2019/2051053

## Complex Dynamical Behaviors of a Fractional-Order System Based on a Locally Active Memristor

Correspondence should be addressed to Bocheng Bao; nc.ude.uzcc@cboab

Received 10 May 2019; Revised 30 August 2019; Accepted 8 October 2019; Published 20 November 2019

Guest Editor: Lazaros Moysis

Copyright © 2019 Yajuan Yu 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

A fractional-order locally active memristor is proposed in this paper. When driven by a bipolar periodic signal, the generated hysteresis loop with two intersections is pinched at the origin. The area of the hysteresis loop changes with the fractional order. Based on the fractional-order locally active memristor, a fractional-order memristive system is constructed. The stability analysis is carried out and the stability conditions for three equilibria are listed. The expression of the fractional order related to Hopf bifurcation is given. The complex dynamical behaviors of Hopf bifurcation, period-doubling bifurcation, bistability and chaos are shown numerically. Furthermore, the bistability behaviors of the different fractional order are validated by the attraction basins in the initial value plane. As an alternative to validating our results, the fractional-order memristive system is implemented by utilizing Simulink of MATLAB. The research results clarify that the complex dynamical behaviors are attributed to two facts: one is the fractional order that affects the stability of the equilibria, and the other is the local activeness of the fractional-order memristor.

#### 1. Introduction

Nonlinear electronic circuits provide powerful and analytical platforms for people to realize and understand the complex dynamical behaviors in physics [1]. Chaotic circuits especially have become effective tools for studying chaos theory. The memristor, originally defined as the forth element of the circuit by Chua in 1971 [2], is a nonlinear circuit device besides the nonlinear resistor, capacitor, and inductor. As a result, many novel memristive circuits have been constructed by integrating the memristors with versatile nonlinearities into some existing linear or nonlinear circuits [3–11]. In these memristive circuits, rich dynamical behaviors have been reported and tested by numerical simulations and hardware experiments, such as chaos and hyperchaos [12, 13], hyperchaotic multiwing attractors [14, 15], coexisting multiple attractors [16, 17], hidden attractors [18, 19], and complex transient chaos and hyperchaos [20]. It should be noted that the simplest chaotic circuit has been proposed based on a locally active nonlinear memristive element [4]. Compared to the chaotic circuit shown in [21], the simplest chaotic circuit has following characteristics: (1) the circuit components are connected in a single way, i.e., in series; (2) the number of the circuit components is decreased from four to three; (3) the memristor is locally active.

At a given moment, the resistance of an ideal memristor is represented by the integration of all states before the current moment. This means that the ideal memristor has no memory loss. But the work [22] shows that the width of the doped layer of the HP TiO_{2} linear model cannot be equal to zero or the whole width of the model. The HP TiO_{2} linear memristor has memory loss. From the definition, the fractional-order derivative depends on the previous history of the variable and is not a strictly local operator [23]. The order of the fractional-order derivative is related to the memory loss or the “proximity effect” of some characteristics [12]. Then the nonideal memristor with memory loss mentioned in [22] can be modeled by a fractional-order derivative with the order between 0 and 1 [24]. According to this, there are many memristors modeled with the fractional-order derivative [25–27]. As shown in [24], the fractional-order memristor in the series circuits has capacitive properties or inductive properties by choosing a suitable fractional order; i.e., the fractional order can be regarded as a parameter which is used to control the memory strength and dynamics of the circuit. In [25], the fractional order can be used to control the time period in which the resistance of the memristor increases from the initial value to its maximum. In addition, a noncommensurate fractional-order autonomous memristor-based circuit is proposed in [27], where the chaotic behavior can be suppressed by applying periodic impulses. In addition, the dynamical system with the locally active equipment can exhibit complexity and emergent behaviors [28, 29]. Then, it is significant to model the memristor or locally active memristor with the fractional-order derivative and display the dynamics induced by these fractional-order memristors.

The main purpose of this paper is to study the complex dynamical behaviors of a fractional-order system based on a locally active voltage-controlled memristor. By theoretical analyses, the stability conditions of the fractional-order memristive system are listed. The complex dynamical behaviors, such as Hopf bifurcation, period-doubling bifurcation, bistability, and chaos, are displayed numerically. The rest of the paper is organized as follows. In Section 2, the mathematical model of the fractional-order memristor is presented. The fractional-order memristor’s fingerprints and local activeness are addressed. In Section 3, an integer-order locally active memristive system is generalized into a fractional-order locally active memristive system. The stability conditions are listed. The complex dynamical behaviors are stated, and numerical simulations are displayed. As an alternative to validating the numerical results, the fractional-order memristive system is implemented by utilizing Simulink of MATLAB. In Section 4, the effect of the local activeness on complex dynamical behaviors is stated. Section 5 ends with some concluding remarks.

#### 2. Fractional-Order Locally Active Nonlinear Memristor

##### 2.1. The Model of the Fractional-Order Memristor

Generally, the memristor can be seen as a sliding resistor whose resistance changes with the charge crossing it. Driven by a bipolar periodic signal, the memristor exhibits a hysteresis loop pinched at the origin in the current-voltage plane. An integer-order nonlinear voltage-controlled memristor is stated as follows [30]:where and *i* are the voltage and current of the memristor, respectively, is the internal state of the memristor and is the memductance, and *p*_{1}, *p*_{2}, *p*_{3}, and *p*_{4} are the system parameters. By using the trial and error method [30], the parameters are decided as *p*_{1} = 1.8, *p*_{2} = 3.9, *p*_{3} = 1.4, and *p*_{4} = 1.5. Considering the memory effect from the memristor, a fractional-order voltage-controlled memristor *M*_{α} corresponding to (1) is modeled as follows:whereis *α*-order derivative of in the sense of Caputo’s definition given in [23], denotes the first-order derivative of with respect to *τ*, and is the memductance of *α*-order memristor *M*_{α}. The integral process in (3) is the memory process of the memristor. In the circuits, the proposed fractional-order memristor is marked as Figure 1(a).