# Coexisting Three-Scroll and Four-Scroll Chaotic Attractors in a Fractional-Order System by a Three-Scroll Integer-Order Memristive Chaotic System and Chaos Control

Authors: Xikui Hu; Ping Zhou
Journal: Complexity
 (2020)

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## Abstract

Based on the integer-order memristive system that can generate two-scroll, three-scroll, and four-scroll chaotic attractors, in this paper, we found other phenomena that two kinds of three-scroll chaotic attractors coexist in this system with different initial conditions. Furthermore, we proposed a coexisting fractional-order system based on the three-scroll chaotic attractors system, in which the three-scroll or four-scroll chaotic attractors emerged with different fractional-ordersq. Meanwhile, with fractional-order q=0.965 and different initial conditions, coexistence of two kinds of three-scroll and four-scroll chaotic attractors is found simultaneously. Finally, we discussed controlling chaos for the fractional-order memristive chaotic system.


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## Body

## 1. Introduction

Due to the typical characteristics of high irregularity, unpredictability, and complexity of chaotic systems, chaotic systems and its applications have been attracted more and more attentions in the last few decades [1–16], e.g., information processing [11], secure communication [12, 13], image encryption [14, 15], machine learning [16], and so on. Memristor—the missing circuit element—has been discovered by Leon Chua in 1971 [17], and it has been successfully realized in 2008 [18]. Recently, some mathematical models of memristor-based systems were proposed. For example, Muthuswamy and Chua reported a memristor-based chaotic system with single-scroll attractor [19], Bao et al. presented a memristor-based chaotic system with double-scroll attractor [20], Teng et al. reported a memristor-based chaotic system with double-scroll and four-scroll attractors [21], Zhou and Ke gave a memristive-based chaotic system with two-scroll to four-scroll attractors [1], Sun et al. suggested a memristor-based chaotic system with infinite chaotic attractors [22], and so on.Chaotic attractors have also been reported in many fractional-order nonlinear systems, e.g., the fractional-order Lorenz chaotic system [23], the fractional-order Chen chaotic system [24], the fractional-order Lu chaotic system [24], the fractional-order brushless DC motor chaotic system[25], the fractional-order micro-electro-mechanical chaotic system [26], the fractional order coronary artery chaotic system [27], etc. On the other hand, some memristor-based fractional-order chaotic systems have been proposed. For example, a fractional-order memristor-based simplest chaotic circuit with double-scroll and four-scroll attractors using fourth-degree polynomial [21] was reported by Teng et al. and a fractional-order memristor-based chaotic system with single-scroll attractor and a stable equilibrium point [28] was reported by Prakash et al., and a fractional-order memristor-based chaotic system with coexisting attractors [1] was reported by Zhou and Ke. Moreover, some fractional-order chaotic systems have been implemented by electronic circuit, e.g., a fractional-order Lorenz hyperchaotic system has been implemented by DSP [29], a 4-D nonequilibrium fractional-order chaotic system has been implemented by EWB [30], and the fractional-order chaotic systems with two equilibriums and no equilibrium have been realized by FPGA [31].On the other hand, due to the important application of stability and control of chaotic systems in engineering science [9, 10, 24–26, 32–34], the stability and control of chaotic systems have attracted more and more attention in recent years. Many control schemes have been suggested, e.g., linear and nonlinear feedback, scalar and vector controller, single-state variable, and multiple-state variables. However, control cost and control effectiveness must be considered in practice. Therefore, in the process of stability control of chaotic system, single-state variable or scalar controller should be the first choice.Based on the integer-order memristor-based chaotic system [1] reported by Zhou and Ke, in this paper, we find some new results in the integer-order memristor-based chaotic system [1], i.e., two kinds of three-scroll chaotic attractor coexist with different initial conditions. Furthermore, its fractional-order version is proposed. We find that not only three-scroll chaotic attractors but also four-scroll chaotic attractors are emerged in its fractional-order version. Meanwhile, two kinds of three-scroll and four-scroll chaotic attractors coexist with different initial conditions for fractional order q=0.965. To the best of our knowledge, our results have rarely been reported before. Finally, in order to stabilize the fractional-order version memristive chaotic system via a single-state variable, a control strategy is suggested.This article is structure as follows: we describe the integer-order memristor-based chaotic system [1] and find some new results in Section 2. In Section 3, based on the integer-order memristor-based chaotic system [1], its fractional-order version is suggested and the four-scroll chaotic attractors are found, and coexisting two kinds of three-scroll and four-scroll chaotic attractors for different initial conditions are found. The Section 4 presents a control strategy via single-state variable. In Section 5, the conclusion is given.


## 2. Two Kinds of Three-Scroll Chaotic Attractors Coexist in an Integer-Order Memristive System

Recently, an integer-order memristive system reported by Zhou and Ke [1] is as follows:(1)dx1dt=x2,dx2dt=−x1+0.5x34−2.4x2L,dx3dt=−x2−34−x22x3,where1≤L≤8.Remark 1.
There is only one equilibrium point in (1), i.e., x1,x2,x3=0,0,0, and the equilibrium point is unstable. More details are in reference [1].Zhou and Ke [1] reported that there are two-scroll to four-scroll chaotic attractors in this system (1) with different L. For example [1], the two-scroll chaotic attractor is emerged for L=4, the four-scroll chaotic attractor is emerged for L=1.4, the three-scroll chaotic attractors are emerged for L=1.734, and another type of three-scroll chaotic attractor is emerged for L=1.8.In this paper, the integer-order memristive system is further studied. By numerical calculations, we find that two kinds of three-scroll chaotic attractors coexist in this system (1) for  L=1.8 with different initial conditions, which has not been reported by Zhou and Ke [1]. Some results are as follows.LetL=1.8 and initial conditions be 2,1,1 and −2,−1,−1, respectively, two kinds of three-scroll chaotic attractors coexist as shown in Figure 1(a).Figure 1
Coexistence of two kinds of three-scroll chaotic attractors.
(a)
(b)LetL=1.8 and initial conditions be 0.2,0.1,0.1 and −0.2,−0.1,−0.1, respectively, two kinds of three-scroll chaotic attractors also coexist as shown in Figure 1(b).After a great deal of numerical calculations, we find that there are the same chaotic attractors (blue line) with initial conditions2,±1,±1 and 0.2,±0.1,±0.1, and the same chaotic attractors (red line) with initial conditions −2,±1,±1 and −0.2,±0.1,±0.1. It must be pointed out that only the chaotic attractors described by blue line are reported in Ref. [1] with  L=1.8. In this paper, new chaotic attractors (red line) are found and two kinds of three-scroll chaotic attractors coexist in system (1) with  L=1.8.


## 3. Two Kinds of Three-Scroll and Four-Scroll Chaotic Attractors Coexist in Fractional-Order Memristive Chaotic System

Based on memristive chaotic system mentioned above withL=1.8, a fractional-order memristive system is constructed, and coexistence of two kinds of three-scroll and four-scroll chaotic attractors is found. The fractional-order memristive system is described as(2)dqx1dtq=x2,dqx2dtq=−x1+0.5x34−2.4x21.8,dqx3dtq=−x2−34−x22x3,where0<q≤1 is fractional-order, and dqxi/dtq=∫0tt−τ−qdxiτ/Γ1−q with Γ1−q=∫0tt−qe−tdt.Now, based on the Adams–Bashforth–Moulton algorithm [1] for fractional-order system, let Tbe the total time of numerical calculation, N be the iterative times, the step length be m=T/N and tn=nmn=0,1,2…,N. So, the fractional-order memristive system (2) is discretized as follows:(3)x1n+1=x10+mqΓq+2x2rn+1+∑k=0nαk,n+1x2k,x2n+1=x20+mqΓq+2−x1rn+1+0.5x3rn+14−2.4x2rn+11.4+∑k=0nak,n+1−x1k+0.5x3k4−2.4x2k1.4,x3n+1=x30+mqΓq+2−x2rn+1−0.75−x2rn+12x3rn+1+∑k=0nak,n+1−x2k−0.75−x2k2x3k,and(4)x1rn+1=x10+1Γq∑k=0nbk,n+1x2k,x2rn+1=x20+1Γq∑k=0nbk,n+1−x1k+0.5x3k4−2.4x2k1.4,x3rn+1=x30+1Γq∑k=0nbk,n+1−x2k−0.75−x2k2x3k,(5)ak,n+1=nq+1−n−qn+1q,k=0,n−k+2q+1+n−kq+1−2n−k+1q+1,1≤k≤n,1,k=n+1,(6)bk,n+1=mqn−k+1q−n−kqq,0≤k≤n.The approximation error is as follows:(7)x1tn−x1n=om1+q,x2tn−x2n=om1+q,x3tn−x3n=om1+q,wherex10,x20,x30 are the initial conditions for fractional-order system (2).In order to study the dynamical behaviors of system (2), we calculate the maximum Lyapunov exponents (Maximum LE) of system (2) firstly. The Maximum LE with respect to fractional-order q is displayed in Figure 2.Figure 2
The maximum LE varies with fractional-orderq.In Figure2, the positive Maximum LE indicates that there are chaotic attractors in fractional-order memristive system (2). Now, some results are as follows:


### 3.1. Four-Scroll Chaotic Attractor Emerges in System (2) with Q = 0.98

Letq=0.98, we can obtain that the Maximum LE is 0.2501. It indicates that the fractional-order memristive system (2) has a chaotic attractor. Let initial conditions be 2,1,1, it is obtained that four-scroll chaotic attractor emerges in system (2) as shown in Figure 3.Figure 3
Four-scroll chaotic attractors in fractional-order memristive system (2).We note that there are only three-scroll chaotic attractors in integer-order memristive system (1). However, the four-scroll chaotic attractors are generated in its fractional-order version system. By numerical calculation, we find that there are same chaotic attractors (as shown in Figure 3) with initial conditions ±2,±1,±1 or ±0.2,±0.1,±0.1.


### 3.2. Two Kinds of Three-Scroll, and Four-Scroll Chaotic Attractors Coexist in System (2) with q = 0.965

Letq=0.965, we can obtain that the maximum LE is 0.2781. It indicates that the fractional-order memristive system (2) has a chaotic attractor. By numerical calculation, we find that a three-scroll chaotic attractor emerges with initial conditions 2,1,1, another type of three-scroll chaotic attractor emerges with initial conditions −2,−1,−1, and a four-scroll chaotic attractor emerges with initial conditions 2,−1,−1. Therefore, two kinds of three-scroll chaotic attractors coexist with four-scroll chaotic attractor in fractional-order memristive chaotic system (2), as shown in Figure 4.Figure 4
Coexisting two kinds of three-scroll, four-scroll chaotic attractors in fractional-order memristive system (2).Please note that only two kinds of three-scroll chaotic attractors coexist in the integer-order memristive system (1), while two kinds of three-scroll chaotic attractors coexist with four-scroll chaotic attractor in the fractional-order memristive system (2).Remark 2.
By numerical calculation, we find that there are only coexisting two kinds of three-scroll chaotic attractors in system (2) for q=0.965 with initial conditions ±0.2,±0.1,±0.1, i.e., there is no four-scroll chaotic attractor in system (2) with these initial conditions. The coexisting two kinds of three-scroll chaotic attractors are shown in Figure 5.Figure 5
Coexisting two kinds of three-scroll chaotic attractors in fractional-order memristive system (2) for q=0.965 with initial conditions ±0.2,±0.1,±0.1.In summary, two kinds of three-scroll chaotic attractors emerge in the integer-order memristive system (1) for  L=1.8. The fractional-order memristive system (2), however, can generate four-scroll chaotic attractors for q=0.98. Moreover, coexisting two kinds of three-scroll and four-scroll chaotic attractors are found in fractional-order memristive system (2) for q=0.965. It indicates that the fractional-order memristive system (2) has richer and more complex chaotic attractors than the integer-order memristive system (1).


## 3.1. Four-Scroll Chaotic Attractor Emerges in System (2) with Q = 0.98

Letq=0.98, we can obtain that the Maximum LE is 0.2501. It indicates that the fractional-order memristive system (2) has a chaotic attractor. Let initial conditions be 2,1,1, it is obtained that four-scroll chaotic attractor emerges in system (2) as shown in Figure 3.Figure 3
Four-scroll chaotic attractors in fractional-order memristive system (2).We note that there are only three-scroll chaotic attractors in integer-order memristive system (1). However, the four-scroll chaotic attractors are generated in its fractional-order version system. By numerical calculation, we find that there are same chaotic attractors (as shown in Figure 3) with initial conditions ±2,±1,±1 or ±0.2,±0.1,±0.1.


## 3.2. Two Kinds of Three-Scroll, and Four-Scroll Chaotic Attractors Coexist in System (2) with q = 0.965

Letq=0.965, we can obtain that the maximum LE is 0.2781. It indicates that the fractional-order memristive system (2) has a chaotic attractor. By numerical calculation, we find that a three-scroll chaotic attractor emerges with initial conditions 2,1,1, another type of three-scroll chaotic attractor emerges with initial conditions −2,−1,−1, and a four-scroll chaotic attractor emerges with initial conditions 2,−1,−1. Therefore, two kinds of three-scroll chaotic attractors coexist with four-scroll chaotic attractor in fractional-order memristive chaotic system (2), as shown in Figure 4.Figure 4
Coexisting two kinds of three-scroll, four-scroll chaotic attractors in fractional-order memristive system (2).Please note that only two kinds of three-scroll chaotic attractors coexist in the integer-order memristive system (1), while two kinds of three-scroll chaotic attractors coexist with four-scroll chaotic attractor in the fractional-order memristive system (2).Remark 2.
By numerical calculation, we find that there are only coexisting two kinds of three-scroll chaotic attractors in system (2) for q=0.965 with initial conditions ±0.2,±0.1,±0.1, i.e., there is no four-scroll chaotic attractor in system (2) with these initial conditions. The coexisting two kinds of three-scroll chaotic attractors are shown in Figure 5.Figure 5
Coexisting two kinds of three-scroll chaotic attractors in fractional-order memristive system (2) for q=0.965 with initial conditions ±0.2,±0.1,±0.1.In summary, two kinds of three-scroll chaotic attractors emerge in the integer-order memristive system (1) for  L=1.8. The fractional-order memristive system (2), however, can generate four-scroll chaotic attractors for q=0.98. Moreover, coexisting two kinds of three-scroll and four-scroll chaotic attractors are found in fractional-order memristive system (2) for q=0.965. It indicates that the fractional-order memristive system (2) has richer and more complex chaotic attractors than the integer-order memristive system (1).


## 4. Control of the Fractional-Order Memristive Chaotic System (2) via a Single State Variable

Firstly, in order to control the fractional-order memristive chaotic system (2), the following lemma for the fractional-order nonlinear system is given. Consider the following fractional-order nonlinear system (8),(8)dqzdtq=Lz+fz,where0<q≤1, z=z1,z2,…,znT is the real state vector, L∈Rn×n is a constant real matrix, Lz and fz denote the linear and nonlinear parts in nonlinear system (8).Lemma 3 (More details are in [35]).
Given the fractional-order nonlinear system (8), if the following conditions are held,(a)
fzz=0=0, and limz→0fz/z=0,(b)
argλiL≥0.5πqi=1,2,…,n, and qL>1, where λiLi=1,2,…,n are the eigenvalues of matrix L, and L is the l2-norm of matrix L, then, system (8) is said to be asymptotically stable.
Secondly, in order to stable the fractional-order memristive chaotic system (2) via single-state variable, the following controlled fractional-order memristive system (9) is considered.(9)dqx1dtq=x2,dqx2dtq=Cx2−x1+0.5x34−2.4x21.8,dqx3dtq=−x2−34−x22x3,
whereC is a real number. According to system (8), we can write out, (10)LC=010−11.8C+4300−1−34,fx=0−5x2x3418x22x3.
Now, one can obtainfxx=0=0 and(11)limx→0fxx=limx→0−5x2x34/182+x22x32x12+x22+x32≤limx→0x38+x22x32=0.Meanwhile, it is obvious that there will be some real numberC, which can meet the needs of both argλiLC≥0.5πqi=1,2,3 and qLC>1. According to the lemma, the controlled fractional-order memristive system (9) is asymptotically stable. This result indicates that the fractional-order memristive chaotic system (2) can be stability controlled via the single-state variable x2.In addition, some numerical simulations are given to verify the validity of the control strategy.For example, choosingq=0.98, and C=−7/3, thus, the eigenvalues of matrix LC are −0.5+11j/6, −0.5−11j/6, −3/4, and qLC=1.7824, respectively. According to the lemma, the controlled fractional-order memristive system (9) is asymptotically stable. The results of state variables vary with time as shown in Figure 6(a). Here, the initial conditions are x10=2, x20=1, and x30=1, respectively.Figure 6
The results of state variables vary with time.
(a)
(b)
(c)For example, choosingq=0.965 and C=−2, the eigenvalues of matrix LC are thus −1+2j/3, −1−2j/3, and −3/4, respectively, as well as qLC=1.6023. According to the lemma, the controlled fractional-order memristive system (9) is still asymptotically stable. The results of state variables vary with time as shown in Figures 6(b) and 6(c). In Figure 6(b), the black curve corresponds to the initial conditions x10,x20,x30=2,−1,−1, and the blue curve corresponds to the initial conditions x10,x20,x30=2,1,1. It should be mentioned that there are four-scroll chaotic attractors (the black curve in Figure 4) in fractional-order memristive system (2) with initial conditions x10,x20,x30=2,−1,−1, while there are three-scroll chaotic attractors (the blue curve in Figure 4) in fractional-order memristive system (2) with initial conditions x10,x20,x30=2,1,1.The results in Figure6(c) indicate that the state variables vary with time for initial conditions 2,−1,−1 and −2,−1,−1. The black curve corresponds to the initial conditions x10,x20,x30=2,−1,−1, and the red curve corresponds to the initial conditions x10,x20,x30=−2,−1,−1, respectively. It should be mentioned again that there are four-scroll chaotic attractors (the black curve in Figure 4) in fractional-order memristive system (2) with initial conditions x10,x20,x30=2,−1,−1, while there are three-scroll chaotic attractors (the red curve in Figure 4) in fractional-order memristive system (2) with initial conditions x10,x20,x30=−2,−1,−1.Proposition 4.
Let the controlled fractional-order memristive chaotic system (2) be,(12)dqx1dtq=x2+Cx2,dqx2dtq=−x1+0.5x34−2.4x21.8,dqx3dtq=−x2−34−x22x3,
if the following conditions are held,(a)
20C + 4 > 0,(b)
arctan20C+4/4≥0.5πq, and qLC>1, where (13)LC=01+C0−11.84300−1−34, and then, system is said to be asymptotically stable.Proof.
By the above Lemma, this Proposition is easy to prove.
As mentioned above, the fractional-order memristive chaotic system (2) can be stability controlled via the single-state variable x2.


## 5. Conclusions

In this paper, some new results for the integer-order memristive system [1] are found. The coexistence of two kinds of three-scroll chaotic tractors emerges in the integer-order memristive system (1) with different initial conditions, which Zhou and Ke have not reported in reference [1]. Furthermore, based on chaotic system (1), a fractional-order memristive system (2) is suggested. The largest Lyapunov exponents are obtained by numerical algorithm, which indicates that there are chaotic attractors in the fractional-order memristive system (2).By numerical calculation, we find that the four-scroll chaotic attractors emerge in the fractional-order memristive system (2) for q=0.98, which is different from the integer-order memristive system (1). It indicates that the fractional-order memristive system (2) generates four-scroll chaotic attractors while the integer-order memristive system (1) generates three-scroll chaotic attractors. To the best of our knowledge, this result is rarely reported. Moreover, let fractional-order q=0.965, it is obtained that the coexisting two kinds of three-scroll and four-scroll chaotic attractors emerge for different initial conditions. Therefore, not only three-scroll chaotic attractors but also four-scroll chaotic attractors are found in fractional-order memristive system (2).Finally, a control strategy for the fractional-order memristive chaotic system (2) is proposed via single-state variable, and numerical simulations are employed to verify the validity of the proposed control strategy.



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