Wang-Peng Huang(黃旺鵬) and Qiang Lai(賴強(qiáng)),2,?

1School of Tian You,East China Jiaotong University,Nanchang 330013,China

2School of Electrical and Automation Engineering,East China Jiaotong University,Nanchang 330013,China

Keywords: chaos,memristive chaotic system,multi-wing attractors,coexisting attractors

1.Introduction

In 1963, Lorenz discovered the classic butterfly attractor and first proposed the concept of chaotic systems.[1]Many theories on chaotic systems have been established in the decades since then.Chaos has drawn much attention from various academics due to chaos behaving randomly and unpredictably.What is more, it had a lot of essential applications in image encryption,security,and other engineering fields.[2-8]

In the past few decades, researchers have been exploring the chaos that can produce complex dynamical behavior, and it is worth mentioning that multi-scroll attractors and coexisting attractors have become a hot topic in nonlinear research.[9-19]The state variable can go through multiple orbital states and jump randomly in different transitions for the multi-scroll attractors.The coexisting attractors can provide multiple optional steady states for the system to respond to different needs, so it is very meaningful to construct multi-scroll systems with abundant coexisting attractors.On the other hand, memristors are the fourth fundamental circuit component,reflecting the nonlinear relationship between current and voltage, expressed via a shrinking hysteresis loop.[20]Besides,the memristor has lower power consumption,smaller size,and memory characteristics,making it particularly useful for studying nonlinear science, especially chaos.[21-26]Thus,memristors are often added to chaotic systems as nonlinear terms to construct more complex dynamical behavior and more multi-scroll attractors.[27,28]

In recent years, much research has been done on the non-memristor-based multi-scroll chaotic systems, but the memristor-based chaotic systems have infinite scrolls, which are still rare.[29-33]Xiaet al.[25]proposed two kinds of novel non-ideal voltage-controlled memristors.Adding these memristor models to a three-dimensional (3D) jerk system establishes a novel memristive multi-scroll hyperchaotic jerk system with 2N+2-scroll and 2M+1-scroll hyperchaotic attractors.Laiet al.[35]based on a simple 3D chaotic system,gridscroll chaotic attractors were constructed by adding a fluxcontrolled non-volatile memristor.Yuet al.[36]designed a novel local active and non-volatile memristor and constructed a four-dimensional (4D) memristive Hopfield neural network with the memristor,which have controllable double-scroll attractors.Laiet al.[37]presented a new Hopfield neural network(HNN) that can generate multi-scroll attractors by utilizing a new memristor as a synapse in the HNN.

The above review of the MMCS shows that there are still two issues for further consideration.For one issue, although the number of scrolls in the MMCS has increased,the size of expanded scrolls is not changed.For another issue,the coexisting attractors of the MMCS mentioned above are not rich due to the coexisting attractors are homogeneous.Inspired by the above-mentioned considerations,we propose a bipolargrowth multi-wing memristive chaotic system(BMMCS),numerically drawing the phase planes with different parameters of BMMCS and analyzing the dynamic behavior of BMMCS by the bifurcation diagrams and Lyapunov exponents(LEs).

2.Chaotic system with a memristor model

2.1.System and memristor description

We obtained the following numerical simulation results through Matlab2018a.Setting the step size of phase planes,bifurcation diagrams and LEs to 0.01,0.5,and 0.1.

Considering a simple chaotic system[38]

wherex,y,zare different state variables.We propose a nonideal flux-controlled memristor model.By coupling the memristor and adding a linear term to system (1), we can obtain BMMCS

wherea,bare positive parameters,φis the internal state variable of the memristor.The memristor model can be expressed as

with the currenti, voltagev,fluxφ.The memductance functionW(φ)is described as

wherek ＞0 andN=0, 1, 2....Letk=1 andN=0,1, we obtain

and the curves of Eq.(5) are displayed in Fig.1.We numerically calculate the relationship between voltage and current,the input voltagev=vqsin(2π ft) with voltage amplitudevqand power frequencyf.Letvq=5 V,k=5,N=1, andfrespectively set as 0.1 Hz, 1 Hz, 10 Hz, then keepf=1 Hz,setvq=5 V,8 V,10 V,the results are presented in Fig.2.

Fig.1.Function W(φ)for(a)k=1,N=0;(b)k=1,N=1.

Fig.2.Pinched hysteresis loops of the memristor for (a) frequency f =0.1 Hz,1 Hz,10 Hz;(b)amplitude vq=5 V,8 V,10 V.

In addition,we design the equivalent circuit model of the memristor as shown in Fig.3.One selects the IN4148 diode,operational amplifier TL082 is powered by±15 V-dc,and the multiplier AD633JN gains 0.1.LetC1=100 nF,R0=100 kΩ,R1=10 MΩ,R2=10 MΩ, the memristor circuit is powered by sinusoidal voltage sourceVpsin(2πFt), whichVpis voltage amplitude,andFis power frequency.SelectingVp=5 V,k= 5,N= 1, and letF= 0.1 Hz, 10 Hz, the relationship between voltage and current is displayed in Fig.4.The numerical simulation and circuit experiment both show that the flux-controlled memristor can exhibit pinched hysteresis loops that pinched at the origin.The lobe area decreases with the increase in frequency.

Fig.3.(a)Equivalent memristor circuit,(b)detailed W(φ).

Fig.4.The relationship between voltage and current of the memristor for: (a) f =0.1 Hz,(b) f =10 Hz.

2.2.Equilibrium points and stability analysis

SupposeS(?x,?y,?z, ?φ) is an equilibrium point of BBMCS,then

By solving Eq.(6), one has ?x= ?φ,W(?φ)?φ=±b, ?y=0, and ?z=±1.Linearizing BBMCS atS(?φ,0,±1, ?φ),the Jacobian matrix is obtained as

By calculating|λI-J|=0,we can get the characteristic equation is

wherem1=b+1,m2= 2a+b,m3= 2a+2W(?φ),m4=2W(?φ)+2W＇(?φ)?φ.For the parametersa= 0.74,b= 2.8,k= 5,N= 1, BBMCS can produce 10 equilibrium points.Taking them into Eq.(7),their corresponding eigenvalues and equilibrium point types are shown in Table 1.

Table 1.The eigenvalues and types of equilibrium points.

Table 2.The relationship between ?φm and BFn.

Fig.5.The corresponding relationship between the BMMCS with a 6-wing attractor and ?φm.

Here ?φm,BFn(|m|=1, 2, 3,...,|n|=0, 1, 2,...) respectively represent the unstable index-2 saddle-focus points on theφ-axis and the expanded 2-wing attractors, the relationship between ?φmandBFnis described in Table 2,and their corresponding phase plane is shown in Fig.5.It is obvious that the ?φofBF0is bipolar,and the others are unipolar.HenceBF0differs in shape fromBFn(n/=0).LetN=3,d|n|is the distance of the two equilibrium points ofBFnon theφ-axis,and it can be seen thatd|n|is smaller as|n|increase from Fig.6.

Ifd|n|is too small,BFnwill not appear.And we can increasebto broaden the distance so that one can obtain moreBFnasNincreases, thus letb=6,N=7, the phase plane is displayed in Fig.7.Moreover, we find that with the increase of|n|, the amplitude ofBFnbecomes larger.So it can generate moreBFnwith different amplitude ranges by controllingNandb.Meanwhile, figure 8 displays more phase planes of BBMCS withN=0,3.

Fig.6.The d|n| with W(φ)φ =±1.

Fig.7.BMMCS with a 26-wing attractor.

Fig.8.BMMCS of the phase planes for (a) 2-wing attractor, (b) 14-wing attractor.

3.Dynamic analysis

3.1.Bifurcation diagrams and LEs

The parameters have a great influence on the dynamic behavior of chaotic systems, so we analyze the bifurcation diagrams and the LEs with a step size is 0.001.

Fixingb=2.8,k=5,N=2, anda ∈[0.8, 1.4], the bifurcation diagram from Fig.9(a)indicates that BBMCS has a 10-wing attractor.The LEs from Fig.9(b)show that BBMCS has experienced chaos, quasi-period, and stable point.a ∈[0.8, 1.32)with a positive LE indicates that BBMCS is chaos.BBMCS witha ∈[1.32, 1.34]is quasi-period due to the first two LEs are zero.And all LEs less than zero shows that BBMCS is a stable point whena ∈(1.34, 1.4].Accordingly,settinga=1,1.21,1.34,1.4,the phase planes of BBMCS are shown in Fig.10.It should be noted that the multi-wing attractor may break,[39]especiallya ∈[1.1, 1.25],and we can find a 10-wing attractor and a 3-wing attractor from Figs.10(a)and 10(b),respectively.

Fig.9.(a)The bifurcation diagram about φ,(b)the LEs of the coexisting attractors with N=2.

Fig.10.The phase planes of BBMCS for (a) the multi-wing attractor with a=1,(b)the broken chaotic attractor with a=1.21,(c)the quasi-period with a=1.34,(d)the stable point with a=1.4.

3.2.Coexisting attractors

Adjustinga=1.8,b=3,k=6,N=2, and initial conditions set as (x(0),1,1,1),x(0)∈[-20, 20], the bifurcation diagram ofφand the LEs are displayed in Fig.11.From Fig.11(a), it is obvious that BBMCS has coexisting attractors.And figure 11(b) shows that BBMCS has chaos, stable point, and quasi-period.Accordingly, letx(0)=0,±5,±10,±15,the phase planes are shown in Fig.12.One can observe that the multi-wing attractor breaks up,BF0of BBMCS breaks up into a quasi-period and two stable points,the others break up into singleBFn.

Fig.11.(a)The bifurcation diagram of φ,(b)the LEs of the coexisting attractors with N=2.

Fig.12.The phase planes of the coexisting attractors with N=2.

Fig.13.The phase planes of the coexisting attractors for(a)N=3 and(b)N=6.

However, ifx(0)is less than-20 or greater than 20, the number ofBFnno longer increases with this parameter.It is found thatNcontrols the number of the coexisting attractors,so setN=3,6,BBMCS can break up into 9 and 15 attractors.Correspondingly,the phase planes for the coexisting attractors can be obtained and shown in Fig.13.From Fig.13(b), it is worth noting that as|n| increases,BFnoffset to the poles along theφ-axis.Meanwhile, the amplitude ofBFnalso enlarges atφ ∈[-33,-7]∪[7,33].What is more,d|n|becomes smaller with|n| increase, resulting in the coexisting attractors that begin to transform into the period asnincrease atφ ∈[-52,-33]∪[33, 52].Accordingly,letx(0)∈[-50, 50],the bifurcation diagram aboutyand the LEs are displayed in Fig.14.

Fig.14.(a) The bifurcation diagram about y and (b) the LEs of the coexisting attractors with N=6.

We further studied the impact of tinydnon the formation of the multi-wing attractors.Leta=1,b=0.8,k=5,N=2,setx(0)=±0.5,±10,±15.From Fig.15(a),the phase planes show that the multi-wing attractor breaks up into periodic attractors.From Fig.15(b),the bifurcation diagram aboutφindicates that BBMCS is always in a periodic state under these parameters atx(0)∈[-20, 20].Besides,with|n|increase,the periodic attractors offset to the poles on thewaxis as well as the positive on theyaxis, and figure 16 shows 2 and 10 periodic attractors withN=0,4 in the phase planes.

Fig.15.Coexisting attractors with N =2: (a) the phase plane on φ-y axis,(b)the bifurcation diagram about φ.

Fig.16.The phase planes of the coexisting attractors for: (a)N=0,(b)N=4.

4.Circuit implementation

We designed the circuit with operational amplifiers TL082,switching diode 1N4148,multiplier AD633JN,and so on,as shown in Fig.17.The powered voltages of operational amplifiers are±15 V-dc,and the output gain of the multiplier is 0.1.

Fig.17.The circuit diagram of BBMCS.

The state variables of BBMCS are compressed to(2x,2y,2z,2φ),and BBMCS yield the following equations:

whereRis a reference resistor,vx,vy,vz,vφare the terminal voltages ofC1,C2,C3,C4, and separately correspond to the state variablesx,y,z,φof BBMCS.

SettingR=10 kΩ,C1=C2=C3=C4=100 nF, and BBMCS with parametersa=0.74,b=2.7,k=5,N=1,2.The circuit parameters can be obtain asR0=R2=10 kΩ,R1= 5 kΩ,R3=R4= 500 Ω,R5= 20 kΩ,R6= 680 Ω,R7=1 kΩ,R8=3.6 kΩ.Figure 18 shows the experimental results with a 3-wing and 10-wing attractor.

Fig.18.The experimental results for (a) 6-wing attractor and (b) 10-wing attractor.

5.Conclusion

This paper proposed a non-ideal flux-controlled memristor with a bisymmetric sawtooth piecewise function.BMMCS was generated based on the memristor.It should be noted thatBFncan expand to the poles, and the amplitude of the wing will gradually enlarge.Another significant feature was that BBMCS had rich coexisting attractors including chaos,quasiperiod,stable point.More interestingly,when the multi-wing attractors break up into periodic attractors,the periodic attractors will offset to the poles on thewaxis as well as the positive on theyaxis with|n| increase.We designed the circuit of BBMCS, and the experimental results showed the reliability of BMMCS.It is worth noting that the amplitude of the scroll enlarges as the number of scroll increase in MMCS.This kind of attractor has a more complex structure and richer dynamic behavior,it has more extensive application value in engineering,and there is still much room for further study on this type of system.

Acknowledgements

Project supported by the National Natural Science Foundation of China(Grant Nos.62366014 and 61961019)and the Natural Science Foundation of Jiangxi Province,China(Grant No.20232BAB202008).