# A New Type of Combination Synchronization among Multiple Chaotic Systems

Authors: Xingxu Wang; Lin Sun; Bingji Wang; Tousheng Huang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

Based on former combination synchronization studies, a new type of combination synchronization approach is developed in this research, with the consideration of parallel combination of drive systems. This new synchronization approach is referred to as combination synchronization-II. As a representative case, the combination synchronization-II between three drive systems and one response system is studied. Applying Lyapunov stability theorem and active backstepping design, sufficient conditions for the proposed combination synchronization approach are derived. Numerical simulations are performed to show the feasibility and effectiveness of the proposed approach. Based on the investigation in this research, the proposed approach provides an applicable method for designing universal combination synchronization among multiple chaotic systems.


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## Body

## 1. Introduction

Chaos phenomena have been widely observed in many systems and chaos synchronization plays an important role in the research on chaos. Great efforts were devoted to chaos synchronization of chaotic systems, a very important subfield of nonlinear science [1]. Since the pioneering work of Pecora and Carroll [2], chaos synchronization has been investigated in a variety of fields, such as ecological systems, chemical reactions, engineering science, and secure communication [3–5].Many research works have been made to achieve various types of chaos synchronization, for example, complete synchronization [6, 7], phase synchronization [8, 9], and generalized synchronization [10, 11]. Nevertheless, most of the research works of chaos synchronization are limited to the synchronization of two chaotic systems, i.e., one drive system and one response system. In 2012, Luo et al. [12] developed a new type of synchronization, combination synchronization, which was applied to investigate the synchronization of two drive systems and one response system. From then on, the synchronization among multiple chaotic systems was extensively studied by many researchers [13–18].The combination synchronization in recent literature develops in various types. Xi et al. proposed the function projective combination synchronization and determined the stability criterion for such combination synchronization of fractional-order chaotic systems [19]. Wang et al. investigated a novel adaptive generalized combination complex synchronization for different real and complex nonlinear systems with unknown parameters [20]. Vincent et al. developed a multiswitching combination synchronization of chaotic systems [18], and this synchronization type is further developed by Ahmad et al. as globally exponential multi-switching-combination synchronization control for chaotic systems in the field of secure communications [21]. Based on the combination synchronization, with the consideration of four or more chaotic systems, the researchers further proposed and explored combination-combination synchronization in the cases where the numbers of drive systems and response systems are both larger than one [13, 22–25].The application of combination synchronization has been widespread for different chaotic systems as recorded in literature. The chaos and combination synchronization of a new fractional-order Lorenz-like system with two stable node-foci was thoroughly studied by Alam et al. [26]. Khan and Shikha used robust adaptive sliding mode control to investigate the combination synchronization of identical Genesio time-delay chaotic system [27]. Moreover, the combination synchronization scheme has also been introduced for complex-variable chaotic systems [17]. Sun et al. launched investigations on the finite-time combination synchronization of complex-variable chaotic systems with unknown parameters via sliding mode control [28, 29]. Singh et al. proposed a novel scheme for the dual combination synchronization among four master systems and two slave systems for the fractional-order complex chaotic systems with stability analysis [30].As so far, the combination synchronization approaches focus on series combination of drive systems; i.e., the drive systems can be regarded as one combined supersystem to synchronize with the response system. Actually, there are two basic ways for combination of three systems, series combination () and parallel combination (). Based on the two combination ways, many complex systems can be composed, such as electric circuit and ecological food web [31, 32]. The combination synchronization based on parallel combination of drive systems with one response system is described as the following: (1) the drive systems are divided into at least two groups, and each group exists as one of the parallel branches in the system combination; (2) the drive systems on the same branch jointly drive the response system; (3) the drive systems on different branches synchronize with the response system separately; and (4) the response system takes merely one controller in the combination synchronization. In this research, we refer the combination synchronization based on such parallel combination of drive systems as combination synchronization-II, whereas we refer to the former Runzi’s combination synchronization as combination synchronization-I [12].In comparing with former research works on the combination synchronization, the combination synchronization-II designed in this research may be a similar but extended type to combination complex synchronization or dual combination synchronization. The combination complex synchronization, for example, as described in the research of Sun et al. [17, 28, 29], suggests that the real part and the imaginary part of the drive systems and the response system, which are all complex systems, achieve combination synchronization, respectively. In this sense, the combination synchronization-II also considers that the synchronization of the chaotic systems can be divided into a few parallel branches or parts, similar to the division of real and imaginary parts in combination complex synchronization.However, the combination synchronization-II exhibits three aspects of difference from the combination complex synchronization. Firstly, the response system commonly exists in every parallel branch divided for the synchronization. Secondly, the division of parallel branches for synchronization relies on the relationship between the chaotic systems in reality, and this is very different from the relation of real and imaginary parts in complex systems. Thirdly, according to the number of systems in each branch, synchronization between two chaotic systems or combination synchronization among three or more chaotic systems is achieved, implying that the synchronization in different braches can be unidentical. Based on the comparison with previous approaches in literature, the combination synchronization-II may be more general than the combination complex synchronization or the combination synchronization-I and it shows advantages in practical applications for more system synchronization modes. Therefore, the research on the combination synchronization-II still remains open and deserves investigation.The organization of this research is organized as follows. Section2 shows the scheme of combination synchronization-II with three drive systems and one response system, which shows two cases. In Section 3, we study the first case of the combination synchronization-II among four chaotic systems. Section 4 focuses on the achievement of the second case of the combination synchronization-II. At last, conclusions are described in Section 5.


## 2. The Scheme of Combination Synchronization-II

The combination synchronization-II is a drive-response type synchronization. Consider multiple drive systems and one response system, the parallel combination for drive systems may have various cases. It is easy to identify that the parallel combination of two drive systems has a simple structure as Figure1(a). For three drive systems, there exist two cases for the parallel combination of drive systems, as shown in Figures 1(b) and 1(c). The case of Figure 1(b) describes three parallel branches, each one of which has merely one drive system whereas the case of Figure 1(c) demonstrates two branches, one of which has two serial drive systems. It should be noticed that the parallel combination of two drive systems can be simplified from the two cases of Figure 1(b) or Figure 1(c). Moreover, more complex cases of parallel combinations can be extended similarly from these two cases.Figure 1
The parallel combination of two and three drive systems. DS: drive system; RS: response system.
(a) (b) (c)Based on the above description, the two cases of Figures1(b) and 1(c) can be regarded as representative cases. Therefore, we focus on the combination synchronization-II of three drive systems and one response system. In this section, the combination synchronization-II scheme for three drive systems and one response system is designed. The three drive systems are given as follows: (1)x˙i=fixii=1,2,3,and the response system is given by(2)z˙=hz+ux1,x2,x3,z.In the above equations, the symbols have the following meaning: xi=x1(i),x2(i),...,xn(i)Ti=1,2,3, z=z1,z2,...,znT∈Rn are the state vectors of systems (1) and (2), respectively; fii=1,2,3, h: Rn→Rn are four continuous vector functions; u=Rn×Rn×Rn×Rn→Rn is a controller to be designed.Definition 1.
Combination synchronization-II between the drive systems (1) and the response system (2) is achieved if there exist constant matrixes Ai, Bj∈Rni=1,2,3;j=1,2,⋯,l and Bj≠0 such that (3)∑j=1llimt→+∞∑i=kj-1+1kjAixi-Bjz=0,in which · represents the matrix norm; l and kjj=0,1,2,⋯,l are integers and satisfy 1<l≤3 and 0=k0<k1<⋯<kl=3.Remark 2.
The constant matrixesAii=1,2,3, Bjj=1,2,⋯,l are called scaling matrixes. Moreover, these matrixes can be extended to functional matrixes of the state variables xii=1,2,3 and z.Remark 3.
If(1)l=1; or (2)l=2, k1=1, A1=B1=0; or (3)l=2, k1=2, A3=B2=0, the combination synchronization-II problem will be turned into the combination synchronization-I problem.Remark 4.
If(1)l=2, k1=1, (1i) A1=0 or (1ii) A2=0; or (2)l=2, k1=2, (2i) A2=0 or (2ii) A3=0; or (3)l=3, k1=1, k2=2, (3i) A1=B1=0 or (3ii) A2=B2=0 or (3iii) A3=B3=0, the combination synchronization-II problem between three drive systems and one response system will be reduced to the combination synchronization-II problem between two drive systems and one response system.Remark 5.
If(1)l=1, B1=I, (1i) A1=A2=0 or (1ii) A1=A3=0 or (1iii) A2=A3=0; or (2)l=2, k1=1, (2i) B1=I and A2=A3=B2=0 or (2ii) B2=I and A1=A2=B1=0 or (2iii) B2=I and A1=A3=B1=0; or (3)l=2, k1=2, (3i) B1=I and A1=A3=B2=0 or (3ii) B1=I and A2=A3=B2=0 or (3iii) B2=I and A1=A2=B1=0; or (4)l=3, k1=1, k2=2, (4i) B1=I and A2=A3=B2=B3=0 or (4ii) B2=I and A1=A3=B1=B3=0 or (4iii) B3=I and A1=A2=B1=B2=0, the combination synchronization-II problem will be reduced to the projective synchronization, where I is an n×n identity matrix.Remark 6.
If(1)l=1, A1=A2=A3=0; or (2)l=2, k1=1or2, A1=A2=A3=0, (2i) B1=0 or (2ii) B2=0;or (3)l=3, k1=1, k2=2, A1=A2=A3=0, (3i) B1=B2=0 or (3ii) B1=B3=0 or (3iii) B2=B3=0, the combination synchronization-II problem will be turned into a chaos control problem.Remark 7.
The combination synchronization-II scheme as described in Definition1 can be extended to the cases which have more than three drive systems.According to Definition1, l can have two values, 2 and 3, for the combination synchronization-II between three drive systems and one response system. In this research, we define the case of l=3 as Case 1, and the case of l=2 as Case 2, corresponding to the two cases of Figures 1(b) and 1(c). The applications of the combination synchronization-II scheme for the two cases are provided in the following.


## 3. Combination Synchronization-II of Case 1

The combination synchronization-II scheme is applied to four chaotic systems. Such application can adapt to a variety of chaotic systems in engineering science, secure communication, etc. As widely known, Lorenz system, Chen system, Rössler system, and Lü system are four basic and classical chaotic systems, which have been studied by many researchers [33–35] and can be extended for designing combination synchronization-II scheme. Also, the memristor chaotic systems, for example, are also suitable cases for the investigation of combination synchronization-II. Recently, autonomous memristor systems of infinite chaotic attractors were proposed and electronic circuits of such memristor chaotic systems have been designed [36, 37]. With circuitry realization where memristor chaotic systems show parallel combination, the combination synchronization-II scheme can be extended for investigating multiple memristor systems of infinite chaotic attractors [36, 37].In this section, the combination synchronization-II of Case 1 is realized among the four classic chaotic systems, Lorenz system, Chen system, Rössler system, and Lü system. We consider the first three systems as drive systems and Lü system as response system. The Lorenz system, Chen system, and Rössler system are described by the following systems (4), (5), and (6), respectively, (4)x˙1=a1x2-x1,x˙2=b1x1-x2-x1x3,x˙3=x1x2-c1x3,(5)y˙1=a2y2-y1,y˙2=c2-a2y1+c2y2-y1y3,y˙3=y1y2-b2y3.(6)w˙1=-w2-w3,w˙2=w1+a3w2,w˙3=b3+w3w1-c3,As described in literature [12, 13], Lorenz system shows chaotic behavior when a1=10, b1=28, and c1=8/3; Chen system exhibits chaotic behavior when a2=35, b2=3, and c2=28; Rössler system exhibits chaotic behavior when a3=0.2, b3=0.2, and c3=5.7. On the other hand, the controlled Lü system is given by (7)z˙1=a4z2-z1+u1,z˙2=c4z2-z1z3+u2,z˙3=z1z2-b4z3+u3.Lü system exhibits chaotic behavior when a4=36, b4=3, and c4=20. In (7), u1, u2, and u3 are controllers to be designed.Notice that to avoid confusions in the calculations, the state variables of the three drive systems in Definition1 are redenoted as x=x1,x2,x3T, y=y1,y2,y3T and w=w1,w2,w3T in (4), (5), and (6). Simultaneously, the constant matrixes in Definition 1 are also redenoted as A, B, C, D, E, and H. According to Definition 1, the combination synchronization-II of Case 1 can be redescribed by(8)limt→+∞Ax-Bz+limt→+∞Cy-Dz+limt→+∞Ew-Hz=0.For convenience of the calculations, we write the constant matrixesA, B, C, D, E, and H as A=diagα1,α2,α3, B=diagβ1,β2,β3, C=diagγ1,γ2,γ3, D=diagδ1,δ2,δ3, E=diagε1,ε2,ε3, and H=diagη1,η2,η3, and let (9)e1=Bz-Ax,e2=Dz-Cy,e3=Hz-Ew,in which e1=e11,e12,e13T, e2=e21,e22,e23T, and e3=e31,e32,e33T. Hence, the error system can be written as the following: (10)e˙11=a4β1β2e12-a4e11+f1+β1u1,e˙12=c4e12-β2β1β3e11e13-α3β2β1β3x3e11-α1β2β1β3x1e13+g1+β2u2,e˙13=β3β1β2e11e12+α2β3β1β2x2e11+α1β3β1β2x1e12-b4e13+h1+β3u3,e˙21=a4δ1δ2e22-a4e21+f2+δ1u1,e˙22=c4e22-δ2δ1δ3e21e23-γ3δ2δ1δ3y3e21-γ1δ2δ1δ3y1e23+g2+δ2u2,e˙23=δ3δ1δ2e21e22+γ2δ3δ1δ2y2e21+γ1δ3δ1δ2y1e22-b4e23+h2+δ3u3,e˙31=a4η1η2e32-a4e31+f3+η1u1,e˙32=c4e32-η2η1η3e31e33-ε3η2η1η3w3e31-ε1η2η1η3w1e33+g3+η2u2,e˙33=η3η1η2e31e32+ε2η3η1η2w2e31+ε1η3η1η2w1e32-b4e33+h3+η3u3,in which (11)f1=a4α2β1β2x2-a4α1x1-a1α1x2-x1,g1=c4+1α2x2-b1α2x1+α2-α1α3β2β1β3x1x3,h1=α1α2β3β1β2-α3x1x2+c1-b4α3x3,f2=a4γ2δ1δ2y2-a4γ1y1-a2γ1y2-y1,g2=c4-c2γ2y2-c2-a2γ2y1+γ2-γ1γ3δ2δ1δ3y1y3,h2=γ1γ2δ3δ1δ2-γ3y1y2+b2-b4γ3y3,f3=-a4ε1w1+a4ε2η1η2+ε1w2+ε1w3,g3=-ε1ε3η2η1η3w1w3-ε2w1+c4-a3ε2w2,h3=ε1ε2η3η1η2w1w2-ε3w1w3+c3-b4ε3w3-b3ε3.Based on the active backstepping design method, the controllers u1, u2, and u3 can be designed in the following for realizing the combination synchronization-II of Case 1 among Lorenz system, Chen system, Rössler system, and Lü system. For the details of the active backstepping design method, one can refer to Luo et al. [12].Theorem 8.
If the controllers are chosen as(12)u1=-s1β1f1-s2δ1f2-s3η1f3,u2=-s1a4-1c4+1a4β1+a4β1β22-α3x3β1β3v11-s2a4-1c4+1a4δ1+a4δ1δ22-γ3y3δ1δ3v21-s3a4-1c4+1a4η1+a4η1η22-ε3w3η1η3v31-c4-a4+2s1β2v12+s2δ2v22+s3η2v32-s1β2g1-s2δ2g2-s3η2g3,u3=-s1a4-1a4β12v11+1β11β2-β2β32v12v11+α1x1+α2β1β2x2v11-s2a4-1a4δ12v21+1δ11δ2-δ2δ32v22v21+γ1y1+γ2δ1δ2y2v21-s3a4-1a4η12v31+1η11η2-η2η32v32v31+ε1w1+ε2η1η2w2v31+b4-1s1β3v13+s2δ3v23+s3η3v33-s1β3h1-s2δ3h2-s3η3h3,in which (13)s1=sgnx1-sgny1-sgnw,s2=sgny1-sgnx1-sgnw,s3=sgnw1-sgnx1-sgny,(14)v11v12v13=1001-a4β2a4β110001e11e12e13,v21v22v23=1001-a4δ2a4δ110001e21e22e23,v31v32v33=1001-a4η2a4η110001e31e32e33,the drive systems (4), (5), and (6) will achieve combination synchronization-II (Case 1) with the response system (7).Proof.
Based on the linear transformation described in (14), we construct v11,v12,v13-subsystem. According to the definition of combination synchronization-II, we have y=0 and w=0 for this subsystem. Then direct calculations lead to (15)v˙11=a4β1β2v12-1-a4β2a4β1v11-a4v11+f1+β1u1x>0,y=0,w=0=-v11+a4β1β2v12,v˙12=1-a4β2a4β1-v11+a4β1β2v12+c4v12-1-a4β2a4β1v11-β2β1β3v11v13-α3β2β1β3x3v11-α1β2β1β3x1v13+g1+β2u2x>0,y=0,w=0=-a4β1β2v11-v12-β2β1β3v11+α1x1v13,v˙13=β3β1β2v11+α1β3β1β2x1v12-1-a4β2a4β1v11+α2β3β1β2x2v11-b4v13+h1+β3u3x>0,y=0,w=0=β2β1β3v11+α1x1v12-v13.
We can take a Lyapunov function as(16)V1=12v112+12v122+12v132
for thev11,v12,v13-subsystem. The derivative of V1 is given by (17)V˙1=v11v˙11+v12v˙12+v13v˙13=v11-v11+a4β1β2v12+v12-a4β1β2v11-v12-β2β1β3v11+α1x1v13+v13β2β1β3v11+α1x1v12-v13=-v112-v122-v132≤0.
According to the Lyapunov stability theorem,V˙1≤0 as shown in (17) suggests that the equilibrium 0,0,0 is globally asymptotically stable for the v11,v12,v13-subsystem. Likewise, we construct v21,v22,v23-subsystem and we have x=0 and w=0 for this subsystem. Meanwhile, the following can be obtained: (18)v˙21=a4δ1δ2v22-1-a4δ2a4δ1v21-a4v21+f2+δ1u1x=0,y>0,w=0=-v21+a4δ1δ2v22,v˙22=1-a4δ2a4δ1-v21+a4δ1δ2v22+c4v22-1-a4δ2a4δ1v21-δ2δ1δ3v21v23-γ3δ2δ1δ3y3v21-γ1δ2δ1δ3y1v23+g2+δ2u2x=0,y>0,w=0=-a4δ1δ2v21-v22-δ2δ1δ3v21+γ1y1v23,v˙23=δ3δ1δ2v21+γ1δ3δ1δ2y1v22-1-a4δ2a4δ1v21+γ2δ3δ1δ2y2v21-b4v23+h2+δ3u3x=0,y>0,w=0=δ2δ1δ3v21+γ1y1v22-v23,
Choosing a candidate Lyapunov function,(19)V2=12v212+12v222+12v232,
for thev21,v22,v23-subsystem. The derivative of V2 is given by (20)V˙2=v21v˙21+v22v˙22+v23v˙23=v21-v21+a4δ1δ2v22+v22-a4δ1δ2v21-v22-δ2δ1δ3v21+γ1y1v23+v23δ2δ1δ3v21+γ1y1v22-v23=-v212-v222-v232≤0,which suggests that v21,v22,v23 asymptotically approaches 0,0,0 as time increases to infinite.
Constructv31,v32,v33-subsystem which satisfies x=0 and y=0. Then we have (21)v˙31=a4η1η2v32+a4-1η2a4η1v31-a4v31+f3+η1u1x=0,y=0,w>0=-v31+a4η1η2v32,v˙32=1-a4η2a4η1-v31+a4η1η2v32+c4v32+a4-1η2a4η1v31-η2η1η3v31v33-ε3η2η1η3w3v31-ε1η2η1η3w1v33+g3+η2u2x=0,y=0,w>0=-a4η1η2v31-v32-η2η1η3v31+ε1w1v33,v˙33=η3η1η2v31+ε1η3η1η2w1v32+a4-1η2a4η1v31+ε2η3η1η2w2v31-b4v33+h3+η3u3x=0,y=0,w>0=η2η1η3v31+ε1w1v32-v33.
Choosing a candidate Lyapunov function,(22)V3=12v312+12v322+12v332,
for thev31,v32,v33-subsystem. The derivative of V3 is given by (23)V˙3=v31v˙31+v32v˙32+v33v˙33=v31-v31+a4η1η2v32+v32-a4η1η2v31-v32-η2η1η3v31+ε1w1v33+v33η2η1η3v31+ε1w1v32-v33=-v312-v322-v332≤0,which suggests that v31,v32,v33 asymptotically approaches 0,0,0.
Taking the above results together, we havev11,v12,v13→0,0,0, v21,v22,v23→0,0,0, and v31,v32,v33→0,0,0 as t→+∞. Notice the linear transformation described by (14), we know eij→0i=1,2,3;j=1,2,3. It means that the drive systems (4), (5), and (6) achieve combination synchronization-II with the response system (7).From Theorem8, the following corollaries can be deduced. Since the proofs of the corollaries are similar to Theorem 8, we omit the corollary proofs for the sake of simplification.Corollary 9.
If the controllers are chosen as(24)u=β1u1x>0,y=0,w=0+δ1u1x=0,y>0,w=0+η1u1x=0,y=0,w>0β1+δ1+η1β2u2x>0,y=0,w=0+δ2u2x=0,y>0,w=0+η2u2x=0,y=0,w>0β2+δ2+η2β3u3x>0,y=0,w=0+δ3u3x=0,y>0,w=0+η3u3x=0,y=0,w>0β3+δ3+η3,in which u1, u2, and u3 are described by (10), we will have (25)limt→+∞Ax+Cy+Ew-B+D+Hz=0.Hence, the drive systems (4), (5), and (6) can achieve combination synchronization-I with the response system (7).Corollary 10.
(i) Letα1=α2=α3=0 and β1=β2=β3=0; if the controllers are chosen as (10) or (24) with removing all the terms related to v11, v12, v13, the drive systems (5) and (6) can achieve combination synchronization-II or combination synchronization-I with the response system (7).
(ii) Letγ1=γ2=γ3=0 and δ1=δ2=δ3=0; if the controllers are chosen as (10) or (24) with removing all the terms related to v21, v22, v23, the drive systems (4) and (6) can achieve combination synchronization-II or combination synchronization-I with the response system (7).
(iii) Letε1=ε2=ε3=0 and η1=η2=η3=0; if the controllers are chosen as (10) or (24) with removing all the terms related to v31, v32, v33, the drive systems (4) and (5) can achieve combination synchronization-II or combination synchronization-I with the response system (7).Corollary 11.
(i) Letβ1=β2=β3=1, γ1=γ2=γ3=0, δ1=δ2=δ3=0, ε1=ε2=ε3=0, and η1=η2=η3=0; if the controllers are chosen as (26)u=-f1-a4-1c4+1a4+a4-α3x3v11-c4-a4+2v12-g11-a4a4v11v11+α1x1-α2x2v11+b4-1v13-h1,the drive system (4) will achieve projective synchronization with the response system (7).
(ii) Letα1=α2=α3=0, β1=β2=β3=0, δ1=δ2=δ3=1, ε1=ε2=ε3=0, and η1=η2=η3=0; if the controllers are chosen as (27)u=-f2-a4-1c4+1a4+a4-γ3y3v21-c4-a4+2v22-g21-a4a4v21v21+γ1y1-γ2y2v21+b4-1v23-h2,the drive system (5) will achieve projective synchronization with the response system (7).
(iii) Letα1=α2=α3=0, β1=β2=β3=0, γ1=γ2=γ3=0, δ1=δ2=δ3=0, and η1=η2=η3=1; if the controllers are chosen as (28)u=-f3-a4-1c4+1a4+a4-ε3w3v31-c4-a4+2v32-g31-a4a4v31v31+ε1w1-ε2w2v31+b4-1v33-h3,the drive system (6) will achieve projective synchronization with the response system (7).Corollary 12.
Letα1=α2=α3=0, γ1=γ2=γ3=0, ε1=ε2=ε3=0, and (i) β1=β2=β3=1, δ1=δ2=δ3=0, η1=η2=η3=0, or (ii) β1=β2=β3=0, δ1=δ2=δ3=1, η1=η2=η3=0, or (iii) β1=β2=β3=0, δ1=δ2=δ3=0, η1=η2=η3=1; if the controllers are chosen as (29)u=0-a4-1c4+1a4+a4v1-c4-a4+2v21-a4a4v12+b4-1v3,in which v1=z1, v2=z2-a4-1z1/a4, v3=z3, the equilibrium 0,0,0 of the response system (7) is globally asymptotically stable.Numerical simulations are performed to illustrate the above combination synchronization-II among systems (4), (5), (6), and (7). Fourth-order Runge-Kutta method is applied with time step equal to 0.001, and the initial values of variables x, y, w, and z are provided randomly. We choose αi=βi=γi=δi=ηi=1, εi=3 (i=1,2,3). The parameter values of systems (4), (5), (6), and (7) are the same as described along with the systems, i.e., a1=10, b1=28, c1=8/3, a2=35, b2=3, c2=28, a3=0.2, b3=0.2, c3=5.7, a4=36, b4=3, and c4=20. With these given conditions, the controller can be calculated as (30)u1=26s1x1-x2+s2y1-y2+s3108w1-111w2-3w3,u2=-s156+512-x3e11-s256+512-y3e21-s356+512-3w3e31+14s1e12-3536e11+14s2e22-3536e21+14s3e32-3536e31-7s13x2-4x1-s27y1+12y2-3s319.8w2-3w1w3-w1,u3=-s1e113536e11+x1+x2-s2e213536e21+y1+y2-s3e313536e31+3w1+3w2+2s1e13+s2e23+s3e33+13s1x3-3s33w1w2-w1w3+2.7w3-0.2.Substituting the equations of controller into the error system, we then have (notice thats1 = 1,s2 = 0, ands3 = 0 for the dynamics ofe11,e12, ande13,s1 = 0,s2 = 1, ands3 = 0 for the dynamics ofe21,e22, ande23,s1 = 0,s2 = 0, ands3 = 1 for the dynamics ofe31,e32, ande33), (31)e˙11=36e12-e11,e˙12=34e12-70+136e11-e11e13-x1e13,e˙13=e11e12+x1e12-e13-3536e11e11+x1,e˙21=36e22-e21,e˙22=34e22-70+136e21-e21e23-y1e23,e˙23=e21e22+y1e22-e23-3536e21e21+y1,e˙31=36e32-e31,e˙32=34e32-70+136e31-e31e33-3w1e33,e˙33=e31e32+3w1e32-e33-3536e31e31+3w1.Via numerical calculations on the above error system, Figure2 demonstrates the time response of the synchronization errors eij (i=1,2,3 and j=1,2,3), which converges to zero, suggesting that the combination synchronization-II among systems (4), (5), (6), and (7) indeed achieves. Figure 3 depicts time response of the states, xi, yi, wi, and zi, showing the synchronization between the drive systems and the response system.Figure 2
Time response of synchronization errorseij,i=1, 2, 3 andj=1, 2, 3, for the combination synchronization-II of Case 1 among systems (4), (5), (6), and (7).Figure 3
Time response of statesxi,yi,wi, and zi,i=1,2,3, and the comparison of states between the drive systems and the response system.


## 4. Combination Synchronization-II of Case 2

In this section, the combination synchronization-II of Case 2 among the four chaotic systems, Lorenz system, Chen system, Rössler system, and Lü system, is realized. Lorenz system, Chen system, and Rössler system are also taken as drive systems, in which Lorenz system and Chen system are on the same branch as shown in Figure1(c); Lü system is taken as the response system.According to Definition1, the combination synchronization of Case 2 among the four chaotic systems can be described by (32)limt→+∞Ax+By-Cz+limt→+∞Dw-Ez=0.Let(33)e1=Cz-Ax-By,e2=Ez-Dw,and then the error system can be described by (34)e˙11=a4γ1γ2e12-a4e11+f1+γ1u1,e˙12=c4e12-γ2γ1γ3e11e13+α3x3+β3y3e11+α1x1+β1y1e13+g1+γ2u2,e˙13=γ3γ1γ2e11e12+α2x2+β2y2e11+α1x1+β1y1e12-b4e13+h1+γ3u3,e˙21=a4ε1ε2e22-a4e21+f2+ε1u1,e˙22=c4e22-ε2ε1ε3e21e23-δ3ε2ε1ε3w3e21-δ1ε2ε1ε3w1e23+g2+ε2u2,e˙23=ε3ε1ε2e21e22+δ2ε3ε1ε2w2e21+δ1ε3ε1ε2w1e22-b4e23+h2+ε3u3,in which(35)f1=a4α2γ1γ2x2+a4β2γ1γ2y2-a4α1x1+β1y1-a1α1x2-x1-a2β1y2-y1,g1=-b1α2x1+c4+1α2x2+a2-c2β2y1+c4-c2β2y2+α2x1x3+β2y1y3-γ2γ1γ3α1x1+β1y1α3x3+β3y3,h1=c1-b4α3x3+b2-b4β2y3+γ3γ1γ2α1x1+β1y1α2x2+β2y2-α3x1x2-β3y1y2,f2=-a4δ1w1+a4δ2ε1ε2+δ1w2+δ1w3,g2=-δ1δ3ε2ε1ε3w1w3-δ2w1+c4-a3δ2w2,h2=δ1δ2ε3ε1ε2w1w2-δ3w1w3+c3-b4δ3w3-b3δ3.The active backstepping design method is also applied for designing the controllers u1, u2, and u3 for the error system (34).Theorem 13.
If the controllersu1, u2, and u3 are designed as (36)
u
1
=
-
s
1
γ
1
f
1
-
s
2
ε
1
f
2
,
u
2
=
-
s
1
a
4
-
1
c
4
+
1
a
4
γ
1
+
a
4
γ
1
γ
2
2
-
α
3
x
3
+
β
3
y
3
γ
1
γ
3
v
11
-
c
4
-
a
4
+
2
s
1
γ
2
v
12
+
s
2
ε
2
v
22
-
s
2
a
4
-
1
c
4
+
1
a
4
ε
1
+
a
4
ε
1
ε
2
2
-
δ
3
w
3
ε
1
ε
3
v
21
-
s
1
γ
2
g
1
-
s
2
ε
2
g
2
,
u
3
=
-
s
1
a
4
-
1
a
4
γ
1
2
v
11
2
+
1
γ
1
α
2
x
2
+
β
2
y
2
γ
2
+
a
4
-
1
α
1
x
1
+
β
1
y
1
a
4
γ
1
v
11
-
s
2
a
4
-
1
a
4
ε
1
2
v
21
+
1
ε
1
1
ε
2
-
ε
2
ε
3
2
v
22
v
21
+
δ
1
w
1
+
δ
2
ε
1
ε
2
w
2
v
21
-
s
1
γ
1
1
γ
2
-
γ
2
γ
3
2
v
11
+
α
1
x
1
+
β
1
y
1
v
12
+
b
4
-
1
s
1
γ
3
v
13
+
s
2
ε
3
v
23
-
s
1
γ
3
h
1
-
s
2
ε
3
h
2
,in which f1, g1, h1, f2, g2, and h2 are described by (35) and (37)s1=sgnx+y1-sgnw,s2=sgnw1-sgnx+y,(38)v11v12v13=1001-a4γ2a4γ110001e11e12e13,v21v22v23=1001-a4ε2a4ε110001e21e22e23,the combination synchronization-II of Case 2 will be achieved among systems (4), (5), (6), and (7).Proof.
Based on the linear transformation described in (38), we construct v11,v12,v13-subsystem. For this subsystem, we have w=0. Then direct calculations lead to (39)v˙11=a4γ1γ2v12-1-a4γ2a4γ1v11-a4v11+f1+γ1u1x+y>0,w=0=-v11+a4γ1γ2v12,v˙12=1-a4γ2a4γ1-v11+a4γ1γ2v12+c4v12-1-a4γ2a4γ1v11-γ2γ1γ3v11v13+α3x3+β3y3v11+α1x1+β1y1v13+g1+γ2u2x+y>0,w=0=-γ2γ1γ3v11+α1x1+β1y1v13-a4γ1γ2v11-v12,v˙13=γ3γ1γ2α2x2+β2y2v11+v11+α1x1+β1y1v12-1-a4γ2a4γ1v11-b4v13+h1+γ3u3x+y>0,w=0=γ2γ1γ3v11+α1x1+β1y1v12-v13.
Take a Lyapunov function as(40)V1=12v112+12v122+12v132
for thev11,v12,v13-subsystem. The derivative of V1 is given by (41)V˙1=v11v˙11+v12v˙12+v13v˙13=v11-v11+a4γ1γ2v12+v12-γ2γ1γ3v11+α1x1+β1y1v13-a4γ1γ2v11-v12+v13γ2γ1γ3v11+α1x1+β1y1v12-v13=-v112-v122-v132≤0,which suggests that the equilibrium 0,0,0 is globally asymptotically stable for the v11,v12,v13-subsystem. Likewise, we construct v21,v22,v23-subsystem which satisfies x+y=0. Then we have (42)v˙21=a4ε1ε2v22+a4-1ε2a4ε1v21-a4v21+f2+ε1u1x+y=0,w>0=-v21+a4ε1ε2v22,v˙22=1-a4ε2a4ε1-v21+a4ε1ε2v22+c4v22+a4-1ε2a4ε1v21-ε2ε1ε3v21v23-δ3ε2ε1ε3w3v21-δ1ε2ε1ε3w1v23+g2+ε2u2x+y=0,w>0=-a4ε1ε2v21-v22-ε2ε1ε3v21+δ1w1v23,v˙23=ε3ε1ε2v21+δ1ε3ε1ε2w1v22+a4-1ε2a4ε1v21+δ2ε3ε1ε2w2v21-b4v23+h2+ε3u3x+y=0,w>0=ε2ε1ε3v21+δ1w1v22-v23.
Choosing a candidate Lyapunov function,(43)V2=12v212+12v222+12v232,
for thev21,v22,v23-subsystem. The derivative of V2 is given by (44)V˙2=v21v˙21+v22v˙22+v23v˙23=v21-v21+a4ε1ε2v22+v22-a4ε1ε2v21-v22-ε2ε1ε3v21+δ1w1v23+v23ε2ε1ε3v21+δ1w1v22-v23=-v212-v222-v232≤0,which suggests that v21,v22,v23 asymptotically approaches 0,0,0.
Considering the above results together, we havev11,v12,v13→0,0,0 and v21,v22,v23→0,0,0 as t→+∞. Via the linear transformation described by (38), eij→0i=1,2;j=1,2,3 is obtained. It means the establishment of (32).The following corollary of combination synchronization-I between three drive systems and one response system can be directly obtained from Theorem13. For the other cases, one can refer to Corollaries 9~12.Corollary 14.
If the controllers are chosen as(45)u=γ1u1x+y>0,w=0+ε1u1x+y=0,w>0γ1+ε1γ2u2x+y>0,w=0+ε2u2x+y=0,w>0γ2+ε2γ3u3x+y>0,w=0+ε3u3x+y=0,w>0γ3+ε3,in which u1, u2, and u3 are described by (36), we will have (46)limt→+∞Ax+By+Dw-C+Ez=0.Hence, the drive systems (4), (5), and (6) can achieve combination synchronization-I with the response system (7).For demonstrating the second case of combination synchronization-II, an example of numerical simulation is provided in the following. The parameter values for the combination synchronization-II and the dynamics of the four chaotic systems are given asαi=βi=γi=δi=ηi=εi=1 (i=1,2,3), a1=10, b1=28, c1=8/3, a2=35, b2=3, c2=28, a3=0.2, b3=0.2, c3=5.7, a4=36, b4=3, and c4=20. Then based on the above theoretical calculations, the controller can be determined as (47)u1=s126x1-26x2+y1-y2+s236w1-37w2-w3,u2=-s156+512-x3-y3e11+14s1e12-3536e11+s2e22-3536e21-s256+512-w3e21+s128x1-21x2-7y1+8y2+x1y3+x3y1+s2w1w3+w1-19.8w2,u3=-s13536e112+x2+y2+3536x1+y1e11-s23536e21e21+w1+w2e21+2s1e13+s2e23+s113x3-x1y2-x2y1-s2w1w2-w1w3+2.7w3-0.2.With this controller, the error system can be changed as the following (notice thats1 = 1 ands2 = 0 for the dynamics ofe11,e12, ande13,s1 = 0 ands2 = 1 for the dynamics ofe21,e22, ande23): (48)e˙11=36e12-e11,e˙12=34e12-70+136e11-e11e13-x1+y1e13,e˙13=e11e12-e13-3536e112+x1+y1e12-3536x1+y1e11,e˙21=36e22-e21,e˙22=34e22+43-736e21-e21e23-w1e23,e˙23=e21e22-3536e21e21+w1e22-3536w1e21-e23.By numerically solving the above error system, Figure4 is plotted. Figure 4 shows that the synchronization errors eij (i=1,2 and j=1,2,3) approach zero as time increases. This suggests that the combination synchronization-II of Case 2 as described above is realized. The time response of states xi+yi, wi, and zi as shown in Figure 5 and their comparison further prove that.Figure 4
Time response of synchronization errorseij,i=1, 2 andj=1, 2, 3, for the combination synchronization-II of Case 2 among systems (4), (5), (6), and (7).Figure 5
Comparison of states betweenxi+yi andzi, andwi and zi,i=1,2,3.


## 5. Discussion and Conclusions

This research develops a new type of combination synchronization, which is referred as combination synchronization-II. Based on the active backstepping design method, the combination synchronization-II between three drive systems and one response system is achieved. Numerical simulations are provided to show the feasibility and effectiveness of the proposed combination synchronization approach, with applications in four classical chaotic systems. Moreover, it should be noticed that such application can be extended to other chaotic systems. For example, when multiple memristor systems of infinite chaotic attractors stay at different chaotic states, the combination synchronization-II can take these memristor chaotic systems into the same oscillation.The proposed combination synchronization approach develops the synchronization techniques among multiple chaotic systems. On one hand, the proposed approach potentially has wider applications than former combination synchronizations. On the other hand, the proposed approach can be applied for more complex system combinations. Via comparison with previous approaches, the following advantages of combination synchronization-II can be emphasized:(1)
The combination synchronization-II is more general than the combination synchronization-I or combination complex synchronization. Based on the norm properties and the conditions of combination synchronization-II, the sufficient conditions for the combination synchronization-I or the combination complex synchronization can be deduced.(2)
The combination synchronization-II can be applied for a variety of system combination modes than the combination synchronization-I or the combination complex synchronization. This shows advantages in practical applications. For example, in secure communication when several signals are transmitted, diversified modes of system combinations can effectively increase antitranslated capability.(3)
Based on the proposed approach, an applicable universal combination synchronization scheme can be designed for multiple chaotic systems which feature series and/or parallel system combinations. This enables wider potential applications of the combination synchronization approach in many fields.



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*Source: 8262654-2019-05-23.xml*