# The Synchronization ofN Cascade-Coupled Chaotic Systems

Authors: Pengyu Li; Juan Du; Shouliang Li; Yazhao Zheng; Bowen Jia
Journal: Complexity
 (2019)

---

## Abstract

In this paper, we investigate a novel synchronization method, which consists ofn n≥2 cascade-coupled chaotic systems. Furthermore, as the number of chaotic systems decreases from n to 2, the proposed synchronization will transform into bidirectional coupling synchronization. Based on Lyapunov stability theory, a general criterion is proposed for choosing the appropriate coupling parameters to ensure cascading synchronization. Moreover, 4 Lü systems are taken as an example and the corresponding numerical simulations demonstrate the effectiveness of our idea.


---

## Body

## 1. Introduction

Since Pecora and Carrol reported the discovery of synchronization for two chaotic systems by circuits implementation [1], the exciting phenomenon has gained much attention. In various fields, such as secure communication, signal processing, and life sciences, many types of chaotic synchronization are proving to be increasingly useful [2–5]. The design of novel synchronization models is necessary in such applications and thus motivating researchers to develop it, such as phase synchronization [6], projective synchronization [7], generalized synchronization [8], complex synchronization [9], modulus synchronization [10], and hybrid dislocated synchronization [11].It is known that chaotic systems are extremely sensitive to initial values [12]. For two chaotic systems with the same structure, minor deviations in their initial values will lead to significantly different chaotic state [13]. After the transmission through the cascade-coupled systems, the difference becomes more difficult to be analyzed. In these cases, when a signal to be transmitted is loaded into a cascade of chaotic systems, the security of the transmitted information and the difficulty of being deciphered will be greatly enhanced. However, the existing synchronization has been acquired for two chaotic systems, and the synchronization problems of n chaotic systems are yet to be investigated.In fact, the synchronization for two chaotic systems can be defined as the following two types: unidirectional coupling synchronization and bidirectional coupling synchronization [14]. Unidirectional coupling synchronization includes a drive system and a response system. The two systems are synchronized by introducing the state of the drive system on the response system [15–17]. However, bidirectional coupling synchronization, also known as mutual coupling synchronization, can be obtained by introducing the state of a system to that of another system, in which each system can be considered as drive system or response system [18–20]. When compared with unidirectional coupling synchronization, bidirectional (or mutual) coupling synchronization is more attractive due to the fact that it is ready to implement in practice. Recently, Sivaganesh et al. proposed the synchronization of a network of mutually coupled chaotic systems, in which the synchronization behaviour changes as the coupling parameters change [21].Interestingly, Carroll and Pecora not only discovered chaos synchronization phenomenon but also studied cascading synchronized chaotic systems [22]. Their model consists of a 3D drive system and two 2-D subresponse systems, where the subresponse systems can be constructed by copying two of the expressions of the drive systems. By combining the cascading synchronized chaotic system with projective synchronization schemes, An and Chen reported the function cascade synchronization method [23, 24], which presented 3 chaotic systems. Indeed, the two 2-D response systems are not chaotic systems. And, the status signal of the response system is not fed back to the drive system.By the motivation of the above discussion, a novel synchronization method is investigated in this paper, which consists ofn n≥2 cascade-coupled chaotic systems. This implies that the chaotic systems are in linear topology and cascade-coupled with each other. For example, the i-th system will be synchronized with the i−1-th and i+1-th systems 2≤i≤n−1. Further, as the number of chaotic systems decreases from n to 2, the proposed synchronization will transform into bidirectional coupling synchronization. Based on global control strategy, the synchronization scheme can be realized by choosing the appropriate coupling parameters, and the corresponding numerical simulations are presented to verify the effectiveness of our idea. In short, the proposed synchronization of n cascade-coupled chaotic systems can provide a novel choice for secure communication and signal processing.The reminder of this paper is organized as follows. In Section2, the principle of the synchronization for n cascade-coupled chaotic systems is introduced. And 4 Lü systems are chosen as an example to illustrate the effectiveness of our idea in Section 3. Conclusions are drawn in Section 4.


## 2. Model Description of the Synchronization ofN Cascade-Coupled Chaotic Systems

This section mainly introduces the principle of the synchronization ofN cascade-coupled chaotic systems.We consider the followingn chaotic systems.The first system can be depicted as(1)X˙1=AX1+fX1−D1X1−X2.The second system can be depicted as(2)X˙2=AX2+fX2−D1′X1−X2−D2X2−X3.The third system can be depicted as(3)X˙3=AX3+fX3−D2′X2−X3−D3X3−X4,⋮The (n − 1)-th system can be depicted as(4)X˙n−1=AXn−1+fXn−1−Dn−2′Xn−2−Xn−1−Dn−1Xn−1−Xn.Then-th system can be depicted as(5)X˙n=AXn+fXn−Dn−1′Xn−1−Xn,where X1, X2, …, and Xn denote m-dimensional state vectors of the chaotic systems. A denotes an m×m parameters matrix. fX1, fX2, …, fXn−1, and fXn denote m×1 continuous vector functions. D1, D1′, D2, D2′, D3, D3′, …, Dn−1 and Dn−1′ are diagonal matrices which rule the feedback gain.Definition 1.
For then chaotic systems, our goal is that the trajectory of X1 synchronizes with that of X2, and the trajectory of X2 synchronizes with that of X3, …, finally the trajectory of Xn−1 synchronizes with that of Xn. Then(6)limt⟶∞et=0,where et is a m×n−1-dimensional column vector and et=e1t,e2t,…,en−1tT, with e1t=X1−X2,e2t=X2−X3,…,en−1t=Xn−1−Xn, and ⋅ represents the Euclidean norm.Remark 1.
The error dynamical system can be acquired from equations (1)–(6):(7)e˙1t=X˙1−X˙2=A+HX1,X2−D1−D1′e1t+D2e2t,e˙2t=X˙2−X˙3=A+HX2,X3−D2−D2′e2t−D1′e1t+D3e3t,⋮e˙n−2t=X˙n−2−X˙n−1=A+HXn−2,Xn−1−Dn−2−Dn−2′en−2t−Dn−3′en−3t+Dn−1en−1t,e˙n−1t=X˙n−1−X˙n=A+HXn−1,Xn−Dn−1−Dn−1′en−1t−Dn−2′en−2t,where fX1−fX2=HX1,X2⋅e1t, fX2−fX3=HX2,X3⋅e2t, ⋯, and fXn−1−fXn=HXn−1,Xn⋅en−1t. It is clear that fX1, fX2, fX3, …, fXn−1, fXn are bounded matrices, and thus HX1,X2, HX2,X3, …, HXn−1,Xn are also bounded matrices. A general condition for achieving the synchronization of n cascade-coupled chaotic systems is given in the following theorem.Theorem 1.
If there exists a positive definite symmetric matrixP, and n−1 constants λ1>0, λ2>0, …, λn−1>0, we have that(8)A+HX1,X2−D1−D1′TP+PA+HX1,X2−D1−D1′≤−λ1I,A+HX2,X3−D2−D2′TP+PA+HX2,X3−D2−D2′≤−λ2I,⋮A+HXn−1,Xn−Dn−1−Dn−1′TP+PA+HXn−1,Xn−Dn−1−Dn−1′≤−λn−1I,where I is the identity matrix. Then the synchronization for the cascade-coupled systems (1)–(5) is realized.Proof.
Construct a Lyapunov function as(9)Vt=eTtPn−1et,where Pn−1=diagP,P,…,P, with the total is n−1. Then it is easily known that Vt≥0. And the time derivative of Vt is given as(10)V˙t=e˙1TtPe1t+e1TtPet˙1+et˙2TPe2t+e2TtPe˙2t+⋯+e˙n−1TtPen−1t+en−1TPe˙n−1t=e1TtA+HX1,X2−D1−D1′TPe1t+e1TtPA+HX1,X2−D1−D1′e1t+e2TtD2TPe1t+e1TtPD2e2t+e2TtA+HX2,X3−D2−D2′TPe2t+e2TtPA+HX2,X3−D2−D2′e2t−e1TtD1′TPe2t+e3TtD3TPe2t−e2TtPD1′e1t+e2TtPD3e3t+⋯+en−1TtA+HXn−1,Xn−Dn−1−Dn−1′TPen−1t+en−1TtPA+HXn−1,Xn−Dn−1−Dn−1′en−1t−en−2TtDn−2′TPen−1t−en−1TtPDn−2′en−2t=e1TtA+HX1,X2−D1−D2TP+PA+HX1,X2−D1−D2e1t+e2TtA+HX2,X3−D2−D3TP+PA+HX2,X3−D2−D3e2t+⋯+en−1TtA+HXn−1,Xn−Dn−1−DnTP+PA+HXn−1,Xn−Dn−1−Dnen−1t+e1TtPD2−D1′+D2−D1′TPe2+e2TtPD3−D2′+D3−D2′TPe3+⋯+en−2TtPDn−1−Dn−2′+Dn−1−Dn−2′TPen−1.
As long as(11)D1′=D2,D2′=D3,⋮Dn−2′=Dn−1.
Then, equation (12) can be represented as(12)V˙t=e˙1TtPe1t+e1TtPet˙1+et˙2TPe2t+e2TtPe˙2t+⋯+e˙n−1TtPen−1t+en−1TPe˙n−1t=e1TtA+HX1,X2−D1−D2TP+PA+HX1,X2−D1−D2e1t+e2TtA+HX2,X3−D2−D3TP+PA+HX2,X3−D2−D3e2t+⋯+en−1TtA+HXn−1,Xn−Dn−1−DnTP+PA+HXn−1,Xn−Dn−1−Dnen−1t≤−λ1e1Tte1t+−λ2e2Tte2t+⋯+−λn−1en−1Tten−1t<0.
Equation (12) holds if and only if et≠0. Based on Lyapunov stability theory, system (7) is globally asymptotically stable. This implies that the synchronization of n cascade-coupled chaotic systems is achieved. The proof is completed.Remark 2.
From Theorem1, we obtain that e1t=X1−X2=0, e2t=X2−X3=0, ⋯, and en−1t=Xn−1−Xn=0 as t⟶∞, i.e., X1=X2=⋯=Xn. This implies that any two systems in the n chaotic systems can be synchronized with each other. The interesting phenomenon is demonstrated in Figure 1. Take the first and second system in Figure 1 as an example. Firstly, we get the error signal e1t=X1−X2 through an adder. Then, e1t is injected into the first system via multiplier −D1, while being injected into the second system via multiplier −D1′. The coupling cases between the subsequent systems are similar.Figure 1
The schematic representation of (n) cascade-coupled chaotic systems.Remark 3.
If then chaotic systems are degraded to 2 chaotic systems, then synchronization problem in this paper will change into bidirectional coupling synchronization problem, which has been intensively revealed in previous researches.


## 3. Synchronization of 4 Cascade-Coupled Lü Systems

This section demonstrates the effectiveness and flexibility of the model by adopting 4 Lü systems as an example.The first system can be presented as(13)x˙11=ax12−x11−d11x11−x21,x˙12=cx12−x11x13−d12x12−x22,x˙13=x11x12−bx13−d13x13−x23.The second system can be presented as(14)x˙21=ax22−x21−d11′x11−x21−d21x21−x31,x˙22=cx22−x21x23−d12′x12−x22−d22x22−x32,x˙23=x21x22−bx23−d13′x13−x23−d23x23−x33.The third system can be presented as(15)x˙31=ax32−x31−d21′x21−x31−d31x31−x41,x˙32=cx32−x31x33−d22′x22−x32−d32x32−x42,x˙33=x31x32−bx33−d23′x23−x33−d33x33−x43.The fourth system can be presented as(16)x˙41=ax42−x41−d31′x31−x41,x˙42=cx42−x41x43−d32′x32−x42,x˙43=x41x42−bx43−d33′x33−x43,where X1=x11,x12,x13T, X2=x21,x22,x23T, X3=x31,x32,x33T, and X4=x41,x42,x43T are the state vectors of 4 Lü systems; dij and dij′ i=1,2,3;j=1,2,3 are coupling parameters.Lete1i=x1i−x2i, e2i=x2i−x3i, and e3i=x3i−x4i, with i=1,2,3. From Remark 1, the error dynamical system can be described as(17)e˙1t=X˙1−X˙2=A+HX1,X2−D1−D1′e1+D2e2,e˙2t=X˙2−X˙3=A+HX2,X3−D2−D2′e2−D1′e1+D3e3,e˙3t=X˙3−X˙4=A+HX3,X4−D3−D3′e3−D2′e2,where e1=e11,e12,e13T, e2=e21,e22,e23T, and e3=e31,e32,e33T.Then systems model, which was presented in Section 2, illustrates that A, fX1, fX2, fX3, fX4, HX1,X2, HX2,X3, HX3,X4, D1, D2, D3, and, D4 can be described as(18)A=−aa00c000−b,fX1=0−x11x13x11x12,fX2=0−x21x23x21x22,fX3=0−x31x33x31x32,fX4=0−x41x43x41x42,HX1,X2=000−x130−x21x12x210,HX2,X3=000−x230−x31x22x310,HX3,X4=000−x330−x41x32x410,D1=diagd11,d12,d13,D1′=diagd11′,d12′,d13′,D2=diagd21,d22,d23,D2′=diagd21′,d22′,d23′,D3=diagd31,d32,d33,D3′=diagd31′,d32′,d33′.When the coupling parameter matrices are set asD1=D1′=D2=D2′=D3=D3′=0 and the initial condition of systems (13)–(16) are chosen as x11,x12,x13=1,1,1, x21,x22,x23=5,5,5, x31,x32,x33=10,10,10, x41,x42,x43=20,20,20. Figure 2 indicates that systems (13)–(16) do not synchronize with each other.Figure 2
The graph of synchronization when the coupling parameters are all zero. (a)x11, x21, x31 and x41 vs. t. (b) x12, x22, x32 and x42 vs. t. (c) x13, x23, x33 and x43 vs. t.
(a)
(b)
(c)Based on global control strategy, we choose the positive definite symmetric constant matrixP=diagp1,p2,p3, with pi>0 i=1,2,3, and positive constants λ1>0, λ2>0, λ3>0, then we have(19)A+Hx1,x2−D1−D1′TP+PA+Hx1,x2−D1−D1′+λ1I=−2p1a+d11−d11′−λ12p1p1a−p2x13p3x12p1a−p2x132p2c−d12+d12′+λ12p2p3−p2x21p3x12p3−p2x21−2p3b+d13−d13′−λ12p3.According to the basic algebraic theory, the matrix (19) is negative definite as long as(20)Δ1=−2p1a+d11−d11′−λ12p1<0,(21)Δ2=−2p1a+d11−d11′−λ12p1p1a−p2x13p1a−p2x132p2c−d12+d12′+λ12p2=−4p1p2⋅a+d11−d11′−λ12p1c−d12+d12′+λ12p2−p1a−p2x132>0,(22)Δ3=−2p1a+d11−d11′−λ12p1p1a−p2x13p3x12p1a−p2x132p2c−d12+d12′+λ12p2p3−p2x21p3x12p3−p2x21−2p3b+d13−d13′−λ12p3=−2p3⋅b+d13−d13′−λ12p3Δ2−p3x122⋅2p2c−d12+d12′+λ1+p3−p2x21⋅2p1a+d11−d11′−λ1p3−p2x21+2p3x12p1a−p2x13<0.From equations (20)–(22), we can get(23)d11−d11′>−a+λ12p1,d12−d12′>−p1a−p2x1324p1p2⋅a+d11−d11′−λ1/2p1+c+λ12p2,d13−d13′>−b+λ12p3−p3x122⋅2p2c−d12+d12′+λ12Δ2+p3−p2x21⋅2p1a+d11−d11′−λ1p3−p2x21+2p3x12p1a−p2x132p3Δ2.The synchronization can be achieved easily as long as the inequalities (23) holds, ie., we need to choose the appropriate coupling parameters dij and dij′, with i=1,2,3 and j=1,2,3.For convenience, let the matrixp2=p3, then we have(24)d11−d11′>−a+λ12p1,d12−d12′>−p1a−p2x1324p1p2⋅a+d11−d11′−λ1/2p1+c+λ12p2,d13−d13′>−b+λ12p3−p3x122⋅2p2c−d12+d12′+λ12Δ2.In a similar way, we can get the following two inequalities:(25)d21−d21′>−a+λ22p1,d22−d22′>−p1a−p2x2324p1p2⋅a+d21−d21′−λ2/2p1+c+λ22p2,d23−d23′>−b+λ22p3−p3x222⋅2p2c−d22+d22′+λ22Δ2′.(26)d31−d31′>−a+λ32p1,d32−d32′>−p1a−p2x3324p1p2⋅a+d31−d31′−λ3/2p1+c+λ32p2,d33−d33′>−b+λ32p3−p3x322⋅2p2c−d32+d32′+λ32Δ2″,where Δ2′=−4p1p2⋅a+d21−d21′−λ2/2p1c−d22+d22′+λ2/2p2−p1a−p2x232, and Δ2′′=−4p1p2⋅a+d31−d31′−λ3/2p1c−d32+d32′+λ3/2p2−p1a−p2x332.The following numerical simulations are presented to demonstrate the effectiveness of our idea. Leta=36,b=3,c=20, P = diag {1, 1, 1}. λj=60, with j=1,2,3. For the purpose of satisfying above inequalities (24)–(26), the coupling parameters are set as d11=d12=d13=10, d11′=d12′=d13′=5, d21=d22=d23=5, d21′=d22′=d23′=5, d31=d32=d33=1, d31′=d32′=d33′=−10. The corresponding numerical results are illustrated in Figures 3 and 4. Figure 3 indicates that the trajectory of 4 Lü systems can be synchronized with each other. Figure 4 illustrates that the errors e1=e11,e12,e13T, e2=e21,e22,e23T, e3=e31,e32,e33T are finally stabilized to 0 as t⟶∞, i.e., the synchronization for 4 cascade-coupled Lü systems is realized.Figure 3
The graph of synchronization. (a)x11, x21, x31 and x41 vs. t. (b) x12, x22, x32 and x42 vs. t. (c) x13, x23, x33 and x43 vs. t.
(a)
(b)
(c)Figure 4
The errors of synchronization. (a)e11, e21 and e31 vs. t. (b) e12, e22 and e32 vs. t. (c) e13, e23 and e33 vs. t.
(a)
(b)
(c)Figure 5
The largest Lyapunov exponent of system (13).Remark 4.
When fixing the above coupling parameters exceptd11, then we obtain d11>−1. Further, we select d11∈−10,40 to observe the nature of the dynamical state of system (13) before and after synchronization. Here, we note that the largest Lyapunov exponent of system (13) is always greater than zero, which demonstrates that system (13) operates in chaotic state within a large range of coupling parameters (Figure 5).


## 4. Conclusions

Recently, the chaos synchronization has been a subject of increasing research interest due to its potential application in fields of secure communication, signal processing, and life sciences. Numerous researchers have extensively explored the synchronization of two chaotic systems. This paper firstly proposes cascade-coupled synchronization ofn n≥2 chaotic systems, which is proved by rigorous mathematical analysis. Based on Lyapunov stability theory, a general condition is presented and applied to 4 Lü systems. This validates the effectiveness of our idea and then motivates us to develop cascade synchronization in different cases, such as complex chaotic systems, fractional-order chaotic systems, and time-delayed chaotic systems. Moreover, the other coupling parameter types are also worthy of further investigation.



---

*Source: 2709820-2019-12-12.xml*