# Combination-Combination Synchronization of Four Nonlinear Complex Chaotic Systems

Authors: Xiaobing Zhou; Lianglin Xiong; Xiaomei Cai
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

This paper investigates the combination-combination synchronization of four nonlinear complex chaotic systems. Based on the Lyapunov stability theory, corresponding controllers to achieve combination-combination synchronization among four different nonlinear complex chaotic systems are derived. The special cases, such as combination synchronization and projective synchronization, are studied as well. Numerical simulations are given to illustrate the theoretical analysis.


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## Body

## 1. Introduction

In 1982, Fowler et al. [1] generalized the real Lorenz model to a complex Lorenz model, which can be used to describe and simulate the physics of a detuned laser and the thermal convection of liquid flows [2, 3]. After that, many new chaotic and hyperchaotic complex systems have been reported and intensively studied, including the complex Van der Pol oscillators [4], the complex Chen and complex Lü systems [5], complex detuned laser system [6], complex hyperchaotic Lorenz system [7], complex modified hyperchaotic Lü system [8], and a novel hyperchaotic complex-variable system [9] which generates 2-, 3-, and 4-scroll attractors.Since Pecora and Carroll [10] first proposed the drive-response concept for constructing synchronization of coupled chaotic systems, synchronization in chaotic systems has been extensively investigated due to their potential applications in the fields of secure communications; optical, chemical, physical, and biological systems; neural networks; and so forth [11–13]. When applying the complex systems in communications, the complex variables will double the number of variables and can increase the content and security of the transmitted information. Based on the Lyapunov stability theory, linear feedback controller was derived to achieve hybrid projective synchronization in a chaotic complex nonlinear system [14]. The authors [15] achieved adaptive antisynchronization of a class of chaotic complex nonlinear systems described by a united mathematical expression with fully uncertain parameters. In [16], the author investigated the modified projective phase synchronization of chaotic complex nonlinear systems. Based on the passive theory, the authors studied the projective synchronization of hyperchaotic complex nonlinear systems and its application in secure communications [17]. In [18], the authors achieved fast synchronization of a novel hyperchaotic complex system based on finite-time stability theory.However, most of the existing synchronization schemes are based on the usual drive-response synchronization mode, which has one drive system and one response system. In [19], Luo et al. proposed the combination synchronization scheme, which has two drive systems and one response system. Zhou et al. investigated combination synchronization of three nonlinear complex hyperchaotic systems in [20]. Sun et al. [21] extended the combination synchronization scheme to the combination-combination synchronization scheme, where synchronization is achieved between two drive systems and two response systems. This synchronization scheme has advantages over the other synchronization schemes, such that it can provide greater security in secure communication.For the nonlinear complex chaotic or hyperchaotic systems, there are no work on combination-combination synchronization for them. This paper aims to study the combination-combination synchronization of four nonlinear complex chaotic systems. The rest of this paper is organized as follows. Section2 introduces the scheme of combination-combination synchronization. In Section 3, we investigate combination-combination synchronization of four complex nonlinear chaotic systems. Numerical simulations are conducted in Section 4. Finally, conclusions are given in Section 5.


## 2. The Scheme of Combination-Combination Synchronization

In the scheme of combination-combination synchronization, there are four nonlinear dynamical systems, two drive systems, and two response system.The two drive systems are, respectively, given by(1)x˙1=f1(x1),(2)x˙2=f2(x2).
The two response systems are, respectively, described by
(3)y˙1=g1(y1)+φ,(4)y˙2=g2(y2)+φ*,
where x1=(x11,x12,…,x1n)T, x2=(x21,x22,…,x2n)T, y1=(y11,y12,…,y1n)T, and y2=(y21,y22,…,y2n)T are the state vectors of the systems (1), (2), (3), and (4), respectively; f1(·),f2(·),g1(·),g2(·):Rn→Rn are four continuous vector functions and φ,φ*:Rn×Rn×Rn×Rn→Rn are two controller vectors which will be designed.Definition 1 (see [21]).
If there exist four constant matricesA, B, C, and D∈Rn and C≠0 or D≠0 such that
(5)limt→+∞∥Ax1+Bx2-Cy1-Dy2∥=0,
the drive systems (1) and (2) are realized combination-combination synchronization with the response systems (3) and (4), where ∥·∥ represents the matrix norm.Remark 2.
The combination-combination synchronization can be reduced to combination synchronization, projective synchronization, and even control problem, if we choose specific values ofA, B, C, and D.


## 3. Combination-Combination Synchronization of Four Nonlinear Complex Chaotic Systems

In this section, we investigate the combination-combination synchronization of four nonlinear complex chaotic systems.The first drive system [22] is given by
(6)x˙11=α1(x12-x11)+x12x13,x˙12=γ1x11-x12-x11x13,x˙13=-β1x13+12(x¯11x12+x11x¯12),
and the second drive system [23] is described as follows:
(7)x˙21=a1x21+b1x22x23,x˙22=a2x22+b2x21x23,x˙23=a3x23+b32(x¯21x22+x21x¯22).
The first response system [6] takes the following form:
(8)y˙11=σ3y12-σ3(1-iδ3)y11+φ1+iφ2,y˙12=(α3-y13)y11-(1+iδ3)y12+φ3+iφ4,y˙13=-β3y13+12(y¯11y12+y11y¯12)+φ5,
and the second response [9] is given by
(9)y˙21=y22-α4y21+β4y22y23+φ1*+iφ2*,y˙22=γ4y22-y21y23+y23+φ3*+iφ4*,y˙23=δ42(y¯21y22+y21y¯22)-σ4y23+φ5*,
where α1, β1, γ1, a1, a2, a3, b1, b2, b3, α3, β3, δ3, σ3, α4, β4, γ4, σ4, and δ4 are system parameters; x11=u1+iu2, x12=u3+iu4, x21=v1+iv2, x22=v3+iv4, y11=w1+iw2, y12=w3+iw4, y21=μ1+iμ2, and y22=μ3+iμ4 are complex variables; i=-1; and ui, vi, wi, μi (i=1,2,3,4), x13=u5, x23=v5, y13=w5, and y23=μ5 are real variables. The overbar represents complex conjugate function. φi and φi* (i=1,2,3,4,5) are real controllers to be determined. Their chaotic attractors are illustrated in Figures 1, 2, 3, and 4, respectively.Chaotic attractor for system (6). (a)–(c) Projections in 3D space; (d)–(f) projections in 2D plane.
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(f)Chaotic attractor for system (7). (a)–(c) Projections in 3D space; (d)–(f) projections in 2D plane.
(a)
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(f)Chaotic attractor for system (8). (a)–(c) Projections in 3D space; (d)–(f) projections in 2D plane.
(a)
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(f)Chaotic attractor for system (9). (a)–(c) Projections in 3D space; (d)–(f) projections in 2D plane.
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(f)For the convenience of our discussions, we assumeA=diag(k1,k2,k3), B=diag(l1,l2,l3), C=diag(m1,m2,m3), and D=diag(n1,n2,n3) in our synchronization scheme.We define error states between the drive systems (6) and (7) and the response systems (8) and (9) as
(10)e1+ie2=k1x11+l1x21-m1y11-n1y21,e3+ie4=k2x12+l2x22-m2y12-n2y22,e5=k3x13+l3x23-m3y13-n3y23,
such that
(11)limt→∞∥k1x11+l1x21-m1y11-n1y21∥=0,limt→∞∥k2x12+l2x22-m2y12-n2y22∥=0,limt→∞∥k3x13+l3x23-m3y13-n3y23∥=0.Thus, we have the following error dynamical system:(12)e˙1+ie˙2=k1x˙11+l1x˙21-m1y˙11-n1y˙21,e˙3+ie˙4=k2x˙12+l2x˙22-m2y˙12-n2y˙22,e˙5=k3x˙13+l3x˙23-m3y˙13-n3y˙23.Substituting (6)–(9) into (12) and separating the real and imaginary parts yields
(13)e˙1=k1[α1(u3-u1)+u3u5]+l1(a1v1+b1v3v5)-m1[σ3(w3-w1-δ3w2)+φ1]-n1(μ3-α4μ1+β4μ3μ5+φ1*),e˙2=k1[α1(u4-u2)+u4u5]+l1(a1v2+b1v4v5)-m1[σ3(w4-w2+δ3w1)+φ2]-n1(μ4-α4μ2+β4μ4μ5+φ2*),e˙3=k2(γ1u1-u5u1-u3)+l2(a2v3+b2v1v5)-m2[(α3-w5)w1-w3-δ3w4+φ3]-n2(γ4μ3-μ1μ5+μ5+φ3*),e˙4=k2(γ1u2-u5u2-u4)+l2(a2v4+b2v2v5)-m2[(α3-w5)w2-w4-δ3w3+φ4]-n2(γ4μ4-μ2μ5+φ4*),e˙5=k3(-β1u5+u1u3+u2u4)+l3[a3v5+b3(v1v3+v2v4)]-m3(-β3w5+w1w3+w2w4+φ5)-n3[δ4(μ1μ3+μ2μ4)-σ4μ5+φ5*].DenoteUi=miφi+niφi* (i=1,2,3,4,5); then we obtain the following results.Theorem 3.
If the controllers are chosen as follows:(14)U1=k1u1+l1v1-m1w1-n1μ1+a1(k1u2+l1v2-m1w2-n1μ2)+k1[α1(u3-u1)+u3u5]+l1(a1v1+b1v3v5)-m1σ3(w3-w1-δ3w2)-n1(μ3-α4μ1+β4μ3μ5),U2=k1u2+l1v2-m1w2-n1μ2-a1(k1u1+l1v1-m1w1-n1μ1)-a2(k2u3+l2v3-m2w3-n2μ3)+k1[α1(u4-u2)+u4u5]+l1(a1v2+b1v4v5)-m1σ3(w4-w2+δ3w1)-n1(μ4-α4μ2+β4μ4μ5),U3=k2u3+l2v3-m2w3-n2μ3+a2(k1u2+l1v2-m1w2-n1μ2)+a3(k2u4+l2v4-m2w4-n2μ4)+k2(γ1u1-u5u1-u3)+l2(a2v3+b2v1v5)-m2[(α3-w5)w1-w3-δ3w4]-n2(γ4μ3-μ1μ5+μ5),U4=k2u4+l2v4-m2w4-n2μ4-a3(k2u3+l2v3-m2w3-n2μ3)-b1(k3u5+l3v5-m3w5-n3μ5)+k2(γ1u2-u5u2-u4)+l2(a2v4+b2v2v5)-m2[(α3-w5)w2-w4-δ3w3]-n2(γ4μ4-μ2μ5),U5=k3u5+l3v5-m3w5-n3μ5+b1(k2u4+l2v4-m2w4-n2μ4)+k3(-β1u5+u1u3+u2u4)+l3[a3v5+b3(v1v3+v2v4)]-m3(-β3w5+w1w3+w2w4)-n3[δ4(μ1μ3+μ2μ4)-σ4μ5],
then the drive systems (6) and (7) will achieve combination-combination synchronization with the response systems (8) and (9).Proof.
Construct the following Lyapunov function:(15)V=12(e12+e22+e32+e42+e52).
Taking the time derivative of V along the trajectory of the error dynamical system (13) yields
(16)V˙=e1e˙1+e2e˙2+e3e˙3+e4e˙4+e5e˙5=e1{k1[α1(u3-u1)+u3u5]+l1(a1v1+b1v3v5)-m1[σ3(w3-w1-δ3w2)+φ1]-n1(μ3-α4μ1+β4μ3μ5+φ1*)}+e2{k1[α1(u4-u2)+u4u5]+l1(a1v2+b1v4v5)-m1[σ3(w4-w2+δ3w1)+φ2]-n1(μ4-α4μ2+β4μ4μ5+φ2*)}+e3{k2(γ1u1-u5u1-u3)+l2(a2v3+b2v1v5)-m2[(α3-w5)w1-w3-δ3w4+φ3]-n2(γ4μ3-μ1μ5+μ5+φ3*)}+e4{k2(γ1u2-u5u2-u4)+l2(a2v4+b2v2v5)-m2[(α3-w5)w2-w4-δ3w3+φ4]-n2(γ4μ4-μ2μ5+φ4*)}+e5{k3(-β1u5+u1u3+u2u4)+l3[a3v5+b3(v1v3+v2v4)]-m3(-β3w5+w1w3+w2w4+φ5)-n3[δ4(μ1μ3+μ2μ4)-σ4μ5+φ5*]}=e1{k1[α1(u3-u1)+u3u5]+l1(a1v1+b1v3v5)-m1[σ3(w3-w1-δ3w2)]-n1(μ3-α4μ1+β4μ3μ5)-(m1φ1+n1φ1*)}+e2{k1[α1(u4-u2)+u4u5]+l1(a1v2+b1v4v5)-m1[σ3(w4-w2+δ3w1)]-n1(μ4-α4μ2+β4μ4μ5)-(m1φ2+n1φ2*)}+e3{k2(γ1u1-u5u1-u3)+l2(a2v3+b2v1v5)-m2[(α3-w5)w1-w3-δ3w4]-n2(γ4μ3-μ1μ5+μ5)-(m2φ3+n2φ3*)}+e4{k2(γ1u2-u5u2-u4)+l2(a2v4+b2v2v5)-m2[(α3-w5)w2-w4-δ3w3]-n2(γ4μ4-μ2μ5)-(m2φ4+n2φ4*)}+e5{k3(-β1u5+u1u3+u2u4)+l3[a3v5+b3(v1v3+v2v4)]-m3(-β3w5+w1w3+w2w4)-n3[δ4(μ1μ3+μ2μ4)-σ4μ5]-(m3φ5+n3φ5*)}.
Substituting (14) into (16) leads to
(17)V˙=e1{k1[α1(u3-u1)+u3u5]+l1(a1v1+b1v3v5)-m1[σ3(w3-w1-δ3w2)]-n1(μ3-α4μ1+β4μ3μ5)-[k1u1+l1v1-m1w1-n1μ1+a1(k1u2+l1v2-m1w2-n1μ2)+k1[α1(u3-u1)+u3u5]+l1(a1v1+b1v3v5)-m1[σ3(w3-w1-δ3w2)]-n1(μ3-α4μ1+β4μ3μ5)]}+e2{k1[α1(u4-u2)+u4u5]+l1(a1v2+b1v4v5)-m1[σ3(w4-w2+δ3w1)]-n1(μ4-α4μ2+β4μ4μ5)-[k1u2+l1v2-m1w2-n1μ2-a1(k1u1+l1v1-m1w1-n1μ1)-a2(k2u3+l2v3-m2w3-n2μ3)+k1[α1(u4-u2)+u4u5]+l1(a1v2+b1v4v5)-m1[σ3(w4-w2+δ3w1)]-n1(μ4-α4μ2+β4μ4μ5)]}+e3{k2(γ1u1-u5u1-u3)+l2(a2v3+b2v1v5)-m2[(α3-w5)w1-w3-δ3w4]-n2(γ4μ3-μ1μ5+μ5)-[k2u3+l2v3-m2w3-n2μ3+a2(k1u2+l1v2-m1w2-n1μ2)+a3(k2u4+l2v4-m2w4-n2μ4)+k2(γ1u1-u5u1-u3)+l2(a2v3+b2v1v5)-m2[(α3-w5)w1-w3-δ3w4]-n2(γ4μ3-μ1μ5+μ5)]}+e4{k2(γ1u2-u5u2-u4)+l2(a2v4+b2v2v5)-m2[(α3-w5)w2-w4-δ3w3]-n2(γ4μ4-μ2μ5)-[k2u4+l2v4-m2w4-n2μ4-a3(k2u3+l2v3-m2w3-n2μ3)-b1(k3u5+l3v5-m3w5-n3μ5)+k2(γ1u2-u5u2-u4)+l2(a2v4+b2v2v5)-m2[(α3-w5)w2-w4-δ3w3]-n2(γ4μ4-μ2μ5)]}+e5{k3(-β1u5+u1u3+u2u4)+l3[a3v5+b3(v1v3+v2v4)]-m3(-β3w5+w1w3+w2w4)-n3[δ4(μ1μ3+μ2μ4)-σ4μ5]-[k3u5+l3v5-m3w5-n3μ5+b1(k2u4+l2v4-m2w4-n2μ4)+k3(-β1u5+u1u3+u2u4)+l3[a3v5+b3(v1v3+v2v4)]-m3(-β3w5+w1w3+w2w4)-n3(δ4(μ1μ3+μ2μ4)-σ4μ5)]},
and we get
(18)V˙=e1(-e1-a1e2)+e2(-e2+a1e1+a2e3)+e3(-e3-a2e2-a3e4)+e4(-e4+a3e3+b1e5)+e5(-e5-b1e4)=-e12-e22-e32-e42-e52.
Since V˙≤0 as t→∞, according to the Lyapunov stability theory, we know ei→0 (i=1,2,3,4,5); that is, limt→∞∥e∥=0. Therefore, the drive systems (6) and (7) will achieve combination-combination synchronization with the response systems (8) and (9).
This completes the proof.If we choose specific values ofki, li, mi, and ni (i=1,2,3), then we can have the following corollaries.Corollary 4.
(i) Suppose thatn1=n2=n3=0, and if the controllers are chosen as follows:
(19)φ1=1m1{k1u1+l1v1-m1w1+a1(k1u2+l1v2-m1w2)+k1[α1(u3-u1)+u3u5]+l1(a1v1+b1v3v5)-m1σ3(w3-w1-δ3w2)},φ2=1m1{k1u2+l1v2-m1w2-a1(k1u1+l1v1-m1w1)-a2(k2u3+l2v3-m2w3)+k1[α1(u4-u2)+u4u5]+l1(a1v2+b1v4v5)-m1σ3(w4-w2+δ3w1)},φ3=1m2{k2u3+l2v3-m2w3+a2(k1u2+l1v2-m1w2)+a3(k2u4+l2v4-m2w4)+k2(γ1u1-u5u1-u3)+l2(a2v3+b2v1v5)-m2[(α3-w5)w1-w3-δ3w4]},φ4=1m2{k2u4+l2v4-m2w4-a3(k2u3+l2v3-m2w3)-b1(k3u5+l3v5-m3w5)+k2(γ1u2-u5u2-u4)+l2(a2v4+b2v2v5)-m2[(α3-w5)w2-w4-δ3w3]},φ5=1m3{k3u5+l3v5-m3w5+b1(k2u4+l2v4-m2w4)+k3(-β1u5+u1u3+u2u4)+l3[a3v5+b3(v1v3+v2v4)]-m3(-β3w5+w1w3+w2w4)},
then the drive systems (6) and (7) will achieve combination synchronization with the response system (8).
(ii) Suppose that m1=m2=m3=0, and if the controllers are chosen as follows:
(20)φ1*=1n1{k1u1+l1v1-n1μ1+a1(k1u2+l1v2-n1μ2)+k1[α1(u3-u1)+u3u5]+l1(a1v1+b1v3v5)-n1(μ3-α4μ1+β4μ3μ5)},φ2*=1n1{k1u2+l1v2-n1μ2-a1(k1u1+l1v1-n1μ1)-a2(k2u3+l2v3-n2μ3)+k1[α1(u4-u2)+u4u5]+l1(a1v2+b1v4v5)-n1(μ4-α4μ2+β4μ4μ5)},φ3*=1n2{k2u3+l2v3-n2μ3+a2(k1u2+l1v2-n1μ2)+a3(k2u4+l2v4-n2μ4)+k2(γ1u1-u5u1-u3)+l2(a2v3+b2v1v5)-n2(γ4μ3-μ1μ5+μ5)},φ4*=1n2{k2u4+l2v4-n2μ4-a3(k2u3+l2v3-n2μ3)-b1(k3u5+l3v5-n3μ5)+k2(γ1u2-u5u2-u4)+l2(a2v4+b2v2v5)-n2(γ4μ4-μ2μ5)},φ5*=1n3{k3u5+l3v5-n3μ5+b1(k2u4+l2v4-n2μ4)+k3(-β1u5+u1u3+u2u4)+l3[a3v5+b3(v1v3+v2v4)]-n3[δ4(μ1μ3+μ2μ4)-σ4μ5]},
then the drive systems (6) and (7) will achieve combination synchronization with the response system (9).Corollary 5.
(i) Suppose thatk1=k2=k3=0, n1=n2=n3=0, and m1=m2=m3=1, and if the controllers are chosen as follows:
(21)φ1=l1v1-w1+a1(l1v2-w2)+l1(a1v1+b1v3v5)-σ3(w3-w1-δ3w2),φ2=l1v2-w2-a1(l1v1-w1)-a2(l2v3-w3)+l1(a1v2+b1v4v5)-σ3(w4-w2+δ3w1),φ3=l2v3-w3+a2(l1v2-w2)+a3(l2v4-w4)+l2(a2v3+b2v1v5)-[(α3-w5)w1-w3-δ3w4],φ4=l2v4-w4-a3(l2v3-w3)-b1(l3v5-w5)+l2(a2v4+b2v2v5)-[(α3-w5)w2-w4-δ3w3],φ5=l3v5-w5+b1(l2v4-w4)+l3[a3v5+b3(v1v3+v2v4)]-(-β3w5+w1w3+w2w4),
then the drive system (7) will achieve projective synchronization with the response system (8).
(ii) Suppose that l1=l2=l3=0, n1=n2=n3=0, and m1=m2=m3=1, and if the controllers are chosen as follows:
(22)φ1=k1u1-w1+α3(k1u2-w2)+k1[α1(u3-u1)+u3u5]-σ3(w3-w1-δ3w2),φ2=k1u2-w2-α3(k1u1-w1)-β3(k2u3-w3)+k1[α1(u4-u2)+u4u5]-σ3(w4-w2+δ3w1),φ3=k2u3-w3+β3(k1u2-w2)+δ3(k2u4-w4)+k2(γ1u1-u5u1-u3)-[(α3-w5)w1-w3-δ3w4],φ4=k2u4-w4-δ3(k2u3-w3)-σ3(k3u5-w5)+k2(γ1u2-u5u2-u4)-[(α3-w5)w2-w4-δ3w3],φ5=k3u5-w5+σ3(k2u4-w4)+k3(-β1u5+u1u3+u2u4)-(-β3w5+w1w3+w2w4),
then the drive system (6) will achieve projective synchronization with the response system (8).
(iii) Suppose that k1=k2=k3=0, m1=m2=m3=0, and n1=n2=n3=1, and if the controllers are chosen as follows:
(23)φ1*=l1v1-μ1+a1(l1v2-μ2)+l1(a1v1+b1v3v5)-(μ3-α4μ1+β4μ3μ5),φ2*=l1v2-μ2-a1(l1v1-μ1)-a2(l2v3-μ3)+l1(a1v2+b1v4v5)-(μ4-α4μ2+β4μ4μ5),φ3*=l2v3-μ3+a2(l1v2-μ2)+a3(l2v4-μ4)+l2(a2v3+b2v1v5)-(γ4μ3-μ1μ5+μ5),φ4*=l2v4-μ4-a3(l2v3-μ3)-b1(l3v5-μ5)+l2(a2v4+b2v2v5)-(γ4μ4-μ2μ5),φ5*=l3v5-μ5+b1(l2v4-μ4)+l3[a3v5+b3(v1v3+v2v4)]-(δ4(μ1μ3+μ2μ4)-σ4μ5),
then the drive system (7) will achieve projective synchronization with the response system (9).
(iv) Suppose that l1=l2=l3=0, m1=m2=m3=0, and n1=n2=n3=1, and if the controllers are chosen as follows:
(24)φ1*=k1u1-μ1+α3(k1u2-μ2)+k1[α1(u3-u1)+u3u5]-(μ3-α4μ1+β4μ3μ5),φ2*=k1u2-μ2-α3(k1u1-μ1)-β3(k2u3-μ3)+k1[α1(u4-u2)+u4u5]-(μ4-α4μ2+β4μ4μ5),φ3*=k2u3-μ3+β3(k1u2-μ2)+δ3(k2u4-μ4)+k2(γ1u1-u5u1-u3)-(γ4μ3-μ1μ5+μ5),φ4*=k2u4-μ4-δ3(k2u3-μ3)-σ3(k3u5-μ5)+k2(γ1u2-u5u2-u4)-(γ4μ4-μ2μ5),φ5*=k3u5-μ5+σ3(k2u4-μ4)+k3(-β1u5+u1u3+u2u4)-(δ4(μ1μ3+μ2μ4)-σ4μ5),
then the drive system (6) will achieve projective synchronization with the response system (9).Corollary 6.
(i) Suppose thatk1=k2=k3=0, l1=l2=l3=0, n1=n2=n3=0, and m1=m2=m3=1, and if the controllers are chosen as follows:
(25)φ1=-w1-α3w2-σ3(w3-w1-δ3w2),φ2=-w2+α3w1+β3w3-σ3(w4-w2+δ3w1),φ3=-w3-β3w2-δ3w4-[(α3-w5)w1-w3-δ3w4],φ4=-w4+δ3w3+σ3w5-[(α3-w5)w2-w4-δ3w3],φ5=-w5-σ3w4-(-β3w5+w1w3+w2w4),
then system (8) is stabilized to the equilibrium O(0,0,0,0,0).
(ii) Suppose that k1=k2=k3=0, l1=l2=l3=0, m1=m2=m3=0, and n1=n2=n3=1, and if the controllers are chosen as follows:
(26)φ1*=-μ1-α4μ2-(μ3-α4μ1+β4μ3μ5),φ2*=-μ2+α4μ1+β4μ3-(μ4-α4μ2+β4μ4μ5),φ3*=-μ3-β4μ2-γ4μ4-(γ4μ3-μ1μ5+μ5),φ4*=-μ4+γ4μ3+σ4μ5-(γ4μ4-μ2μ5),φ5*=-μ5-σ4μ4-[δ4(μ1μ3+μ2μ4)-σ4μ5],
then system (9) is stabilized to the equilibrium O(0,0,0,0,0).Remark 7.
The above corollaries can be easily obtained from Theorem3, and their proofs are similar to that of Theorem 3, so we omit the proofs here.


## 4. Numerical Simulations

In this section, three numerical examples are presented to illustrate the theoretical analysis. In the following numerical simulations, the fourth-order Runge-kutta method is employed with time step size 0.001. The system parameters are selected asα1=30, γ1=90, β1=11, a1=9.5, a2=-19, a3=-3, b1=-1, b2=1, b3=1, α3=50.0625, β3=0.75, σ3=5, δ3=0.25, α4=3.5, β4=0.599, γ4=3, δ4=2, and σ4=9, so that the four nonlinear complex chaotic systems exhibit chaotic behaviors, respectively.Firstly, consider the combination-combination synchronization of the two drive systems (6) and (7) and the response systems (8) and (9) with the controllers (14). We assume k1=k2=k3=1, l1=l2=l3=1, m1=m2=m3=1, and n1=n2=n3=1, and the initial states for the drive systems and response systems are arbitrarily given by (x11(0),x12(0),x13(0))=(2+4i,1+3i,2), (x21(0),x22(0),x23(0))=(-2-i,5-3i,4), (y11(0),y12(0),y13(0))=(2+i,5+3i,4), and (y21(0),y22(0),y23(0))=(5+2i,-1+i,-4); that is, (u1(0),u2(0),u3(0),u4(0),u5(0))=(2,4,1,3,2), (v1(0),v2(0),v3(0),v4(0),v5(0))=(-2,-1,5,-3,4), (w1(0),w2(0),w3(0),w4(0),w5(0))=(2,1,5,3,4), and (μ1(0),μ2(0),μ3(0),μ4(0),μ5(0))=(5,2,-1,1,-4), respectively. The corresponding numerical results are shown in Figure 5. Figure 5 displays time response of the combination-combination synchronization errors e1, e2, e3, e4, and e5, where ei=ui+vi-wi-μi (i=1,2,3,4,5). The errors converge to zero which implies that the drive systems (6) and (7) and the response systems (8) and (9) have achieved combination-combination synchronization.Combination-combination synchronization errorse1, e2, e3, e4, and e5 between the drive systems (6) and (7) and the response systems (8) and (9), where ei=ui+vi-wi-μi (i=1,2,3,4,5).
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(e)Secondly, consider the combination synchronization of the two drive systems (6) and (7) and the response system (8) with the controllers (19). We assume k1=k2=k3=1, l1=l2=l3=1, m1=m2=m3=1, and n1=n2=n3=0. The corresponding numerical results are shown in Figure 6. Figure 6 displays time response of the combination synchronization errors e1, e2, e3, e4, and e5, where ei=ui+vi-wi (i=1,2,3,4,5). The errors converge to zero which implies that the drive systems (6) and (7) and the response system (8) have achieved combination synchronization.Combination synchronization errorse1, e2, e3, e4, and e5 between the drive systems (6) and (7) and the response system (8), where ei=ui+vi-wi (i=1,2,3,4,5).
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(e)Finally, consider another special case, that is, whenk1=k2=k3=0, l1=l2=l3=0, m1=m2=m3=1, and n1=n2=n3=0, system (8) will be stabilized to its equilibrium O(0,0,0,0,0). Figure 7 shows the time evolution of the states w1, w2, w3, w4, and w5 of system (8) with controller (25), which illustrates that system (8) is stabilized to the equilibrium O(0,0,0,0,0).Time evolution of the states of system (8).
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## 5. Conclusions

In this paper, we investigate the combination-combination synchronization of four nonlinear complex chaotic systems. Based on the Lyapunov stability theory, corresponding controllers to achieve combination-combination synchronization among four different nonlinear complex chaotic systems are derived. The special cases, such as combination synchronization and projective synchronization, are studied as well. This synchronization scheme has advantages over the usual drive-response synchronization, such as being able to provide greater security in secure communication. In [24], the authors applied combination synchronization in secure communication; the signal was divided into two parts, and each part was transmitted by a different chaotic system (the drive system), which implied that the signal transmitted by this model may have stronger antiattack ability and antitranslated capability than that transmitted by the usual transmission model. When applying the nonlinear complex systems in communications, the complex variables will double the number of variables and can increase the content and security of the transmitted information. Thus combination-combination synchronization of complex nonlinear systems can find better applications in security communication, such as wireless communication [25].



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*Source: 953265-2014-02-03.xml*