Aug 12, 2009 - We consider generalized synchronization of complex networks, which are unidirectionally coupled in the drive-response configuration.
PHYSICAL REVIEW E 80, 027201 共2009兲
Generalized synchronization of complex networks 1
Yun Shang,1 Maoyin Chen,2 and Jürgen Kurths3
Institute of Mathematics, AMSS, Academia Sinica, Beijing 100080, China Department of Automation, TNlist, Tsinghua University, Beijing 100084, China 3 Institute of Physics, Humboldt University, 10099 Berlin, Germany 共Received 16 April 2009; published 12 August 2009兲
2
We consider generalized synchronization of complex networks, which are unidirectionally coupled in the drive-response configuration. The drive network consists of linearly and diffusively coupled identical chaotic systems. By choosing suitable driving signals, we can construct the response network to generally synchronize the drive network in a predefined functional relationship. This extends both generalized synchronization of chaotic systems and synchronization inside a network. Theoretical analysis and numerical simulations fully verify our main results. DOI: 10.1103/PhysRevE.80.027201
PACS number共s兲: 05.45.Xt
In the past decades, an increasing interest has been focused on complex networks 关1,2兴. Recently, one of the most interesting topics is to study network synchronization 关3–7兴. Network synchronization can be considered in two ways. 共i兲 Synchronization arises inside a network composed of coupled dynamical systems. Complete synchronization and phase synchronization are two typical types of synchronization phenomena. 共ii兲 Synchronization occurs between two coupled complex networks in the drive-response configuration 关8,9兴. In this case, one network is the drive network, and the other network is the response network. The driving signal should be chosen suitably such that the drive and response networks are synchronized. Generally speaking, synchronization of chaotic systems in the drive-response configuration is one special kind of synchronization of networks 关10–13兴. As far as we know, except for complete synchronization, there exists another wellknown phenomenon, namely, generalized synchronization 关12,13兴. It implies that there exists a functional relationship between the drive and response systems. Recently, researchers began to study synchronization of networks. Li et al. considered synchronization of two unidirectionally coupled networks by the control strategy 关8兴. Complete synchronization can be ensured if controllers are applied to the response network. Even the drive network is uncertain, Yu et al. utilized the adaptive filtering strategy to construct the response network 共to estimate the unknown drive network兲, and two networks can be also completely synchronized 关9兴. However, the above methods can only ensure complete synchronization, and cannot realize generalized synchronization. Compared with complete synchronization, generalized synchronization leads to richer behavior. Though the auxiliary system approach developed for chaos synchronization can be directly applied to analyze whether there exists the phenomenon of generalized synchronization of networks 关13兴, the auxiliary system approach is only one sufficient condition. It cannot provide us the detailed functional relationship between the drive and response networks. In this Brief Report, we also consider generalized synchronization of networks in the drive-response configuration. We first use a simple chaotic system, namely, Genesio-Tesi system, to construct the drive network. By choosing special 1539-3755/2009/80共2兲/027201共4兲
driving signals, we then construct the response network to generally synchronize the drive network in a predefined functional relationship. At last, we extend our main results using the the Lie derivative operator. Suppose that the drive network consists of N linearly and diffusively coupled identical nodes, with each node being an n-dimensional chaotic system, in the following form N
x˙i = f共xi兲 +
兺
GijL共y j − y i兲
共1兲
j=1,j⫽i
for 1 ⱕ i ⱕ N, where xi = 共xi1 , . . . , xin兲T 苸 Rn is the state, y i 苸 R is the scalar output variable, L 苸 Rn is the inner coupling matrix, f : Rn → Rn is a smooth nonlinear vector-valued function, and is the global coupling strength. Matrix G = 共Gij兲 苸 RN⫻N represents the network topology. Gij is defined as follows: If there is a connection between nodes i, j, then Gij = G ji = 1; otherwise Gij = G ji = 0, and the diagonal elements are Gii = −兺Nj=1,j⫽iGij. Here a simple Genesio-Tesi system represents the node dynamics: x˙i = f共xi兲, given by x˙i1 = xi2,x˙i2 = xi3,x˙i3 = − cxi1 − bxi2 − axi3 + 共xi1兲2
共2兲
for 1 ⱕ i ⱕ N, where a , b , c are positive parameters. When a = 0.44, b = 1.1, c = 1, Genesio-Tesi system behaves chaotically 关14兴. Genesio-Tesi system is first proposed in Ref. 关14兴. It is used to illustrate the novel harmonic balance method, which is effective to analyze chaotic dynamics in nonlinear systems. In this Brief Report, we first use Genesio-Tesi system to explain our approach. Then we extend our approach to other chaotic systems. In order to realize generalized synchronization of networks, we assume that 共i兲 the driving signal is chosen to be y i = xi2 + wxi1 with a positive parameter w; 共ii兲 L = 关001兴T; 共iii兲 there exists no isolate cluster in the drive network 关Eq. 共1兲兴, and G is symmetrical and irreducible; 共iv兲 the drive and response networks have the same topology and the same labels of nodes; and 共v兲 all nodes in the drive network 关Eq. 共1兲兴 are chaotic, which ensures that the driving signal y i is bounded. In the following, by the driving signal y i = xi2 + wxi1, we can easily construct the response network, and analyze the condition for generalized synchronization of the drive and
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©2009 The American Physical Society
PHYSICAL REVIEW E 80, 027201 共2009兲
BRIEF REPORTS
response networks. We first transform Genesio-Tesi system 关Eq. 共2兲兴 using the driving signal y i. From the viewpoint of control, y i can be regarded as an input of equation x˙i1 + wxi1 = y i 关12兴. Thus xi1 = e−wtxi1共0兲 + e−wt
冕
t
Differentiating xi1 for 1, 2 and 3 times, together with Genesio-Tesi system 关Eq. 共2兲兴, we get y¨ i + 1y˙ i + 0y i + e−wt
冉冕
冊
where the driving signal y i can be directly injected into the response network 关Eq. 共6兲兴. Define the error i = y i −¯xi1 = xi2 + wxi1 −¯xi1. Hence we get N
¨ i + 1i + 0i −
ewy i共兲d + xi1共0兲 关⌳ − xi1兴 = 0,
0
where ⌳ = 共−w兲3 + a共−w兲2 + b共−w兲 + c, 1 = 共−w兲 + a, and 0 = 共−w兲2 + a共−w兲 + b. Let i = e−wt兰t0ewy i共兲d. So the above equation becomes 关12兴 y¨ i + 1y˙ i + 0y i + i⌳ − 2i = Oi共e−wt兲, 2 共0兲 + 2e−2wtxi1共0兲兰t0ewy i共兲d where Oi共e−wt兲 = e−2wtxi1 −wt − e xi1共0兲⌳. Owning to the boundedness of the driving signal y i, we have limt→⬁关Oi共e−wt兲兴 = 0. Hence we approximately get
y¨ i + 1y˙ i + 0y i + i⌳ − 2i = 0 From ˙ i = −wi + y i, Genesio-Tesi system 关Eq. 共2兲兴 is trans¯ i兲 where ¯xi = 共y i , y˙ i , i兲T formed into the dynamics: ¯x˙ i = g共x 关12兴 ¯x˙ i1 = ¯xi2,x ¯˙ i2 = − 0¯xi1 − 1¯xi2 − ⌳x ¯ i3 + 共x ¯ i3兲2 ,
The parameter w is chosen by 共i兲 w 苸 共0 , a兲, if a2 − b ⬍ 0; 共ii兲 w 苸 共0 , a / 2兲 艛 共a / 2 , a兲, if a2 − 4b = 0; 共iii兲 w 苸 共0 , 共a − 冑a2 − 4b兲 / 2兲 艛 共a + 冑a2 − 4b / 2 , a兲, if a2 − 4b ⬎ 0. Hence we can choose w such that 1 and 0 are positive. In this Brief Report, the response network is constructed as follows:
N
¨ i + 1˙ i + 0i −
兺
¯ j1 − ¯xi1兲, Gij¯L共x
共4兲
j=1,j⫽i
where ¯L = 关010兴T. Now we show that the drive network 关Eq. 共1兲兴 and the response network 关Eq. 共4兲兴 are generally synchronized in the sense that limt→⬁共xi2 + wxi1 −¯xi1兲 = 0 for 1 ⱕ i ⱕ N. The detailed dynamics the drive and response network are described by x˙i1 = xi2 , x˙i2 = xi3 ,
兺
j=1,j⫽i
and ¯x˙ i1 = ¯xi2 ,
Gij共 j − i兲 = 0
for 1 ⱕ i ⱕ N. Obviously, Eq. 共7兲 can be transformed into
冋册冋
0 1 e˙i1 = − 0 − 1 e˙i2
册冋 册
N
冋 册
ei1 e j1 + 兺 Gij⌫ ei2 e j2 j=1
⌫=
冋 册 0 0 1 0
共8兲
.
Note that there exists no isolate cluster in the drive and response networks, and G is symmetrical and irreducible. From Refs. 关3,4,6兴, the stability of Eq. 共8兲 can be transformed into the following N − 1 subsystems:
冋 册冋
0 1 ˙ i1 = − 0 + i − 1 ˙ i2
册冋 册
i1 , i2
共9兲
where i are eigenvalues of G, and 0 = 1 ⬎ 2 ⱖ ¯ ⱖ N. Hence the drive network 关Eq. 共5兲兴 and the response network 关Eq. 共6兲兴 are generally synchronized 共namely, the limit limt→⬁ i共t兲 = 0 for 1 ⱕ i ⱕ N兲 if and i make the polynomial 共s兲 = 2 + 1 + 共0 − i兲
共10兲
be Hurwitz stable. The Hurwitz stability means that 共s兲 = 0 ¯ 兲 ⬍ 0 and Re共 ¯ 兲 ⬍ 0. has two roots ¯1 and ¯2 with Re共 1 2 Since eigenvalues i ⱕ 0, ⱖ 0, 0 ⬎ 0, and 1 ⬎ 0, the drive network 关Eq. 共5兲兴 and the response network 关Eq. 共6兲兴 can be generally synchronized if and nonzero i satisfy
0 − i ⬎ 0. Gij共y j − y i兲 共5兲
共7兲
for 1 ⱕ i ⱕ N, where ei1 = i, ei2 = ˙ i, and
N
x˙i3 = − cxi1 − bxi2 − axi3 + 共xi1兲2 +
兺
j=1,j⫽i
N
¯x˙ i = g共x ¯ i兲 +
¯ 共e−wt兲 Gij共 j − i兲 = O 兺 i j=1,j⫽i
¯ 共e−wt兲 = e−2wtx2 共0兲 + 2e−2wtx 共0兲兰t ewy 共兲 where O i i1 i 0 i1 2 −wt 共0兲 − 2e−2wt¯xi3共0兲兰t0ewy i共兲d ⫻d − e xi1共0兲⌳ − e−2wt¯xi3 ¯ 共e−wt兲 approaches zero as time t + e−wt¯xi3共0兲⌳. Note that O i tends to infinity. Hence the stability of the above equation is equivalent to the stability of the following equation:
共3兲
¯x˙ i3 = − wx ¯ i3 + y i .
共6兲
¯x˙ i3 = − wx ¯ i3 + y i ,
ewy i共兲d
0
t
N
兺 Gij共x¯ j1 − ¯xi1兲, j=1,j⫽i
¯x˙ i2 = − 0¯xi1 − 1¯xi2 − ⌳x ¯ i3 + 共x ¯ i3兲2 +
共11兲
Thought condition 共11兲 holds for arbitrary value , the value should not be large. This is because states of the drive network 关Eq. 共5兲兴 should be bounded, which further ensures the driving signal is bounded. One case is to require all nodes in the drive network 关Eq. 共5兲兴 are chaotic. Accordingly,
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1
0.8 E
the coupling strength should be sufficiently small. From Eq. 共2兲, Genesio-Tesi system is somewhat special. This may restrict the application of the proposed approach. Fortunately, under certain conditions, many chaotic systems can be transformed into equation like Eq. 共2兲. Consider chaotic systems given by
0.6
x˙共t兲 = F关x共t兲兴,s共t兲 = h关x共t兲兴,
0.4
where F : Rn → Rn is a smooth nonlinear vector-valued function, x 苸 Rn is the state and s 苸 R is the scalar output signal. From the scalar signal s and its derivatives of successively higher order, we get the following state: Z = 关s共t兲,s˙共t兲, . . . ,s共m−1兲共t兲兴T = 关h共x兲,LF共h共x兲兲, . . . ,LFm−1h共x兲兴T = H共x兲, where L denotes the Lie derivative operator 关15,16兴, that is, j−1 n 共LF h兲 LFjh共x兲 = 兺i=1 xi Fi共x兲. Further, as long as m is sufficiently large 共for example, m ⬎ 2n兲, H is an embedding and H / x is of full rank 关16兴. It is easy to show that the state Z satisfies 关17兴 z˙1 = z2,z˙2 = z3, ¯ ,z˙m−1 = zm,z˙m = 共z兲,
共12兲
where 共z兲 = LFmh共x兲 = LFm共H−1共Z兲兲. Similar to the above analysis, we can choose some parameters a1 , . . . , am ⬎ 0 such that z˙m = −a1z1 − ¯ −amzm + ¯共z兲 and ¯共z兲 = 共z兲 + a1z1 + ¯ +amzm. If we choose y = z2 + wz1, we can transform Eq. 共12兲. Hence, based on Eq. 共12兲 and its transformation, we can construct the drive network 共1兲 and the response network 共4兲, where L = 关0 , 0 , . . . , 0 , 1兴T and ¯L = 关0 , . . . , 0 , 1 , 0兴T. If ¯共z兲 satisfies the Liptchitz condition and parameter w is chosen suitably, we also ensure the stability of N − 1 subsystems by the Lyapunov stability. Our analysis and simulation are based on Barabasi-Albert 共BA兲 networks 关2兴. The drive and response networks have the same topology generated by the standard algorithm. Initially M nodes with labels i = 1 , . . . , M are fully connected. At every time step a new node is introduced to be connected to M existing nodes. The probability that the new node is connected to node i depends on degree ki, i.e., ⌸i = ki / 兺 jk j. In order to measure generalized synchronization, we define the N average error as E共t兲 = N1 兺i=1 兩y i共t兲 −¯xi1共t兲兩.
0.2 0 0
40
60
80
100 t
120
140
160
180
200
FIG. 2. The average error E1共t兲 versus the time t. Here E1共t兲 1 N 1 N = N 兺i=1 兩xi共t兲 − xˆi共t兲兩 and xˆi共t兲 = N 兺i=1 xi共t兲. All estimates are the results of averaging over 100 realizations.
Throughout our simulations, the number of nodes in networks are N = 500, and M = 3. In addition, the parameter w = 0.2. Hence parameters 0 = 1.0520, 1 = 0.24, and ⌳ = 0.7896. For BA networks, we can compute eigenvalues of matrix G, and the coupling is determined by Eq. 共11兲. After many realizations, we assign the coupling by = 0.005 关satisfying Eq. 共11兲兴. In the drive network 关Eq. 共5兲兴 and the response network 关Eq. 共6兲兴, initial states, xi1共0兲 and ¯xi1共0兲, xi2共0兲 and ¯xi2共0兲, and xi3共0兲 and ¯xi3共0兲, are uniformly distributed in 关−0.2, −0.1兴, 关−0.5, −0.4兴, and 关0.8, 0.9兴, respectively. When there exists no channel disturbance, the average error E共t兲 versus the time t is plotted in Fig. 1. It shows that the drive network and the response network are generally synchronized in the sense that limt→⬁共xi2 + wxi1 −¯xi1兲 = 0 for 1 ⱕ i ⱕ N. However, synchronization insider the drive network cannot be ensured 共please refer to Fig. 2兲. In order to show the effectiveness of our approach, we further consider the robustness to the disturbance in channel 共such as channel noise兲, which often happens during the transition of the driving signal y i. Suppose that the disturbance d共t兲 satisfies 兩d共t兲兩 ⬍ ␦, where ␦ is a positive constant. Hence the transmitted signal becomes y ⬘共t兲 = y共t兲 + d共t兲, and it is directly injected into the response network. Here parameters N, M, w, 0, 1, and ⌫ are chosen as above. The disturbance is chosen to be the white noise with ␦ = 0.5. The average error E共t兲 versus the time t is plotted in Fig. 3. It shows that the drive network and the response network are almost generally 2.5
2
2
1.5
1.5 E
2.5
E
20
1
1
0.5
0.5
0 0
20
40
60
80
100 t
120
140
160
180
0 0
200
FIG. 1. The average error E共t兲 versus the time t when there exists no channel disturbance. All estimates are the results of averaging over 100 realizations.
20
40
60
80
100 t
120
140
160
180
200
FIG. 3. The average error E共t兲 versus the time t when there exists channel disturbance. All estimates are the results of averaging over 100 realizations.
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synchronized. Therefore, our approach is robust to disturbance in channel. In this Brief Report, we consider generalized synchronization of complex networks, which are unidirectionally coupled in the drive-response configuration. For the drive network, we choose suitable driving signals and construct the response network, such that the drive and response networks are generally synchronized in a predefined functional rela-
tionship. This can be regarded as the extension of both generalized synchronization of chaotic systems and chaos synchronization inside a network.
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M.C. was supported by NSFC project under Grant No. 60804046 and Special Doctoral Fund in University by Ministry of Education under Grant No. 20070003129. J.K. was supported by SFB 555 共DFG兲 and BRACCIA 共EU兲.
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