Adaptive Exponential Synchronization of Coupled Complex Networks on General Graphs

Abstract

We investigate the synchronization in complex dynamical networks, where the coupling configuration corresponds to a weighted graph. An adaptive synchronization method on general coupling configuration graphs is given. The networks may synchronize at an arbitrarily given exponential rate by enhancing the updated law of the variable coupling strength and achieve synchronization more quickly by adding edges to original graphs. Finally, numerical simulations are provided to illustrate the effectiveness of our theoretical results.

1. Introduction

In general, complex networks consist of a large number of nodes and links among them, in which a node is a fundamental cell with specific activity, hence, complex networks and graphs closely contact each other. The dynamics on complex networks is one on graphs, though the graphs may have different characteristic, for example, classical random graph model [1], small-world model [2–4], scale-free model [5], or others that are closely related to natural structure.

Synchronization of coupled complex dynamical networks is a universal phenomenon in various fields of science and society. All kinds of synchronization, including adaptive synchronization, global synchronization, antisynchronization, phase synchronization, projection synchronization, and generalized synchronization have been studied [6–14], in particular on the synchronization of an array of linearly coupled identical systems [15–18]. Researches [15, 19] imply that the structural properties of a network must have inevitable effect on the ability and speed of synchronization, but such work still does not see more. In addition, as pointed in [18], the coupling strength may be self-adaptive due to the spontaneousness of updated law, not be calculated numerically in many other works.

In the paper [18], an adaptive synchronization method is introduced, and the networks can synchronize by enhancing the coupling strength automatically under a simple updated law. However, their work is limited to tree-like networks and cannot be applied to general networks. In fact, a tree is a graph without cycles; in this paper, we try to extend the work in [18] to general networks or graphs. We find that their method is in fact effective to general graphs by a rigorous proof, and the networks can also achieve synchronization at an arbitrarily given exponential rate by increasing coupling strength. Based on the knowledge of spectral graph theory, networks can synchronize more quickly by increasing the algebraic connectivity of the graphs, which can be realized by adding edges to original graphs. Finally, numerical simulations are provided to illustrate the effectiveness of theoretical results.

2. Preliminaries

In this section, we now introduce some notations and preliminaries.

Consider the complex dynamical network consisting of 𝑁 linearly and diffusively coupled identical nodes with full diagonal coupling, and each node is an 𝑛-dimensional dynamical oscillator which may be chaotic. The state equations of the network are 𝑥̇_𝑖=𝑓𝑥_𝑖+𝑐^𝑁_𝑗=1𝑎_𝑖𝑗Γ𝑥_𝑗,𝑖=1,2,…,𝑁, (1) where 𝑥_𝑖=(𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑖𝑛)^𝑇∈ℝ^𝑛 is a state vector of node 𝑖, 𝑓(𝑥_𝑖)=(𝑓₁(𝑥_𝑖),𝑓₂(𝑥_𝑖),…,𝑓_𝑛(𝑥_𝑖))^𝑇∶ℝ^𝑛→ℝ^𝑛 is a given nonlinear vector valued function describing the dynamics of the nodes, 𝑐>0 represents coupling strength, and the inner coupling link matrix is a diagonal matrix Γ=diag{𝑟₁,𝑟₂,…,𝑟_𝑛} with 𝑟_𝑖>0. The coupling configuration matrix 𝐴=(𝑎_𝑖𝑗)_𝑁×𝑁 is a zero row sums matrix with nonnegative off-diagonal entries, representing the topological structure of the network.

There is a weighted graph corresponding to the coupling configuration matrix 𝐴, called the coupling configuration graph, defined as a graph 𝐺 on vertices 1,2,…,𝑁 which contains an edge 𝑖𝑗(𝑖≠𝑗) with weight 𝑎_𝑖𝑗 if and only if 𝑎_𝑖𝑗>0. Giving an arbitrary orientation of the edges of 𝐺, so that each edge has a head and a tail, and a labeling of edges as 𝑒₁,𝑒₂,…,𝑒_𝐸, where 𝐸 denotes the number of edges of 𝐺. We obtain an edge-vertex incidence matrix of 𝐺, denoted as 𝑀∶=𝑀(𝐴)=(𝑚_𝑖𝑗)_𝐸×𝑁, which is defined as 𝑚_𝑖𝑗=𝑎_𝑖𝑗  (resp., 𝑚_𝑖𝑗=−𝑎_𝑖𝑗) if the the edge 𝑒_𝑖 has the vertex 𝑗 as a head (resp., a tail), and 𝑚_𝑖𝑗=0 otherwise. The Laplacian matrix of 𝐺 is defined as 𝐿=𝑀^𝑇𝑀=𝐷−𝐵, where 𝐷 is a diagonal matrix, and its 𝑖th diagonal entry of which is exactly the degree of the vertex 𝑖, that is, 𝑑_𝑖=∑_{𝑗∈𝑁(𝑖)}𝑎_𝑖𝑗, where 𝑁(𝑖) is the set of neighbors of 𝑖 in the graph 𝐺 (or the vertices joining 𝑖 by edges), and 𝐵=(𝑏_𝑖𝑗) is a weighted adjacency matrix of 𝐺 such that 𝑏_𝑖𝑗=𝑎_𝑖𝑗 if 𝑖𝑗 is an edge of 𝐺 and 𝑏_𝑖𝑗=0 otherwise. One can find that 𝐿=−𝐴 and is symmetric and positive semidefinite, so its eigenvalues can be arranged as 0=𝜆₀≤𝜆₁≤𝜆₂≤⋯≤𝜆_𝑁−1, (2) where 𝜆₀=0 as 𝐿 has zero row sums, and 𝜆₁>0 if and only if 𝐺 is connected and is called the algebraic connectivity of 𝐺 by Fiedler [20] in the case of 𝐺 is simple (i.e., all edges have weight 1). If 𝐺 is connected, the corresponding eigenvector of the eigenvalue 0 is an all one vector (up to a scalar multiple), denoted by 1. One can refer to Chung [21] and Merris [22] for the details of Laplacian matrices of graphs.

If we denote 𝑥=(𝑥^𝑇₁,𝑥^𝑇₂,…,𝑥^𝑇_𝑁)^𝑇,𝐹(𝑥)=(𝑓^𝑇(𝑥₁),𝑓^𝑇(𝑥₂),…,𝑓^𝑇(𝑥_𝑁))^𝑇 and substitute −𝐿 for 𝐴, (1) is transformed as 𝑥̇=𝐹(𝑥)−𝑐𝐿⊗Γ𝑥. (3) Taking the transformation 𝑦=𝑥𝑒^𝑎𝑡(𝑎>0), (3) is written as 𝑦̇=𝑒^𝑎𝑡𝐹𝑒^−𝑎𝑡𝑦−𝑐𝐿⊗Γ−𝑎𝐼_𝑁𝑛𝑦, (4) where 𝐼_𝑁𝑛=𝐼_𝑁⊗𝐼_𝑛, and 𝐼_𝑚 denotes an identity matrix of order 𝑚.

In this paper, we adopt the 𝑙₂-norm for vectors and the induced spectral norm for matrices. We always suppose that the function 𝐹 in (3) is Lipschitz continuous, or equivalently 𝑓 in (1) is Lipschitz continuous, that is, there exists a constant 𝑙>0 such that for any 𝑥,𝑦∈𝑅^𝑛, 𝑓(𝑥)−𝑓𝑦≤𝑙⋅𝑥−𝑦. (5)

Denote by 𝐌=𝑀⊗𝐼_𝑛, one can obtain the following lemma.

Lemma 1 (see [15]). Suppose that the coupling configuration graph corresponding to 𝐴=−𝐿 is connected. Then, the dynamical networks (4) achieve synchronization if and only if lim_𝑡→∞‖𝐌𝑦‖=0.

3. Main Results

If the network (4) achieves synchronization, surely the network (3) or (1) achieves synchronization with exponential rate 𝑎. So, we directly discuss (4), the main result of this paper is stated as follows.

Theorem 2. Suppose that 𝐹 is Lipschitz continuous, and the coupling configuration graph corresponding to 𝐴=−𝐿 is connected. Then, the network (4) achieves synchronization, and in particular the network (1) achieves exponential synchronization with rate 𝑎, when the coupling strength 𝑐 is adapted duly according to the following updated law: 𝑐̇=𝑘𝑦^𝑇(𝑡)𝐿²⊗Γ𝑦(𝑡), (6) where 𝑘>0 is an arbitrary constant.

Proof. Construct a Lyapunov function 𝑉(𝑡)=12𝐌𝑦(𝑡)²+1()2𝑘𝑐−ℎ², (7) where ℎ is a sufficiently large constant.
Noting that 𝐌^𝑇𝐌=𝐿⊗𝐼_𝑛, and substituting (6) for 𝑐̇, we get the derivative of 𝑉(𝑡) along the trajectories of (4) as follows: 𝑉̇=𝐌𝑦^𝑇𝐌𝑦̇+1𝑘()𝑐̇=𝑐−ℎ𝐌𝑦^𝑇𝐌𝑒^𝑎𝑡𝐹𝑒^−𝑎𝑡𝑦−𝑐𝐿⊗Γ−𝑎𝐼_𝑁𝑛𝑦+(𝑐−ℎ)𝑦^𝑇𝐿²⊗Γ𝑦=𝑒^𝑎𝑡𝐌𝑦^𝑇𝐌𝐹𝑒^−𝑎𝑡𝑦−ℎ𝑦^𝑇𝐿²⊗Γ𝑦+𝑎𝑦^𝑇𝐿⊗𝐼_𝑛𝑦−𝑐𝑦^𝑇𝐌^𝑇𝐌(𝐿⊗Γ)𝑦+𝑐𝑦^𝑇𝐿²⊗Γ𝑦. (8) Noting that 𝐌^𝑇𝐌(𝐿⊗Γ)=𝐿⊗𝐼_𝑛(𝐿⊗Γ)=𝐿²⊗Γ, (9) we have 𝑉̇=𝑒^𝑎𝑡𝐌𝑦^𝑇𝐌𝐹𝑒^−𝑎𝑡𝑦−ℎ𝑦^𝑇𝐿²⊗Γ𝑦+𝑎𝑦^𝑇𝐿⊗𝐼_𝑛𝑦. (10)
Let 𝐺 be the coupling configuration graph corresponding to 𝐴=−𝐿. Let 𝐺 be the directed graph after an arbitrary orientation of the edges of 𝐺, and let ℰ(𝐺) be the edge set of 𝐺. An oriented edge in ℰ(𝐺) is denoted by 𝑒=𝑖→𝑗, where 𝑗 is the head of 𝑒 and 𝑖 is the tail of 𝑒.
Observe that 𝐌𝑦^𝑇𝐌𝐹𝑒^−𝑎𝑡𝑦=_{𝐺𝑒=𝑖→𝑗∈ℰ}𝑚_𝑖𝑗𝑦_𝑗−𝑦_𝑖^𝑇⋅𝑚_𝑖𝑗𝑓𝑒^−𝑎𝑡𝑦_𝑗−𝑓𝑒^−𝑎𝑡𝑦_𝑖≤𝑒^−𝑎𝑡⋅𝑙⋅_{𝑒=𝑖→𝑗∈ℰ(𝐺)}𝑚²_𝑖𝑗𝑦_𝑗−𝑦_𝑖²=𝑙𝑒^−𝑎𝑡𝐌𝑦^𝑇𝐌𝑦=𝑙𝑒^−𝑎𝑡𝑦^𝑇𝐿⊗𝐼_𝑛𝑦. (11) Hence, 𝑉̇≤(𝑙+𝑎)𝑦^𝑇𝐿⊗𝐼_𝑛𝑦−ℎ𝑦^𝑇𝐿²𝑦≤()𝑦⊗Γ𝑙+𝑎^𝑇𝐿⊗𝐼_𝑛𝑦−ℎ𝑟𝑦^𝑇𝐿²⊗𝐼_𝑛𝑦, (12) where 𝑟=min_{1≤𝑖≤𝑛}𝑟_𝑖>0.
Note that 𝐿 is a real and symmetric matrix, so we may let 𝑧₀∶=(1/√𝑛)𝟏,𝑧₁,…,𝑧_𝑁−1 as unit orthogonal eigenvectors corresponding to eigenvalues 𝜆₀=0,𝜆₁,…,𝜆_𝑁−1 of the matrix 𝐿, respectively. Let 𝑧₀,𝑧₁,…,𝑧_𝑛−1 be an arbitrary orthogonal base of ℝ^𝑛. Thus, 𝑧_𝑖⊗𝑧_𝑗, 𝑖=0,1,…,𝑁−1, 𝑗=0,1,…,𝑛−1, form an orthogonal base of ℝ^𝑁𝑛, which are also the unit orthogonal eigenvectors of 𝐿⊗𝐼_𝑛 and 𝐿²⊗𝐼_𝑛. As the associated graph of 𝐿 or 𝐴 is connected, 𝜆₁>0, we now write 𝑦(𝑡)=_𝑖,𝑗𝛼_𝑖𝑗(𝑡)𝑧_𝑖⊗𝑧_𝑗. (13) Then, we obtain 𝑉̇≤(𝑙+𝑎)_{𝑖≥0,𝑗}𝜆_𝑖𝛼_𝑖𝑗(𝑡)²−ℎ𝑟_{𝑖≥0,𝑗}𝜆²_𝑖𝛼_𝑖𝑗(𝑡)²=(𝑙+𝑎)_{𝑖≥1,𝑗}𝜆_𝑖𝛼_𝑖𝑗(𝑡)²−ℎ𝑟_{𝑖≥1,𝑗}𝜆²_𝑖𝛼_𝑖𝑗(𝑡)²≤(𝑙+𝑎)_{𝑖≥1,𝑗}𝜆_𝑖𝛼_𝑖𝑗(𝑡)²−ℎ𝑟_{𝑖≥1,𝑗}𝜆_𝑖𝜆₁𝛼_𝑖𝑗(𝑡)²=𝑙+𝑎−ℎ𝑟𝜆_{1𝑖≥1,𝑗}𝜆_𝑖𝛼_𝑖𝑗(𝑡)²=𝑙+𝑎−ℎ𝑟𝜆₁𝑦^𝑇𝐿⊗𝐼_𝑛𝑦=𝑙+𝑎−ℎ𝑟𝜆₁𝐌𝑦^𝑇𝐌𝑦. (14) Hence, we can choose a sufficiently large ℎ such that 𝛽∶=𝑙+𝑎−ℎ𝑟𝜆₁<0. Thus, 𝑉̇≤𝛽𝐌𝑦(𝑡)^𝑇𝐌𝑦(𝑡). (15)
It is obvious that 𝑉̇=0 if and only if 𝐌𝑦=0. In addition, if 𝐌𝑦=0, then 𝑦=𝟏⊗𝑦₁, and 𝑉̇=0 by (10). Therefore, the set 𝐸={𝑦∣𝐌𝑦=0} is the largest invariant set contained in 𝑉̇=0 for the system (4). Then, we obtain lim_𝑡→∞‖𝐌𝑦‖=0 based on the LaSalle invariant principle of differential equations. Hence, by Lemma 1, the complex dynamical network (4) is synchronized under the updated law (6) of coupling strength 𝑐. In particular, the complex dynamical network (1) is exponentially synchronized with rate 𝑎.

Remark 3. In [18], the updated law of coupling strength 𝑐 is given as follows: 𝑐̇=𝐾𝑥(𝑡)^𝑇𝐌^𝑇𝐌𝐌^𝑇𝐌Γ𝑥(𝑡)=𝐾𝑥(𝑡)^𝑇𝐿²⊗Γ𝑥(𝑡), (16) where Γ=𝐼_𝑁⊗Γ; the synchronization speed of complex dynamical network (1) or (3) can be controlled by the constant 𝐾 in (16). Here, we take 𝑘=𝐾𝑒^2𝑎𝑡; the synchronization is realized with exponential rate 𝑎. So, we make a further illustration of the above statement in [18] from (6) and Theorem 2.

Remark 4. From (15) in the proof of Theorem 2, if 𝛽 is taken smaller, 𝑉(𝑡) will decrease quickly, and then the synchronization of the network will be attained quickly. This will be done if the Laplacian matrix of the coupling configuration graph has a larger eigenvalue 𝜆₁. Based on the knowledge of spectral graph theory (or see [23]), 𝜆₁ is not decreased by adding edges to original graph. Therefore, we can add edges to the coupling configuration graph such that the complex network achieves synchronization more quickly.

Remark 5. As the coupling configuration graph is connected and Lemma 1 holds, the dynamical network (4) achieves synchronization. If the graph is disconnected, the network only can synchronize in each connected component of the graph but not in the whole graph.

4. Numerical Simulations

In this section, we present several numerical examples to show the theoretical results. In particular, we consider complex dynamical networks with the Lorenz system as each node 𝑥̇_𝑖1𝑥̇_𝑖2𝑥̇_𝑖3=𝛼𝑥_𝑖2−𝑥_𝑖1𝛾𝑥_𝑖1−𝑥_𝑖1𝑥_𝑖3−𝑥_𝑖2𝑥_𝑖1𝑥_𝑖2−𝛽𝑥_𝑖3,𝑖=1,2,3,4,5, (17) which is chaotic when 𝛼=10, 𝛽=8/3, 𝛾=28. Three small networks with five nodes are shown in Figure 1, where the weight of every edge in each network is 1. Choose the inner coupling link matrix to be Γ=diag{1,2,3}.

(a)

(b)

(c)

Figure 1 

Three small networks.

The corresponding coupling configuration matrices are listed in the same order as follows: ,,.−110001−210001−210001−210001−1−110001−310101−210001−210101−2−110001−411101−210011−310101−2 (18)

According to Theorem 2, the updated law of coupling strength 𝑐 is chosen as 𝑐̇=𝐾𝑥^𝑇(𝑡)(𝐿²⊗Γ)𝑥(𝑡) (i.e., 𝑘=𝐾, 𝑎=0 in (6)). Here 𝐾=0.0001, the initial coupling strength 𝑐=0, and the initial value of 𝑥 is taken as [−8,2,35,1,−9,−3,−4,5,1,2,−4,1,−7,8,4]^𝑇. The synchronization errors ‖𝐌𝑥‖ of three networks are shown in Figure 2, respectively, where the figure listed below is a local magnification of the above figure, and Figure 2 also shows that the method in [18] is effective to general networks. Numerical simulations show that synchronization can be reached more quickly by increasing the algebraic connectivity of the graph, which can be realized by adding edges to original graph.

(a)

(b)

Figure 2 

Synchronization errors ‖𝐌𝑥‖ of three networks in Figure 1.

Figure 3 shows that adaptive exponential synchronization is achieved for complex dynamical network in Figure 1(a), where 𝑘=0.0001, 𝑎=0, and 𝑎=3 in (6). The same results also hold for the networks in Figures 1(b) and 1(c).

Figure 3 

Exponential Synchronization errors ‖𝐌𝑥‖ of the network in Figure 1(a).

5. Conclusions

In this paper, an adaptive synchronization method on general networks or graphs is obtained. According to our method, networks may synchronize at an arbitrarily given exponential rate by increasing coupling strength and synchronize more quickly by increasing the algebraic connectivity of the corresponding graphs, which can be achieved by adding edges to original graphs. The obtained results also extend the work in [18] as a tree is a graph without cycles. Finally, numerical simulations are provided to illustrate the effectiveness of our theoretical results.

Acknowledgments

This research is supported by National Natural Science Foundation of China (11071001, 11071002), Major Program of Educational Commission of Anhui Province of China (KJ2010ZD02), Program of Natural Science of Colleges of Anhui Province (KJ2013A032, KJ2011A020, KJ2012B040), Starting Research Fund of Anhui University (023033190181), Young Scientist Fund of Anhui University (023033050055), and 211 Project of Anhui University (KJTD002B).