In the past decade, more and more attention has been attracted to the problem of complex networks, which could be regarded as lots of dynamical nodes coupled with each other. As the result of the complex coupling effect, many different types of collective dynamical behaviors can be realized, such as complete synchronization [1, 2], cluster synchronization [3–5], exponential synchronization [6, 7], intermittent synchronization [8, 9], etc. Cluster synchronization, which is discussed in this paper, implies that the oscillators split into several clusters, and the oscillators in the same cluster synchronize with one another. Many researches have shown that there is a close interplay between cluster synchronization and network topologies. There are also some networks which can not realize a particular type of synchronization only via the inherent coupling effect of oneself. In order to realize cluster synchronization in networks without cluster structures, many various control schemes have been proposed during the past decades, for instance, adaptive feedback control [10, 11], pinning control [12, 13], intermittent control [14], and so on. The control approaches concerned in this paper are the famous pinning control and adaptive feedback control, both have been intensively investigated recently. First, we make a brief introduction to the background and related work of pinning control. Generally speaking, the node number of each complex network in the real world may be the massive level, and it is hard to control all nodes to realize a particular type of synchronization. In order to reduce the control gain and the number of controllers, pinning control is proposed as a more effective control approach [13]. The advantage of the pinning control approach is that it is only necessary to control a significantly smaller number of local controllers as compared to the randomly pinning scheme. Since the idea of pinning control was first developed, researchers concentrate their efforts on the popularization and application of the control approach [15–20]. For instance, by using the Schur complement and Lyapunov stability theory, several pinning schemes were designed to realize lag synchronization between two coupled networks in [15]. It is also shown that a directed network can realize pinning synchronization in [17]. Pinning control scheme was also applied to drive a directed dynamical network with non-identical nodes to cluster synchronization [18]. Similarly, the intermittent pinning-control problem of directed heterogeneous dynamical networks has also been discussed in [19]. Second, it is worth to point out that adaptive control scheme is a feasible approach to avoid larger feedback control gains. Notice that the coupling strength of many real network can not be arbitrarily large, adaptive feedback control should be a valid method for achieving synchronization. During the recent years, researchers have made some great achievements on adaptive feedback control. In [21], cluster synchronization of an adaptive dynamical network with non-identical nodes was discussed.

Motivated by the discussions above, this paper aims to provide several novel criteria on cluster synchronization of complex networks with adaptive coupling strength via pinning control. We carry out a new dynamical network model possessing four characteristics, i.e., delayed dynamics, non-identical nodes, directed topology and time-varying coupling strength. To the best of our knowledge, there is no previous research discussing the problem of cluster synchronization of such a network model. However, the problem mentioned above should be an important issue of complex networks because of its wide applications in practical problems. Then we design proper pinning controllers and adaptive coupling strength, and sufficient conditions are derived to realize cluster synchronization by using Lyapunov stability theory and matrix theory. In order to illustrate the feasibility and validity of the obtained theoretical results, several numerical simulations are presented.

The rest of this paper is designed as follows. The network model is described and some necessary lemmas and assumptions are stated in section 2. Via pinning and adaptive controls, cluster synchronization in a delayed network is discussed and then some sufficient conditions are given in section 3. Numerical examples are obtained in section 4. Finally, in section 5, conclusions are presented.

Consider a network consisting of N nodes, which are divided into m communities C₁, C₂, …, C_m, where 2 ≤ m ≤ N. It is easy to see that there holds ∪i=1mCi={1,2,⋯,N}. If the ith node belongs to the jth cluster, then we define ϕ_i = j. The function fϕi:Rn→Rn can be employed to describe the local dynamics of each node in the ith cluster.

Now, we consider a delayed dynamic network described by the following equations,

where xi(t)=(xi1(t),xi2(t),⋯,xin(t))T∈Rn is the state variable of node i; f_ϕi is differentiable and capable of performing abundant dynamical behaviors. If ϕ_i ≠ ϕ_j, i.e., the ith node and the jth node belong to different clusters, then f_ϕi ≠ f_ϕj. δ_i(t) > 0 is a time-varying coupling strength. τ > 0 is a constant. δ_i(t) is the time-varying coupling strength. Γ ∈ R^n×n is an inner-coupling matrix. For simplicity, we suppose that Γ is a positive definite diagonal matrix, that is, Γ = diag(ρ₁, ρ₂, …, ρ_n) with ρ_i > 0. The asymmetric matrix A = (_aij)N×N describes the topology of the directed network, where a_ij ≥ 0, i ≠ j, is the weight of the edge from j to i. Here, the coupling matrix A is supposed to be diffusive, that is, a_ii are defined as aii=-∑j=1,j≠iNaij,i=1,2,⋯,N.

At first, we give an assumption about the function f_ϕi, which has been widely employed in the previous researches on the problem of synchronization in complex networks. Most of the previous researches pointed out that many well-known chaotic systems have been checked to satisfy the following assumption, such as cellular neural networks and Chua's circuit [22].

Assumption 1 ([23]). For the vector-valued function fϕi(t,z1,z̄1), there exist a constant L such that

for any z1,z2,z̄1,z̄2∈Rn,i=1,2,…,N.

By introducing the control inputs ui(t)∈Rn(i=1,2,…,N), the controlled dynamical network with respect to the network (1) can be rewritten as

Definition 1. Let s_ϕi(t) = s_ϕi(t, t₀, s₀) be the solution of the following system

Defining the error variables by e_i(t) = x_i(t) − s_ϕi(t), i = 1, 2, …, N, the oscillator network (2) is said to realize cluster synchronization, if the synchronization errors satisfy

The following error system can be concluded from the Equations (2) and (3),

According to the definition of the diffusive coupling matrix A, one obtains that ∑j=1NaijΓsϕj(t)=0 for i ∈ G_ϕi − Ḡ_ϕi, where G_ϕi denotes the index set consisting of all nodes in the ϕ_ith cluster and Ḡ_ϕi represents the index set consisting of all nodes in the ϕ_ith cluster which have direct connections with some nodes in other clusters. In view of this property, the pinning controller is designed as

where θ_i = 1 for i ∈ Ḡ_ϕi, or θ_i = 0 for i ∈ G_ϕi − Ḡ_ϕi, and d_i > 0. The coupling strengths δ_i(t) are constructed by the following adaptive laws

where the constant σ_i > 0 is the control gain, δ_i(0) > 0, the 0 − 1 switching function γ_t is defined as follows,

T is a positive constant. Here, u_i(t) and δ_i(t) are different from the corresponding functions of the previous papers, such as [3, 12, 14, 21]. It is easy to see that each node in the network has a different coupling strength. By intuition, the first term in the controller (5) is used to synchronize all nodes in the same cluster, while the second term in the controller (5) is to weaken the influences of the couplings among different clusters.

Remark: The controller (5) only works on the nodes which have direct connections with some nodes in other clusters since u_i(t) = 0 for i ∈ G_ϕi − Ḡ_ϕi. That is to say, only the nodes which have direct connections with some nodes in other clusters need to be pinned. In addition, the coupling strengths δ_i(t) are constructed by the adaptive laws. Therefore, the controller (5) is an pinning controller with adaptive strength.

In this section, the problem of cluster synchronization in delayed networks will be investigated by designing proper pinning controllers and adaptive coupling strength. Based on Lyapunov functional method and matrix theory, the following theorem is established.

Theorem 1. Let δ_i ≜ δ_i(t). Under Assumption 1, if there exists a positive definite matrix Q, such that the matrix H₁ is negative definite for t ≤ T, and the matrix H₂ is negative definite for t > T, where

I_N(I_n) is the N(n)-dimensional identity matrix, U=12(B+BT),

then, by the local control (5) and the corresponding adaptive laws (6), the solutions e₁(t), e₂(t), …, e_N(t) of the system (4) satisfy limt→+∞ei(t)=0,i=1,2,…,N. That is, cluster synchronization in the network system (2) is realized.

Proof. Let

Apparently, V(t) = 0 if and only if e_i(t) ≡ 0, t ∈ [t − τ, t] and δ_i(T) − δ_i(t) = 0. Then, the derivate of V(t) along the solution of error system (4) under control (5) is

Denoting e(t)=(e1T(t),e2T(t),…,eNT(t))T, we obtain

That is, for U=12(B+BT), we have

According to Assumption 1, we have

Denote η^T(t) = (e^T(t), e^T(t − τ)). For t ≤ T, γ_t = 1, that is δ∙i(t)=σieiT(t)Γei(t). Therefore, we have

Because of H₁ < 0, we know that V∙(t)≤0, and the equality holds if and only if e_i(t) = 0 and e_i(t − τ) = 0 for i = 1, 2, …, N. For t > T, γ_t = 0, that is δ∙i(t)=0. Then we obtain

As H₂ < 0, we know that V∙(t)≤0, and the equality holds if and only if e(t) = 0 and e_i(t − τ) = 0 for i = 1, 2, …, N.

Therefore, the solutions e₁(t), e₂(t), …, e_N(t) of the system (4) satisfy limt→+∞ei(t)=0 for i = 1, 2, …, N. Thus, cluster synchronization in the network system (2) is realized under the local control (5) and the corresponding adaptive laws (6).

Let Q = Γ = I_n, we obtain the following corollary.

Corollary 1. Let δ_i ≜ δ_i(t). Under Assumption 1, if Γ = I, H₃ is negative definite for t ≤ T, and H₄ is negative definite for t > T, where

I_N(I_n) is the N(n)-dimensional identity matrix, U=12(B+BT),

F = diag(θ₁δ₁d₁, θ₂δ₂d₂, ⋯ , θ_Nδ_Nd_N), W = diag(δ₁(T) − δ₁, δ₂(T) − δ₂, ⋯ , δ_N(T) − δ_N), then, by the local control (5) and the corresponding adaptive laws (6), the solutions e₁(t), e₂(t), …, e_N(t) of the system (4) satisfy limt→+∞ei(t)=0,i=1,2,…,N. That is, cluster synchronization in the network system (2) is realized.

In this section, we give a numerical example to verify the theorem given in the previous section.

Consider a directed network consisting of 10 nodes with two communities, the topological structure of which is shown in Figure 1. Choose the node dynamics of the five nodes in the first cluster as the time-delayed Chua's circuit, which is described by the following equations,

where xi(t)=(xi1(t),xi2(t),xi3(t))T, w(xi(t))=(27xi1(t)-314(|xi1(t)+1|-|xi1(t)-1|),0,0)T, and v(xi(t-τ))=0.3(xi1(t-τ),0,0)T, i = 1, 2, …, 5. Choose the node dynamics of the five nodes in the second cluster as the time-delayed 3-D neural network which is described by the following equations,

where xi(t)=(xi1(t),xi2(t),xi3(t))T, g(xi(t))=((|xi1(t)+1|-|xi1(t)-1|)/2,(|xi2(t)+1|-|xi2(t)-1|)/2,(|xi3(t)+1|-|xi3(t)-1|)/2)T, h(xi(t-τ))=0.3(xi1(t-τ),0,0)T, i = 6, 7, ⋯, 10.

Matrices V₁ and V₂ are, respectively, defined by

Recalling that Ḡ_ϕi denotes the set consisting of all nodes in the ϕ_ith cluster which have direct connections to the nodes in other clusters. It is easy to see that Ḡ₁ = {1, 5}, Ḡ₂ = {10} and

According to the discussion in [18] and some simple calculations, Assumption 1 is satisfied for systems (7) and (8). And then we obtain L = 21. Let σ_i = 2, T = 2, δ_i(0) = 15, τ = 0.2, d₁ = d₅ = d₁₀ = 20, d₂ = d₃ = d₄ = d₆ = d₇ = d₈ = d₉ = 0, Γ = 12I₃, Q = 22I₃. It is easy to verify that H₁ < 0 for 0 ≤ t ≤ 2 and H₂ < 0 for t > 2.

Let the state of the Chua's circuit under initial condition be s1(t)=(-0.1,-0.2,-0.3)T, t ∈ [−0.2, 0], and the state of the 3-D neural network under initial condition be s2(t)=(0.1,0.2,0.3)T, t ∈ [−0.2, 0]. Initial states of all nodes are constant functions on [−0.2, 0]. Here, the values of the initial constant functions are randomly chosen on [−1, 1]. Now we define E(t)=∑i=110|xi(t)-sϕi(t)|,   Eij(t)=|xi(t)-xj(t)|,i=1,2,…,5,j=6,…,10, where the norm |·| of vector x is defined as |x| = (x^Tx)^1/2. It is clear that the cluster synchronization is achieved if E(t) converges to zero and E_ij(t) do not converge to zero as t → +∞.

By the aid of Matlab, we obtain Figure 2, which shows the time evolutions of the component x_i1, i = 1, …, 10. In Figure 2, the red lines represent the first five nodes, and the blue lines represent the latter five nodes. Figures 3, 4 show the time evolutions of E(t) and E₅₈(t), respectively. It is easy to see that the error E(t) tends to zero, but E₅₈ doesn't tend to zero when t → +∞. That is to say, 2-cluster synchronization is achieved, and the effectiveness of the theoretic results is verified.

In this paper, the problem of cluster synchronization in a complex network via pinning control has been investigated, and several novel criteria have been derived by utilizing the matrix theory and the stability analysis technique. The object of this research is a dynamical network possessing four characteristics, delayed dynamics, non-identical nodes, directed topology and time-varying coupling strength. And from the control schemes of this research, we present a novel mixed method, which integrated the pinning control and adaptive feedback control, to design proper pinning controllers and adaptive coupling strengths to realize cluster synchronization in such a dynamical network. For convenience, we also carried out a succinct and utilitarian corollary by simplifying the obtained conditions. Finally, numerical simulations are provided to illustrate the effectiveness of the proposed control methods.

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author/s.

JZ performed the theory derivation and wrote the first draft of the paper. ZM and XL provided the numerical experiments and contributed to the modification of this paper. JQ revised the first draft and made many helpful suggestions, which improve this paper greatly. All authors contributed to the article and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.