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Cluster linear generalized outer synchronization in community networks via pinning control with two different switch periods
Advances in Difference Equations volumeÂ 2017, ArticleÂ number:Â 117 (2017)
Abstract
This study investigates the problem of cluster generalized outer synchronization in community networks via pinning control with two different switch periods. Several pinning controllers have been designed to achieve linear generalized outer synchronization. Using Lyapunov stability theory, sufficient linear generalized outer synchronization criteria for community networks are derived. Finally, numerical examples are presented to demonstrate the effectiveness of the theoretical results.
1 Introduction
Recently, complex networks have drawn increasing attention from researchers and engineers in virtue of its wide applications in many fields, such as World Wide Web, communication networks, social networks, neural networks, epidemic networks, traffic networks, etc. Lots of network models, such as weighted networks [1, 2], directed networks [3, 4], hierarchical networks [5], community networks [6â€“8] are introduced to explore the potential applications better. As is well known, the research on network synchronization is very important due to its potential applications in many fields including secure communication, laser transmission, image identification, information science, and so on [9â€“14]. In recent years, much literature reported the research results of network synchronization, and it has become a frontier issue [15â€“19]. As a result, different types of network synchronization have been put forward, for example, complete synchronization [20â€“22], phase synchronization [23, 24], projective synchronization [25, 26] and cluster synchronization [27, 28].
Furthermore, many real complex networks cannot synchronize themselves or synchronize with the desired orbits. Therefore, proper controllers should be designed to achieve the goals by adopting some control schemes, such as adaptive control [29], feedback control [30], observerbased control [31], impulsive control [32], intermittent control [33â€“35], pinning control [36, 37] and so on. As a matter of fact, there are many examples of relationships between different networks, which indicates that it is necessary and significant to investigate the dynamical systems between different networks. Recently, [38] investigated the synchronization between two unidirectionally coupled complex networks with identical topological structures. [39] discussed the synchronization between two complex dynamical networks with nonidentical topological structures via using adaptive control method. [40] discussed adaptive projective synchronization between two complex networks with timevarying coupling delay. In the above papers, it is assumed that each node in driveresponse networks has identical dynamics. Later, [41] studied the problem of generalized outer synchronization between two complex dynamical networks with different topologies and diverse node dynamics. Reference [42] discussed the linear generalized synchronization between two complex networks with the nondelay coupling and the same topological structure, each network has identical dynamics. However, detailed analysis of the linear generalized synchronization between two networks of different topological structures and timevarying coupling delay has not been attempted in [42].
Motivated by the above discussions, this paper investigates the problem of cluster linear generalized outer synchronization (CLGOS) in community networks via pinning control with two different switch periods. Several pinning controllers have been designed to achieve linear generalized outer synchronization. Using Lyapunov stability theory, sufficient linear generalized outer synchronization criteria for community networks are derived. Finally, numerical examples are presented to demonstrate the effectiveness of the theoretical results. Contributions of this paper can be summarized as follows:

By adding adaptive semiperiodically intermittent controllers to a small fraction of nodes in response network, several sufficient conditions are derived based on the Lyapunov stability theory and strict mathematical proofs.

Both community networks with identical nodes and nonidentical nodes are investigated. Therefore, our proposed control schemes are more applicable technically.
The rest of the current paper is organized as follows. Section 2 introduces the problem formulation and some necessary definitions, lemmas, and hypotheses. Some sufficient conditions for the linear generalized outer synchronization are obtained in Section 3. Section 4 gives some numerical examples to demonstrate the effectiveness of our main results. Finally, Section 5 draws the conclusion.
Notation
The superscripts T and \((1) \) stand for matrix transposition and matrix inverse, respectively; \(\mathbb{R}^{n}\) denotes the ndimensional Euclidean space; \(I_{l}\) means the ldimensional identity matrix. The notation \(X>Y\) (\(X\ge Y\)), where X, Y are symmetric matrices, means that \(XY\) is positive definite (positive semidefinite). âˆ— denotes the term that is induced by symmetry. \(\Vert \xi \Vert \) indicates the 2norm of a vector Î¾, i.e., \(\Vert \xi \Vert =\xi^{T}\xi\). \(\operatorname{col}\{x_{1}, x_{2}, \ldots, x_{n}\}\) means \([x_{1}^{T},x_{2}^{T}, \ldots,x_{n}^{T}]^{T}\) and \(\operatorname{Sym}\{X\}\) means \(X+X^{T}\). The shorthand notation \(\operatorname{diag}\{ M_{1}, M_{2}, \ldots, M_{n}\}\) denotes a block diagonal matrix with diagonal blocks being the matrices \(M_{1},M_{2}, \ldots, M_{n}\). \(\lambda_{\min}(\cdot)\) and \(\lambda_{\max}(\cdot)\) denote the smallest and largest eigenvalue of â‹…. The symbol âŠ— denotes the Kronecker product. Matrices, if their dimensions are not explicitly stated, are assumed to have appropriate dimensions for algebraic operations.
2 Problem formulation and preliminaries
Consider the following complex networks with timevarying coupling delay consisting of N nodes and s communities with \(2\leq s < N\)
where \(x_{i}\in\mathbb{R}^{n}\) is the state variables of node i in networks X. \(f_{\varphi i}(\cdot):\mathbb{R}^{n}\longrightarrow \mathbb{R}^{n}\) is a continuously differentiable nonlinear function. c is the coupling strength, and Î“ is an innercoupling matrix. \(\tau(t)\) is the timevarying coupling delay satisfying \(0\leq \dot{\tau }(t)\leq \mu <1\). \(A=(a_{ij})_{N\times N}\in\mathbb{R}^{N\times N}\) and \(B=(b_{ij})_{N\times N}\in\mathbb{R}^{N\times N}\) are the outercoupling matrices with the sum of each row being zero. If there is a connection from node i to node j (\(j\neq i\)), then the coupling \(a_{ij}(b_{ij})\neq 0\); otherwise, \(a_{ij}(b_{ij})=0\) (\(j\neq i\)), and the diagonal elements of matrix are defined as \(a_{ii}=\sum_{j=1,j\neq i}^{N}a_{ij}\) or \(b_{ii}=\sum_{j=1,j\neq i}^{N}b_{ij}\). The function Ï† is defined as Ï†: \(\{1,2,\ldots,N\}\rightarrow \{1,2,\ldots,s\} \); if a node \(i\in V_{k}\), then \(\varphi_{i}=k\); \(V_{k}\) (\(k=1,2,\ldots,s\)) denotes the set of all nodes belong to the sth community.
Consider the controller response complex dynamical network as follows:
where \(y_{i}\in\mathbb{R}^{n}\) is the response state variables of node i in networks Y. \(\tilde{f}_{\varphi i}(\cdot):\mathbb{R}^{n}\longrightarrow \mathbb{R}^{n}\) is a continuously differentiable nonlinear function. \(u_{i}(t)\in\mathbb{R}^{n}\) (\(i=1,2,\ldots,N\)) are the nonlinear controllers ro be designed later, and the other parameters involved in system (2) all have the same meaning with the corresponding parameters in system (1).
Remark 2.1
The nonlinear vectorvalued functions \(f_{\varphi i}\) and \(\tilde{f}_{\varphi i}\) can be identical or nonidentical.
Remark 2.2
There are no limitations for the division of the clusters, the number of nodes in each cluster and the connections between nodes.
Remark 2.3
All nodes within a cluster have the same dynamics, and the dynamics of the nodes in different clusters can be different.
Remark 2.4
The proposed approach on the case with undirected topology is similar to the one that on the case with directed topology. So in this paper the underlying topology is assumed to be undirected.
Suppose that the networks (2) will be controlled onto some desired inhomogeneous state as \(\{y_{1}(t),\ldots,y_{m_{1}}(t)\}\rightarrow \phi_{1}(t)\), \(\{y_{m_{1}+1}(t),\ldots,y_{m_{2}}(t)\}\rightarrow \phi_{2}(t)\), â€¦â€‰, \(\{y_{m_{s1}+1}(t),\ldots,y_{m_{s}}(t)\}\rightarrow \phi_{s}(t)\), i.e., \(\mathcal{M}=\{\{\phi_{1}(t),\ldots,\phi_{1}(t)\},\{\phi_{2}(t),\ldots,\phi_{2}(t)\}, \ldots,\{\phi_{s}(t),\ldots,\phi_{s}(t)\}\}\in \mathbb{R}^{n\times N}\) is desired cluster synchronization pattern under the pinning control.
Definition 2.1
Let \(\phi_{\varphi_{i}}(t)=Px_{i}(t)+Q:\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) (\(i=1,2,\ldots,N\)) be continuously differentiable vector maps. If \(\varphi_{i}\neq \varphi_{j}\), \(\phi_{\varphi_{i}} \neq \phi_{\varphi_{j}}\). Generalized outer synchronization between the driveresponse networks are achieved if
where P and Q are constant matrices with proper dimension.
Assumption 2.1
Assuming that there are positive constants L, LÌƒ such that f and fÌƒ satisfy the following inequalities:
where Î“ and Î“Ìƒ are positive definite matrix, \(i=1,2,\ldots,N\). Here, x and y are timevarying vectors.
Lemma 2.1
For a diagonal matrix \(D=\operatorname{diag}\{\underbrace{d_{1},d_{2},\ldots,d_{l}}_{i= \{1,2,\ldots,l \} \subseteq \bar{V}_{\varphi i}}0,0,\ldots,0\}\) with \(d_{i}>0\), (\(i=1,2,\ldots,l\); \(1\leq l\leq N\)) and a symmetric matrix \(M\in \mathbb{R}^{N\times N}\), let \(MD= \bigl[ {\scriptsize\begin{matrix}{} E\bar{D}&S\cr \ast & M_{l} \end{matrix}} \bigr] \), where \(M_{l}\) is the minor matrix of M by removing its first l (\(1\leq l\leq N\)) rowcolumn pairs, E and S are matrices with appropriate dimensions, \(\bar{D}=\operatorname{diag}\{d_{1},d_{2},\ldots,d_{l}\}\). If \(d_{i}> \lambda_{\max}(ESM_{l}^{1}S^{T})\), then \(MD<0\) is equivalent to \(M_{l}<0\).
Proof
Let \(\bar{D}=\operatorname{diag}\{\underbrace{d_{1},d_{2},\ldots,d_{l}}_{i= \{1,2,\ldots,l \} \subseteq \bar{V}_{\varphi i}}\}\). Using matrix decomposition, \(M\in \mathbb{R}^{N\times N}\), let \(MD=\bigl[ {\scriptsize\begin{matrix}{} E\bar{D}&S\cr \ast&M_{l}\end{matrix}} \bigr]\), where \(M_{l}\) is the minor matrix of M by removing its first l (\(1\leq l\leq N\)) rowcolumn pairs, E and S are matrices with appropriate dimensions.
Using the Schur complement, it is easy to see that \(MD<0\) is equivalent to \(M_{l}<0\). We only need to prove that if \(M_{l}<0\), then \(MD<0\). When \(d_{i}>0\) (\(i=1,2,3,\ldots, l\)) are sufficiently large such that \(d_{i}> \lambda_{\max}(ESM_{l}^{1}S^{T})\) hold, it is easy to see that \(E\bar{D}SM_{l}^{1}S^{T}<0\). Then, using the Schur complement, we can conclude that \(MD<0\), so the proof is finished.â€ƒâ–¡
Lemma 2.2
[37]
Assume that A, B are N by N Hermitian matrices. Let \(\alpha_{1}\geq \alpha_{2}\geq \cdots \geq \alpha_{N}\), \(\beta_{1}\geq \beta_{2}\geq \cdots \geq \beta_{N}\) and \(\gamma_{1}\geq \gamma_{2} \geq \cdots \gamma _{N}\) be eigenvalues of A, B and \(A+B\), respectively. Then one has \(\alpha_{i}+\beta_{N}\leq\gamma_{i}\leq \alpha_{i}+\beta_{1}\), \(i=1,2,\ldots, N\).
3 Main results
In this section, the CLGOS of the driveresponse community networks (1) and (2) will be investigated in three cases.
Case I. Assuming that the nodes dynamics in both community networks are identical, i.e., nonlinear vectorfunctions \(f_{\varphi i}=\tilde{f}_{\varphi i}=f\) for all \(1\leq i\leq N\). Then the driveresponse networks (1) and (2) can be written as
In this subsection, the intermittent control with two different switched periods is considered. The sketch of such control strategy is given by Figure 1. As shown in Figure 1, \(T_{1}\) and \(T_{2}\) are two periods appearing alternately. \(\eta_{1}\) (\(0<\eta_{1}<1\)) and \(\eta_{2}\) (\(0<\eta_{2}<1\)) are called the rates of control duration in each control period. \((1\eta_{1})T_{1}\) and \((1\eta_{2})T_{2}\) are called nonfeedback control widths in control periods \(T_{1}\) and \(T_{2}\), respectively. The rates of control duration may be different, i.e., \(\eta_{1}\neq \eta_{2}\), while they are assumed to be the same in [21]. In this regards, the semiperiodically intermittent control scheme considered here is more general than [21].
We denote \(\Xi_{1}^{m}=[mT,mT+\eta_{1}T_{1}]\) is the control width in period \(T_{1}\), \(\Xi_{2}^{m}=[mT+\eta_{1}T_{1},mT+T_{1}]\) is the nonfeedback control width in period \(T_{1}\), \(\Xi_{3}^{m}=[mT+T_{1},mT+T_{1}+\eta_{2}T_{2}]\) is the control width in period \(T_{2}\), \(\Xi_{4}^{m}=[mT+T_{1}+\eta_{2}T_{2}, (m+1)T]\) the nonfeedback control width in period \(T_{2}\), where \(m=0,1,2,\ldots\)â€‰.
The adaptive semiperiodically intermittent controllers are defined as follows:
where \(\bar{V}_{\varphi i}\) denotes the set of the nodes in the \(\varphi _{i}\)th community which have direct connections to the nodes in other communities and the updating laws
and
where \(\varepsilon_{i}\) (\(i\in \bar{V}_{\varphi i}\)) and Î² are positive constants.
According to the definition of the coupling matrix A, B, it is easy to see that
Let \(e_{i} (t)=y_{i} (t)\phi_{\varphi_{i}}(t)\), \(g(e_{i}(t))=f(y_{i}(t))f(\phi_{\varphi i}(t))\), \(g(\cdot):\mathbb{R}^{n}\longrightarrow \mathbb{R}^{n}\) is a continuously differentiable nonlinear function. With the aid of equations (5)(10), the error systems can be rewritten as
Theorem 3.1
Suppose that Assumption 2.1 holds. Using the adaptive controllers and updated laws (7)(9), then the response networks (6) can linear generalized synchronize with the drive networks (5) if there exist positive constants \(\alpha >\beta >0\) such that the following conditions are satisfied:
where
\(\hat{A}_{l}\) is the minor matrix of A by removing its first l (\(1\leq l\leq N\)) rowcolumn pairs.
Proof
Construct the following Lyapunov function candidate:
where
where \(k_{i}^{\ast}\) are sufficiently large positive constant to be determined. We write \(\tilde{e}_{j}(\theta)=(\tilde{e}_{1j}(\theta),\tilde{e}_{2j}(\theta),\ldots,\tilde{e}_{Nj}(\theta))^{T}\). \(\dot{\tilde{p}}_{ij}=e_{i}^{T} (t)\Gamma y_{i} (t\tau (t))\).
When \(t\in \Xi_{1}^{m}\cup \Xi_{3}^{m}\), differentiating \(V(t)\) with respect to time along the solution of (11) yields
From (17)(20), it is easy to see that
where
Let \(\Omega=\kappa I_{N}+c (\hat{A}K^{\ast})=\big[ {\scriptsize\begin{matrix}{} EK^{\ast\ast}&S\cr \ast&\Omega_{l} \end{matrix}} \big]\), in which \(\Omega_{l}\) is the minor matrix of Î© by removing its \(l(l\in \bar{V}_{\varphi i})\) rowcolumn pairs, E and S are matrices with appropriate dimensions, \(K^{\ast\ast}=\operatorname{diag}\{\underbrace{ck_{1},ck_{2},\ldots,ck_{l}}_{i= \{1,2,3,\ldots,l \}\subseteq \bar{V}_{\varphi i}}\}\). It is obvious that Î© is symmetric. According to Lemma 2.1, we know that if one can select \(k_{i}> \lambda_{\max}(ES\Omega_{l}^{1}S^{T})\), then \(\Omega<0\) is equivalent to \(\Omega_{l}<0\). Based on Lemma 2.2 and the condition (12), we have \(\lambda_{\max}(\Omega_{l})\leq \kappa +c\lambda_{\max}(A_{l})<0\), which implies that \(\Omega_{l}<0\). Then we obtain
When \(t\in \Xi_{2}^{m}\cup \Xi_{4}^{m}\), differentiating \(V(t)\) with respect to time along the solution of (11) and using the condition in (13) yields
Then we have
From (24), it is easy to see that:

When \(t\in \Xi_{1}^{m}\), i.e., \(\frac{t\eta_{1}T_{1}}{T}< m\leq \frac{t}{T}\)
$$\begin{aligned} V (t) \leq & V (m T)\operatorname{exp}\bigl(\beta (tmT) \bigr) \\ \leq & V (0)\operatorname{exp}\bigl(\beta mT +m\alpha \bigl( (1\eta_{1}) T_{1}+ (1\eta_{2})T_{2} \bigr) \bigr) \\ \leq & V (0) \operatorname{exp}\bigl( (\beta+\alpha\theta)t+\beta\eta_{1}T_{1} \bigr). \end{aligned}$$(25) 
When \(t\in \Xi_{2}^{m}\), i.e., \(\frac{tT_{1}}{T}< m\leq \frac{t\delta_{1}T_{1}}{T}\),
$$\begin{aligned} V (t) \leq &V (mT+T_{1}) \operatorname{exp}\bigl(t (mT+T_{1}) \bigr) \\ \leq&V (0)\operatorname{exp}\bigl( (\alpha\beta) (mT+T_{1})\alpha \bigl( (m+1)\eta_{1}T_{1}+m\eta_{2}T_{2} \bigr) \bigr) \\ \leq & V (0) \operatorname{exp}\biggl( (a_{\ast}\alpha\rho)t\alpha ( \eta_{1}\eta_{2}) \frac{T_{1}T_{2}}{T}+a_{\ast} (1 \eta_{1})T_{1} \biggr). \end{aligned}$$(26) 
When \(t\in \Xi_{3}^{m}\), i.e., \(\frac{tT_{1}\eta_{2}T_{2}}{T}< m\leq \frac{tT_{1}}{T}\),
$$\begin{aligned} V (t) \leq &V ( mT+T_{1}) \operatorname{exp}\bigl(\beta \bigl(t (mT+T_{1}) \bigr) \bigr) \\ \leq & V (0)\operatorname{exp}\bigl(\beta (mT+T_{1}) +\alpha \bigl( (m+1) (1 \eta_{1})T_{1} +m (1\eta_{2})T_{2} \bigr) \bigr) \\ \leq & V (0) \operatorname{exp}\biggl( (\beta+\alpha\theta)+\beta\eta_{2}T_{2} +\alpha\frac{\eta_{2}\eta_{1}}{T}T_{1}T_{2} \biggr). \end{aligned}$$(27) 
When \(t\in \Xi_{4}^{m}\), i.e., \(\frac{t}{T}< m+1\leq \frac{t+TT_{1}\eta_{2}T_{2}}{T}\),
$$\begin{aligned} V (t) \leq &V \bigl( (m+1)T \bigr) \operatorname{exp}\bigl(a_{\ast} \bigl(t (m+1)T \bigr) \bigr) \\ \leq &V (0)\operatorname{exp}\bigl(a_{\ast} (m+1)T  (m+1)\alpha ( \eta_{2}T_{2}+\eta_{1}T_{1}) \bigr) \\ \leq &V (0)\operatorname{exp}\bigl( (a_{\ast}\alpha\rho)t+a_{\ast} (1 \eta_{2})T_{2} \bigr). \end{aligned}$$(28)
Therefore, when \(t\in \Xi_{1}^{m}\cup \Xi_{3}^{m}\), if \(\beta\alpha\theta >0\) is satisfied, one has \(\lim_{t\rightarrow \infty}V(t)=0\); when \(t\in \Xi_{2}^{m}\cup \Xi_{4}^{m}\), if \(\alpha\rho a_{\ast}>0\) is satisfied, one has \(\lim_{t\rightarrow \infty}V(t)=0\). The conclusion of Theorem 3.1 holds. This completes the proof.â€ƒâ–¡
Case II. Assume that the nodes dynamics in both community networks are nonidentical; in view of the special property, the adaptive semiperiodically intermittent controllers are defined as follows:
Theorem 3.2
Suppose that Assumption 2.1 holds. Using the adaptive controllers and updated laws (29)(30), then the response networks (2) can linear generalized synchronize with the drive networks (1) if there exist positive constants \(\alpha >\beta >0\) such that the following conditions are satisfied:
where
\(\hat{A}_{l}\) is the minor matrix of A by removing its first l (\(1\leq l\leq N\)) rowcolumn pairs.
Proof
The proof is omitted here, as it is similar to that of Theorem 3.1.â€ƒâ–¡
4 Numerical examples and simulation
In this section, two numerical examples will be provided to verify and demonstrate the effectiveness of the proposed method.
Example 1
Theorem 3.1 is verified. The node dynamics of the first community are chosen as the wellknown Lorenz system:
and the node dynamics of the second community are chosen as the wellknown Chen system:
and the node dynamics of the third community are chosen as the wellknown Lv system:
Similar to the discussion of the Chen system in [24], one can discuss the Lorenz and Lv systems and choose the positive constant \(L=65\) such that Assumption 2.1 holds.
For simplicity, in the numerical simulations, assuming \(P=\left[{\scriptsize\begin{matrix}{} 1&0&0\cr 0&1&0\cr 0&0&1 \end{matrix}} \right]\), \(Q=\left[{\scriptsize\begin{matrix}{} 0\cr 0\cr 0 \end{matrix}} \right]\), \(\Gamma=\left[{\scriptsize\begin{matrix}{} 1&0&0\cr 0&1&0\cr 0&0&1 \end{matrix}} \right]\), \(T_{1}=0.2\), \(T_{2}=0.1\), \(\eta_{1}=0.9\), \(\eta_{2}=0.8\), the timevarying \(\tau(t)=10.5e^{t}\), then \(\dot{\tau}(t)=0.5e^{t}\in(0,0.5]\leq \frac{1}{2}\doteq \mu< 1 \). A complex network consisting of 19 nodes with three communities is shown in Figure 2 (\(A=B\)).
The feedback control gain is \(k_{i}=20\) for \(i\in \bar{V}_{\varphi i}\). By simple calculation, \(\theta=\frac{1}{15}\), \(\rho=0.867\), for Theorem 3.1, one can choose \(\alpha=155\), \(\beta=22\), and \(c<\frac{\kappa}{\lambda_{\max}(\hat{A}_{l})}\) such that conditions (13)(15) hold. Therefore, the CLGOS can be achieved for any initial values. Figures 35 show the orbits of linear generalized outer synchronization errors.
Example 2
Choosing the node dynamics as the following wellknown timedelayed Chua oscillator:
where
\(a=10\), \(b=19\), \(d=0.1636\), \(m_{1}=1.4325\), \(m_{2}=0.7831\), \(\upsilon=0.5\), \(\chi_{0}=0.2\). Choose the positive constant \(L=12\) such that Assumption 2.1 holds. For simplicity, in the numerical simulations, assuming \(P=\left[{\scriptsize\begin{matrix}{} 1&0&0\cr 0&1&0\cr 0&0&1 \end{matrix}} \right]\), \(Q=\left[{\scriptsize\begin{matrix}{} 0\cr 0\cr 0 \end{matrix}} \right]\), \(\Gamma=\left[{\scriptsize\begin{matrix}{} 1&0&0\cr 0&1&0\cr 0&0&1 \end{matrix}} \right]\), \(T_{1}=0.2\), \(T_{2}=0.1\), \(\eta_{1}=0.9\), \(\eta_{2}=0.8\), the timevarying \(\tau(t)=10.5e^{t}\), then \(\dot{\tau}(t)=0.5e^{t}\in(0,0.5]\leq \frac{1}{2}\doteq \mu< 1 \). A complex network consisting of 19 nodes with three communities is shown in Figure 2 (\(A=B\)). The community network is constructed by integrating three BA networks consisting of 50 nodes with \(m_{0}=m=3\). For any pair of communities, four edges are chosen to connect them randomly (Figure 6). The feedback control gains as \(k_{i}=20\) for \(i\in \bar{V}_{\varphi i}\). By simple calculations, \(\theta=\frac{1}{15}\), \(\rho=0.867\), for Theorem 3.1, one can choose \(\alpha=35\), \(\beta=5\), and \(c<\frac{\kappa}{\lambda_{\max}(\hat{A}_{l})}\) such that condition (13)(15) hold. Therefore, the CLGOS can be achieved for any initial values. Figures 79 show the orbits of linear generalized outer synchronization errors.
5 Conclusions
In this paper, we investigated the problems of CLGOS in community networks via pinning control with two different switch periods. Using Lyapunov stability theory, linear matrix inequality (LMI), sufficient CLGOS criteria for community networks are derived. Both community networks with identical nodes and nonidentical nodes are investigated. Therefore, our proposed control schemes are better applicable technically. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed method.
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Acknowledgements
The authors greatly appreciate the reviewers suggestions and the editors encouragement. The work is partially supported by the Sichuan Science and Technology Plan (2017GZ0165).
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YHL carried out the main part of this article, corrected and revised the manuscript, HL, SMZ, QSZ brought forward some suggestions on this article. All authors have read and approved the final manuscript.
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Liu, Y., Li, H., Zhong, Q. et al. Cluster linear generalized outer synchronization in community networks via pinning control with two different switch periods. Adv Differ Equ 2017, 117 (2017). https://doi.org/10.1186/s1366201711737
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DOI: https://doi.org/10.1186/s1366201711737
Keywords
 community networks
 linear generalized outer synchronization
 pinning control