Cluster linear generalized outer synchronization in community networks via pinning control with two different switch periods
 Yuhong Liu^{1}Email author,
 Hui Li^{1},
 Qishui Zhong^{1} and
 Shouming Zhong^{2, 3}
https://doi.org/10.1186/s1366201711737
© The Author(s) 2017
Received: 20 December 2016
Accepted: 6 April 2017
Published: 20 April 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.
Keywords
community networks linear generalized outer synchronization pinning control1 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 [6–8] 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 [9–14]. In recent years, much literature reported the research results of network synchronization, and it has become a frontier issue [15–19]. As a result, different types of network synchronization have been put forward, for example, complete synchronization [20–22], 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 [33–35], 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].

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). ∗ denotes the term that is induced by symmetry. \(\Vert \xi \Vert \) indicates the 2norm of a vector ξ, 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 ⋅. The symbol ⊗ 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
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 \(\{y_{1}(t),\ldots,y_{m_{1}}(t)\}\rightarrow \phi_{1}(t)\), \(\{y_{m_{1}+1}(t),\ldots,y_{m_{2}}(t)\}\rightarrow \phi_{2}(t)\), … , \(\{y_{m_{s1}+1}(t),\ldots,y_{m_{s}}(t)\}\rightarrow \phi_{s}(t)\), 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
Assumption 2.1
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. □
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.
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 \(m=0,1,2,\ldots\) .
Theorem 3.1
Proof

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. □
Theorem 3.2
Proof
The proof is omitted here, as it is similar to that of Theorem 3.1. □
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
Example 2
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.
Declarations
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).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Huang, DW, Yu, ZG, Anh, V: Multifractal analysis and topological properties of a new family of weighted Koch networks. Phys. A, Stat. Mech. Appl. 469, 695705 (2017) MathSciNetView ArticleGoogle Scholar
 Chen, Y, Wang, X, Xiang, X, Tang, B, Chen, Q, Fan, S, Bu, J: Overlapping community detection in weighted networks via a Bayesian approach. Phys. A, Stat. Mech. Appl. 468, 790801 (2017) MathSciNetView ArticleGoogle Scholar
 Huang, C, Li, H, Xia, D, Xiao, L: Quantized subgradient algorithm with limited bandwidth communications for solving distributed optimization over general directed multiagent networks. Neurocomputing 185, 153162 (2016) View ArticleGoogle Scholar
 Wang, X, Yang, GH: Distributed consensus tracking control for multiagent networks with switching directed topologies. Neurocomputing 207, 693699 (2016) View ArticleGoogle Scholar
 Hu, X, Li, Y, Xu, H: Multimode clustering model for hierarchical wireless sensor networks. Phys. A, Stat. Mech. Appl. 469, 665675 (2017) View ArticleGoogle Scholar
 Yang, L, Jiang, J, Liu, X: Cluster synchronization in community network with hybrid coupling. Chaos Solitons Fractals 86, 8291 (2016) MathSciNetView ArticleMATHGoogle Scholar
 Kim, P, Kim, S: Detecting community structure in complex networks using an interaction optimization process. Phys. A, Stat. Mech. Appl. 465, 525542 (2017) View ArticleGoogle Scholar
 Chen, Y, Wang, X, Xiang, X, Tang, B, Chen, Q, Fan, S, Bu, J: Overlapping community detection in weighted networks via a Bayesian approach. Phys. A, Stat. Mech. Appl. 468, 790801 (2017) MathSciNetView ArticleGoogle Scholar
 Kaviarasan, B, Sakthivel, R, Lim, Y: Synchronization of complex dynamical networks with uncertain inner coupling and successive delays based on passivity theory. Neurocomputing 186, 127138 (2016) View ArticleGoogle Scholar
 Li, WL, Zhang, FY, Li, C, Song, HS: Quantum synchronization in a startype cavity QED network. Commun. Nonlinear Sci. Numer. Simul. 42, 121131 (2017) MathSciNetView ArticleGoogle Scholar
 Sakthivel, R, Anbuvithya, R, Mathiyalagan, K, Ma, YK, Prakash, P: Reliable antisynchronization conditions for BAM memristive neural networks with different memductance functions. Appl. Math. Comput. 275, 213228 (2016) MathSciNetGoogle Scholar
 Thuan, MV, Trinh, H, Hien, LV: New inequalitybased approach to passivity analysis of neural networks with interval timevarying delay. Neurocomputing 194, 301307 (2016) View ArticleGoogle Scholar
 Ray, A, Roychowdhury, A: Outer synchronization of networks with different node dynamics. Eur. Phys. J. Spec. Top. 223, 15091518 (2014) View ArticleGoogle Scholar
 Ahmadizadeh, S, Freestone, DR, Grayden, DB: On synchronization of networks of WilsonCowan oscillators with diffusive coupling. Automatica 71, 169178 (2016) MathSciNetView ArticleMATHGoogle Scholar
 Celikovsky, S, Lynnyk, V, Chen, G: Robust synchronization of a class of chaotic networks. J. Franklin Inst. 350, 29362948 (2013) MathSciNetView ArticleMATHGoogle Scholar
 Franceschelli, M, Giua, A, Pisano, A, Usai, E: Finitetime consensus for switching network topologies with disturbances. Nonlinear Anal. Hybrid Syst. 10, 8393 (2013) MathSciNetView ArticleMATHGoogle Scholar
 Li, WL, Li, C, Song, HS: Synchronization between uncertain nonidentical networks with quantum chaotic behavior. Physica A 461, 270277 (2016) MathSciNetView ArticleGoogle Scholar
 Zhou, GY, Li, CR, Li, TT, Yang, Y, Wang, C, He, FJ, Sun, JC: Outer synchronization investigation between WS and NW smallworld networks with different node numbers. Physica A 457, 506513 (2016) MathSciNetView ArticleGoogle Scholar
 Rakkiyappan, R, Sakthivel, N: Pinning sampleddata control for synchronization of complex networks with probabilistic timevarying delays using quadratic convex approach. Neurocomputing 162, 2640 (2015) View ArticleGoogle Scholar
 Lü, L, Chen, LS, Bai, SY, Li, G: A new synchronization tracking technique for uncertain discrete network with spatiotemporal chaos behaviors. Physica A 460, 314325 (2016) MathSciNetView ArticleGoogle Scholar
 Li, WL, Li, C, Song, HS: Synchronization transmission of target signal within the coupling network with quantum chaos effect. Physica A 462, 579585 (2016) MathSciNetView ArticleGoogle Scholar
 Skardal, PS, Taylor, D, Sun, J, Arenas, A: Erosion of synchronization in networks of coupled oscillators. Phys. Rev. E 91(1), 010802 (2015) View ArticleGoogle Scholar
 Dörfler, F, Bullo, F: Synchronization in complex networks of phase oscillators: a survey. Automatica 50, 15391564 (2014) MathSciNetView ArticleMATHGoogle Scholar
 AlMahbashi, G, Noorani, MSM, Bakar, SA: Projective lag synchronization in driveresponse dynamical networks. Int. J. Mod. Phys. C 25, 771776 (2014) View ArticleGoogle Scholar
 Bao, HB, Cao, JD: Projective synchronization of fractionalorder memristorbased neural networks. Neural Netw. 63(3), 19 (2015) View ArticleMATHGoogle Scholar
 Jalan, S, Singh, A, Acharyya, S, Kurths, J: Impact of a leader on cluster synchronization. Phys. Rev. E 91(2), 022901 (2015) MathSciNetView ArticleGoogle Scholar
 Hou, HZ, Zhang, QL, Zheng, M: Cluster synchronization in nonlinear complex networks under sliding mode control. Nonlinear Dyn. 83(12), 739749 (2016) MathSciNetView ArticleMATHGoogle Scholar
 Rakkiyappan, R, Sakthivel, N: Cluster synchronization for TS fuzzy complex networks using pinning control with probabilistic timevarying delays. Complexity 21(1), 5977 (2015) MathSciNetView ArticleGoogle Scholar
 Cui, G, Zhuang, G, Lu, J: Neuralnetworkbased distributed adaptive synchronization for nonlinear multiagent systems in purefeedback form. Neurocomputing 218, 234241 (2016) View ArticleGoogle Scholar
 Xu, Y, Zhang, J, Zhou, W, Tong, D: Adaptive synchronization of complex dynamical networks with bounded delay feedback controller. Optik 131, 467474 (2017) View ArticleGoogle Scholar
 Wei, W, Wang, M, Li, D, Zuo, M, Wang, X: Disturbance observer based active and adaptive synchronization of energy resource chaotic system. ISA Trans. 65, 164173 (2016) View ArticleGoogle Scholar
 Zhang, WY, Yang, C, Guan, ZH, Liu, ZW, Chi, M, Zheng, GL: Bounded synchronization of coupled Kuramoto oscillators with phase lags via distributed impulsive control. Neurocomputing 218, 216222 (2016) View ArticleGoogle Scholar
 Feng, J, Yang, P, Zhao, Y: Cluster synchronization for nonlinearly timevarying delayed coupling complex networks with stochastic perturbation via periodically intermittent pinning control. Appl. Math. Comput. 291, 5268 (2016) MathSciNetGoogle Scholar
 Wen, GH, Duan, ZS, Chen, GR, Yu, WW: Consensus tracking of multiagent systems with Lipschitztype node dynamics and switching topologies. IEEE Trans. Circuits Syst. 61(2), 499511 (2014) MathSciNetView ArticleGoogle Scholar
 Wen, GH, Yu, WW, Hu, GH, Cao, GD, Yu, XH: Pinning synchronization of directed networks with switching topologies: a multiple Lyapunov functions approach. IEEE Trans. Neural Netw. Learn. Syst. 26(12), 32393250 (2015) MathSciNetView ArticleGoogle Scholar
 Lei, X, Cai, S, Jiang, S, Liu, Z: Adaptive outer synchronization between two complex delayed dynamical networks via aperiodically intermittent pinning control. Neurocomputing 222, 2635 (2017) View ArticleGoogle Scholar
 Song, Q, Cao, J: On pinning synchronization of directed and undirected complex dynamical networks. IEEE Trans. Circuits Syst. I 57, 672680 (2010) MathSciNetView ArticleGoogle Scholar
 Li, CP, Sun, WG, Kurths, J: Synchronization between two coupled complex networks. Phys. Rev. E 76, 046204 (2007) View ArticleGoogle Scholar
 Tang, HW, Chen, L, Lu, JA, Tse, CK: Adaptive synchronization between two complex networks with nonidentical topological structures. Physica A 387, 56235630 (2008) View ArticleGoogle Scholar
 Zheng, S, Bi, Q, Cai, G: Adaptive projective synchronization between two complex networks with timevarying coupling delay. Phys. Lett. A 373, 15531559 (2009) MathSciNetView ArticleMATHGoogle Scholar
 Wu, X, Zheng, W, Zhou, J: Generalized outer synchronization between complex dynamical networks. Chaos 19, 013109 (2009) MathSciNetView ArticleMATHGoogle Scholar
 Sun, M, Zeng, CY, Tian, LX: Linear generalized synchronization between two complexes. Commun. Nonlinear Sci. Numer. Simul. 15, 21622167 (2010) MathSciNetView ArticleMATHGoogle Scholar