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With the development of the networks, the coupling between networks has become increasingly significant. Here, the networks can be described as interdependent networks. An interdependent network can have two different kinds of links, a connectivity link and a dependency link, which are fundamental properties of interdependent networks. During the past several years, interdependent complex network science has attracted a great deal of attention. This is mainly because the rapid increase in computing power has led to an information and communication revolution. Investigating and improving our understanding of interdependent networks will enable us to make the networks (such as infrastructures) we use in daily life more efficient and robust. As a significant collective behavior, synchronization phenomena and processes are common in nature and play a vital role in the interaction between dynamic units. At the same time, the time delay problem is an important issue to be investigated, especially in biological and physical networks. As a matter of fact, time delays exist commonly in the real networks. A signal or influence traveling through a network is often associated with time delay. In this paper, the local adaptive heterogeneous synchronization is investigated for interdependent networks with delayed coupling consisting of two sub-networks, which are one-by-one inter-coupled. The delays exist both in the intra-coupling and in the inter-coupling between two sub-networks, the intra-coupling and inter-coupling relations of the networks satisfy the requirements for nonlinearity and smoothness, and the nodes between two sub-networks have different dynamical systems, namely heterogeneous systems. Based on the Lyapunov stability theory, linear matrix inequality, and adaptive control technique, with proper controllers and adaptive laws for the networks, the sufficient conditions are proposed to synchronize the sub-networks of the interdependent networks into heterogeneous isolated systems, respectively. In order to illustrate the main results of the theoretical analysis clearly, some numerical simulations for an interdependent network with NW small world sub-network and BA sub-network are presented, in which each sub-network has 100 nodes and the heterogeneous systems are Lorenz and Rössler systems. The numerical simulations show that using the controllers and adaptive laws proposed, the network obtains the local heterogeneous synchronization quickly, that is, the nodes of two sub-networks are synchronized into Lorenz and Rössler systems separately. Thus, they verify the feasibility and correctness of the proposed techniques. It is worth noting that the presented results are delay-independent. In the future, our research will be directed to the further investigation of the delay-dependent synchronization of interdependent networks by using the current results as a basis.
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Keywords:
- delayed interdependent networks /
- linear matrix inequality /
- Lyapunov stability theory /
- adaptive heterogeneous synchronization
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[28] Zheng Y G, Bao L J 2017 Chaos. Soliton Fract. 98 145
[29] Yang S F, Guo Z Y, Wang J 2017 IEEE Trans. Neur. Net. Lear. 28 1657
[30] He W L, Chen G R, Han Q L, et al. 2017 IEEE Trans. Syst. Man. Cy-S. 47 1655
[31] Zhang X Y, Boccaletti S, Guan S G 2015 Phys. Rev. Lett. 114 038701
[32] Li Y, Wu X Q, Lu J A, L J H 2016 IEEE Trans. Circuits Syst. Ⅱ 63 206
[33] Xu Q, Zhuang S X, Hu D, Zeng Y F, Xiao J 2014 Abst. Appl. Anal. 10.1155 453149
[34] Boyd S, Ghaoui L E, Feron E, Balakrishnan V 1994 Linear Matrix Inequalities in System and Control Theory (Philadelphia: SIAM) pp7-14
[35] Tu L L, Liu H F, Yu L 2013 Acta Phys. Sin. 62 140506 (in Chinese) [涂俐兰, 刘红芳, 余乐 2013 物理学报 62 140506]
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[1] Havlin S, Kenett D Y, Ben-Jacob E, et al. 2012 Eur. Phys. J. Spec. Top. 214 273
[2] Feng A, Gao X Y, Guan J H, Huang S P, Liu Q 2017 Physica A 483 57
[3] Buldyrev S V, Parshani R, Paul G, Stanley H E, Havlin S 2010 Nature 464 1025
[4] Cardillo A, Zanin M, Gómez-Gardeñes J, et al. 2013 Eur. Phys. J. Spec. Top. 215 23
[5] Ang L M, Seng K P, Zungeru A M 2016 IJSIR 7 52
[6] Stasiuk A I, Hryshchuk R V, Goncharova L L 2017 Cybernet. Syst. Analysis 53 476
[7] Bauch C T, Galvani A P 2013 Science 342 47
[8] Chen W, Wu T, Li Z W, Wang L 2017 Physica A 479 542
[9] Um J, Minnhagen P, Kim B J 2011 Chaos 21 025106
[10] Lee K, Kim J, Lee S, et al. 2014 Multiplex networks// D'Agostino G, Scala A Networks of Networks: The Last Frontier of Complexity. (1st Ed.) (Berlin: Springer) pp3-36
[11] Albert R, Barabási A L 2002 Rev. Mod. Phys. 74 47
[12] Wang X F, Chen G 2002 IEEE Trans. Circuits Syst. I 49 54
[13] Wang X F, Li X, Chen G R 2006 Theory and Application of Complex Networks (Beijing: Tsinghua University Press) p7 (in Chinese) [汪小帆, 李翔, 陈关荣 2006 复杂网络理论及其应用(北京: 清华大学出版社) 第7页]
[14] Doyle J C, Alderson D L, Li L 2005 PNAS 102 14497
[15] Wang X F, Chen G R 2002 Physica A 310 521
[16] Kocarev L, Amato P 2005 Chaos 15 024101
[17] Zhou J, Chen T 2006 IEEE Trans. Circuits Syst. I 53 733
[18] Tu L L, Lu J A 2009 Comput. Math. Appl. 57 28
[19] Zhang Q J, Lu J A, Lv J H 2008 IEEE Trans. Circuits Syst. Ⅱ 55 183
[20] Liu J L 2013 Acta Phys. Sin. 62 040503 (in Chinese) [刘金良 2013 物理学报 62 040503]
[21] Liang Y, Wang X Y 2013 Acta Phys. Sin. 62 018901 (in Chinese) [梁义, 王兴元 2013 物理学报 62 018901]
[22] Wu W, Zhou W, Chen T 2009 IEEE Trans. Circuits Syst. I 56 829
[23] Ma J, Mi L, Zhou P, et al. 2017 Appl. Math. Comput. 307 321
[24] Liu J, Chen S H, Lu J A 2003 Acta Phys. Sin. 52 1595 (in Chinese) [刘杰, 陈士华, 陆君安 2003 物理学报 52 1595]
[25] Wong W K, Zhen B, Xu J, Wang Z 2012 Chaos 22 033146
[26] Rosenblum M G, Pikovsky A S, Kurth J 1997 Phys. Rev. Lett. 78 4193
[27] Zhang H G, Liu Z W, Huang G B, Wang Z S 2010 IEEE Trans. Neural. Netw. 21 91
[28] Zheng Y G, Bao L J 2017 Chaos. Soliton Fract. 98 145
[29] Yang S F, Guo Z Y, Wang J 2017 IEEE Trans. Neur. Net. Lear. 28 1657
[30] He W L, Chen G R, Han Q L, et al. 2017 IEEE Trans. Syst. Man. Cy-S. 47 1655
[31] Zhang X Y, Boccaletti S, Guan S G 2015 Phys. Rev. Lett. 114 038701
[32] Li Y, Wu X Q, Lu J A, L J H 2016 IEEE Trans. Circuits Syst. Ⅱ 63 206
[33] Xu Q, Zhuang S X, Hu D, Zeng Y F, Xiao J 2014 Abst. Appl. Anal. 10.1155 453149
[34] Boyd S, Ghaoui L E, Feron E, Balakrishnan V 1994 Linear Matrix Inequalities in System and Control Theory (Philadelphia: SIAM) pp7-14
[35] Tu L L, Liu H F, Yu L 2013 Acta Phys. Sin. 62 140506 (in Chinese) [涂俐兰, 刘红芳, 余乐 2013 物理学报 62 140506]
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