Cluster synchronization of nonlinearly-coupled complex networks with nonidentical nodes and asymmetrical coupling matrix. (English) Zbl 1242.93009

Summary: In this paper, we investigate the cluster synchronization problem for networks with nonlinearly coupled non-identical dynamical systems and asymmetrical coupling matrix by using pinning control. We derive sufficient conditions for cluster synchronization for any initial values through a feedback scheme and propose an adaptive feedback algorithm that adjusts the coupling strength. Some numerical examples are then given to illustrate the theoretical results.


93A15 Large-scale systems
93C10 Nonlinear systems in control theory
93B52 Feedback control
Full Text: DOI


[1] Strogatz, S.H., Stewart, I.: Coupled oscillators and biological synchronization. Sci. Am. 269(6), 102–109 (1993)
[2] Gray, C.M.: Synchronous oscillations in neural systems. J. Comput. Neurosci. 1, 11–38 (1994)
[3] Glass, L.: Synchronization and rhythmic processes in physiology. Nature 410(6825), 277–284 (2001)
[4] Vieira, M.D.: Chaos and synchronized chaos in an earthquake model. Phys. Rev. Lett. 82(1), 201–204 (1999)
[5] Kunbert, L., Agladze, K.I., Krinsky, V.I.: Image processing using light-sensitive chemical waves. Nature 337, 244–247 (1989)
[6] Wang, S.H., Kuang, J.Y., Li, J.H., Luo, Y.L., Lu, H.P., Hu, G.: Chaos-based secure communications in a large community. Phys. Rev. E 66, 065202(R) (2002)
[7] Yu, D.C., Righero, M., Kocarev, L.: Estimating topology of networks. Phys. Rev. Lett. 97, 188701 (2006)
[8] Boccaletti, S., Kurths, J., Osipov, G., Valladares, D.L., Zhou, C.S.: The synchronization of chaotic systems. Phys. Rep. 366, 1–101 (2002) · Zbl 0995.37022
[9] Zheng, Z.G., Hu, G.: Generalized synchronization versus phase synchronization. Phys. Rev. E 62, 7882–7885 (2000)
[10] Rosenblum, M.G., Pikovsky, A.S., Kurths, J.: From phase to lag synchronization in coupled chaotic oscillators. Phys. Rev. Lett. 78, 4193–4196 (1997) · Zbl 0896.60090
[11] Belykh, V.N., Belykh, I.V., Mosekilde, E.: Cluster synchronization modes in an ensemble of coupled chaotic oscillators. Phys. Rev. E 63, 036216 (2001) · Zbl 1097.37055
[12] Rosenblum, M.G., Pikovsky, A., Kurths, J.: Phase synchronization of chaotic oscillators. Phys. Rev. Lett. 76, 1804–1807 (1996)
[13] Vreeswijk, C.: Partial synchronization in populations of pulse-coupled oscillators. Phys. Rev. E 54, 5522–5537 (1996)
[14] Kaneko, K.: Relevance of dynamic clustering to biological networks. Physica D 75, 55–73 (1994) · Zbl 0859.92001
[15] Yoshioka, M.: Cluster synchronization in an ensemble of neurons interacting through chemical synapses. Phys. Rev. E 71, 061914 (2005)
[16] Wu, W., Zhou, W.J., Chen, T.P.: Cluster synchronization of linearly coupled complex networks under pinning control. IEEE Trans. Circuits Syst. I, Regul. Pap. 56(4), 829–839 (2009)
[17] Ma, Z.J., Liu, Z.R., Zhang, G.: A new method to realize cluster synchronization in connected chaotic networks. Chaos 16, 023103 (2006) · Zbl 1146.37330
[18] Lu, W.L., Liu, B., Chen, T.P.: Cluster synchronization in networks of coupled noidentical dynamical system. Chaos 20, 013120 (2010)
[19] Lu, W.L., Liu, B., Chen, T.P.: Cluster synchronization in networks of distinct groups of maps. Eur. Phys. J. B 77(2), 257–264 (2010)
[20] Wang, K.H., Fu, X.C., Li, K.Z.: Cluster synchronization in community networks with nonidentical nodes. Chaos 19, 023106 (2009) · Zbl 1309.34107
[21] Liu, X., Chen, T.: Synchronization of identical neural networks and other systems with an adaptive coupling strength. Int. J. Circuit Theory Appl. 38, 631–648 (2010) · Zbl 1202.94231
[22] Chen, T.P., Liu, X.W., Lu, W.L.: Pinning complex networks by a single controller. IEEE Trans. Circuits Syst. I, Regul. Pap. 54(6), 1317–1326 (2007) · Zbl 1374.93297
[23] Guo, W.L., Austin, F., Chen, S.H.: Global synchronization of nonlinearly coupled complex networks with non-delayed and delayed coupling. Commun. Nonlinear Sci. Numer. Simul. 15, 1631–1639 (2010) · Zbl 1221.34213
[24] Liu, X.W., Chen, T.P.: Synchronization analysis for nonlinearly-coupled complex networks with an asymmetrical coupling matrix. Physica A 387, 4429–4439 (2007)
[25] Li, K.Z., Small, M., Fu, X.C.: Generation of clusters in complex dynamical networks via pinning control. J. Phys. A, Math. Theor. 41, 505101 (2008) · Zbl 1152.93002
[26] Li, D.M., Lu, J.A., Wu, X.Q., Chen, G.R.: Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system. J. Math. Anal. Appl. 323, 844–853 (2006) · Zbl 1104.37024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.