×

zbMATH — the first resource for mathematics

Adaptive synchronization in nonlinearly coupled dynamical networks. (English) Zbl 1154.93424
Summary: Recently, it has been demonstrated that many large-scale complex dynamical networks display a collective synchronization motion. In this paper, synchronization in nonlinearly coupled dynamical networks is studied. By using the invariance principle of differential equations, some simple linear feedback controllers with dynamical updated strengths are constructed to make the dynamical network synchronize with an isolate node. The feedback strength can be automatically enhanced to make the dynamical network collectively synchronized. The structure of the network can be random, regular, small-world, or scale-free. A numerical example is given to demonstrate the validity of the proposed method, in which the famous Lorenz system is chosen as the node of the network.

MSC:
93D21 Adaptive or robust stabilization
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Watts, D.J.; Strogatz, S.H., Collective dynamics of ‘small-world’ networks, Nature, 393, 6684, 409-410, (1998)
[2] Barabási, A.L.; Albert, R., Emergence of scaling in random networks, Science, 286, 5439, 509-512, (1999) · Zbl 1226.05223
[3] Boccaletti, S.; Latora, V.; Moreno, Y.; Chavez, M.; Hwang, D.U., Complex networks: structure and dynamics, Phys rep, 424, 4-5, 175-308, (2006) · Zbl 1371.82002
[4] Strogatz, S.H., Exploring complex networks, Nature, 410, 268-276, (2001) · Zbl 1370.90052
[5] Wang, X.F., Complex networks: topology, dynamics and synchronization, Int J bifurcat chaos, 12, 5, 885-916, (2002) · Zbl 1044.37561
[6] Lü, J.; Yu, X.; Chen, G., Chaos synchronization of general complex dynamical networks, Phys A: statist mech applicat, 334, 1-2, 281-302, (2004)
[7] Wang, X.F.; Chen, G., Synchronization in scale-free dynamical networks: robustness and fragility, IEEE trans circuits systems - I, 49, 1, 54-62, (2002) · Zbl 1368.93576
[8] Timme, M.; Wolf, F.; Geisel, T., Topological speed limits to network synchronization, Phys rev lett, 92, 7, 74101, (2004)
[9] Boccaletti, S.; Kurths, J.; Osipov, G.; Valladares, D.L.; Zhou, C.S., The synchronization of chaotic systems, Phys rep, 366, 1, 1-101, (2002) · Zbl 0995.37022
[10] Pecora, L.M.; Carroll, T.L., Synchronization in chaotic systems, Phys rev lett, 64, 8, 821-824, (1990) · Zbl 0938.37019
[11] Lu, J.; Cao, J., Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters, Chaos, 15, 4, 43901-43910, (2005)
[12] Rosenblum, M.G.; Pikovsky, A.S.; Kurths, J., Phase synchronization of chaotic oscillators, Phys rev lett, 76, 11, 1804-1807, (1996)
[13] Rulkov, N.F.; Sushchik, M.M.; Tsimring, L.S.; Abarbanel, H.D.I., Generalized synchronization of chaos in directionally coupled chaotic systems, Phys rev E, 51, 2, 980-994, (1995)
[14] Rosenblum, M.G.; Pikovsky, A.S.; Kurths, J., From phase to lag synchronization in coupled chaotic oscillators, Phys rev lett, 78, 22, 4193-4196, (1997)
[15] Wu, C.W.; Chua, L.O., Synchronization in an array of linearly coupled dynamical systems, IEEE trans circuits systems - I, 42, 8, 430-447, (1995) · Zbl 0867.93042
[16] Chen, G.; Zhou, J.; Liu, Z., Global synchronization of coupled delayed neural networks and applications to chaotic CNN models, Int J bifurcat chaos, 14, 7, 2229-2240, (2004) · Zbl 1077.37506
[17] Li, C.; Chen, G., Synchronization in general complex dynamical networks with coupling delays, Phys A: statist mech applicat, 343, 263-278, (2004)
[18] Zhou, J.; Lu, J., Adaptive synchronization of an uncertain complex dynamical network, IEEE trans automat control, 51, 4, 652-656, (2006) · Zbl 1366.93544
[19] Cao, J.; Li, P.; Wang, W., Global synchronization in arrays of delayed neural networks with constant and delayed coupling, Phys lett A, 353, 4, 318-325, (2006)
[20] Wang, W.; Cao, J., Synchronization in an array of linearly coupled networks with time-varying delay, Phys A: statist mech applicat, 366, 197-211, (2006)
[21] Song, Qiankun; Cao, Jinde, Synchronization and anti-synchronization for chaotic systems, Chaos, solitons & fractals, 33, 2, 929-939, (2007) · Zbl 1133.37313
[22] Cao, Jinde; Ho, Daniel W.C., A general framework for global asymptotic stability analysis of delayed neural networks based on LMI approach, Chaos, solitons & fractals, 24, 5, 1317-1329, (2005) · Zbl 1072.92004
[23] Lu, Jianquan; Ho Daniel, W.C., Local and global synchronization in general complex dynamical networks with delay coupling, Chaos, solitons & fractals, 37, 5, 1497-1510, (2008) · Zbl 1142.93426
[24] Cao, Jinde; Li, H.X.; Ho, Daniel W.C., Synchronization criteria of lure systems with time-delay feedback control, Chaos, solitons & fractals, 23, 4, 1285-1298, (2005) · Zbl 1086.93050
[25] Wang, X.F.; Chen, G., Synchronization in small-world dynamical networks, Int J bifurcat chaos, 12, 1, 187-192, (2002)
[26] LaSalle, J.P., The stability of dynamical systems, (1976), Society for Industrial and Applied Mathematics Philadelphia · Zbl 0364.93002
[27] Pecora, L.M.; Carroll, T.L., Master stability functions for synchronized coupled systems, Phys rev lett, 80, 10, 2109-2112, (1998)
[28] Lorenz, E.N., Deterministic nonperiodic flow, J atmos sci, 20, 130-141, (1963) · Zbl 1417.37129
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.