zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Exponential synchronization of weighted general delay coupled and non-delay coupled dynamical networks. (English) Zbl 1205.93124
Summary: Time delays commonly exist in the real world. In the present work, we investigate the exponential synchronization of weighted general delay coupled and non-delay coupled complex dynamical networks with different topological structures. Based on the Lyapunov stability theory, the suitable controllers are designed to make the controlled dynamical network exponentially synchronize an isolated node with any pre-specified exponential convergence rate, and proved theoretically. The synchronization scheme is applicable to the undirected networks as well as the directed ones. The derived controllers are simple and can be readily used in practical applications. Furthermore, the coupling matrix is not necessary to be irreducible and the network node dynamics need not satisfy the very strong and conservative uniform Lipschitz condition. Numerical simulations further validate the effectiveness and feasibility of our synchronization method.

93D15Stabilization of systems by feedback
37N35Dynamical systems in control
93C23Systems governed by functional-differential equations
Full Text: DOI
[1] Watts, D. J.; Strogatz, S. H.: Collective dynamics of small-world networks, Nature 393, No. 6684, 409-410 (1998)
[2] Barabási, A. L.; Albert, R.: Emergence of scaling in random networks, Science 286, No. 5439, 509-512 (1999) · Zbl 1226.05223 · doi:10.1126/science.286.5439.509
[3] Strogatz, S. H.: Exploring complex networks, Nature 410, No. 6825, 268-276 (2001)
[4] Albert, R.; Barabási, A. L.: Statistical mechanics of complex networks, Reviews of modern physics 74, No. 1, 47-97 (2002) · Zbl 1205.82086 · doi:10.1103/RevModPhys.74.47
[5] Lü, J.; Yu, X. H.; Chen, G.; Cheng, D. Z.: Characterizing the synchronizability of small-world dynamical networks, IEEE transactions on circuits and systems I 51, No. 4, 787-796 (2004)
[6] Boccaleti, S.; Latora, V.; Moreno, Y.; Chavez, M.; Hwang, D. U.: Complex networks: structure and dynamics, Physics reports 424, No. 4--5, 175-308 (2006)
[7] Kaneko, K.: Theory and applications of coupled map lattices, (1993) · Zbl 0777.00014
[8] Gelover-Santiago, A. L.; Lima, R.; Matinez-Mekler, G.: Synchronization and cluster periodic solutions in globally coupled maps, Physica A 283, No. 1--2, 131-135 (2000) · Zbl 1090.37517
[9] Gu, Y. Q.; Shao, C.; Fu, X. C.: Complete synchronization and stability of star-shaped complex networks, Chaos, solitons fractals 28, No. 2, 480-488 (2006) · Zbl 1083.37025 · doi:10.1016/j.chaos.2005.07.002
[10] Wang, X. F.; Chen, G.: Synchronization in small-world dynamical networks, International journal of bifurcation and chaos 12, No. 1, 187-192 (2002)
[11] Wang, X. F.; Chen, G.: Synchronization in scale-free dynamical networks: robustness and fragility, IEEE transactions on circuits and systems I 49, No. 1, 54-62 (2002)
[12] Li, X.; Wang, X.; Chen, G.: Pinning a complex dynamical network to its equilibrium, IEEE transactions on circuits and systems I 51, No. 3, 2074-2087 (2004)
[13] Lü, J.; Chen, G.: A time-varying complex dynamical network models and its controlled synchronization criteria, IEEE transactions on automatic control 50, No. 6, 841-846 (2005)
[14] Wu, C. W.: Synchronization and convergence of linear dynamics in random directed networks, IEEE transactions on automatic control 51, No. 7, 1207-1210 (2006)
[15] Yu, W.; Cao, J.; Lü, J.: Global synchronization of linearly hybrid coupled networks with time-varying delay, SIAM journal on applied dynamical systems 7, No. 1, 108-133 (2008) · Zbl 1161.94011 · doi:10.1137/070679090
[16] Cao, J.; Li, P.; Wang, W.: Global synchronization in arrays of delayed neural networks with constant and delayed coupling, Physics letters A 353, No. 4, 318-325 (2006)
[17] He, G.; Yang, J.: Adaptive synchronization in nonlinearly coupled dynamical networks, Chaos, solitons fractals 38, No. 5, 1254-1259 (2008) · Zbl 1154.93424 · doi:10.1016/j.chaos.2007.07.067
[18] Arenas, A.; Diaz-Guilera, A.; Kurths, J.; Moreno, Y.; Zhou, C.: Synchronization in complex networks, Physics reports 469, No. 1, 93-153 (2008)
[19] Yu, W.; Chen, G.; Lü, J.: On pinning synchronization of complex dynamical networks, Automatica 45, No. 2, 429-435 (2009) · Zbl 1158.93308 · doi:10.1016/j.automatica.2008.07.016
[20] Song, Q.; Cao, J.; Liu, F.: Synchronization of complex dynamical networks with nonidentical nodes, Physics letters A 374, No. 4, 544-551 (2010) · Zbl 1234.05218
[21] Wu, X.; Zheng, W.; Zhou, J.: Generalized outer synchronization between complex dynamical networks, Chaos 19, No. 1, 013109 (2009) · Zbl 1311.34119
[22] Dhamala, M.; Jirsa, V. K.; Ding, M.: Enhancement of neural synchrony by time delay, Physical review letters 92, No. 7, 074104 (2004)
[23] Thomas, M.; Morari, M.: Delay effects on stability: A robust control approach, Lecture notes in control and information science 269 (2001)
[24] Xu, X.; Chen, Z.; Si, G.; Hu, X.; Luo, P.: A novel definition of generalized synchronization on networks and a numerical simulation example, Computers mathematics with applications 56, No. 11, 2789-2794 (2008) · Zbl 1165.34379 · doi:10.1016/j.camwa.2008.07.036
[25] Liu, Z. X.; Chen, Z. Q.; Yuan, Z. Z.: Pinning control of weighted general complex dynamical networks with time delay, Physica A 375, No. 1, 345-354 (2007)
[26] Wen, S.; Chen, S.; Guo, W.: Adaptive global synchronization of a general complex dynamical network with non-delayed and delayed coupling, Physics letters A 372, No. 42, 6340-6346 (2008) · Zbl 1225.05223 · doi:10.1016/j.physleta.2008.08.059
[27] Wang, L.; Dai, H. P.; Dong, H.; Shen, Y. H.; Sun, Y. X.: Adaptive synchronization of weighted complex dynamical networks with coupling time-varying delays, Physics letters A 372, No. 20, 3632-3639 (2008) · Zbl 1220.90041 · doi:10.1016/j.physleta.2008.02.010
[28] Wu, C. W.: Synchronization in networks of nonlinear dynamical systems coupled via a directed graph, Nonlinearity 18, No. 3, 1057-1064 (2005) · Zbl 1089.37024 · doi:10.1088/0951-7715/18/3/007
[29] Li, P.; Yi, Z.: Synchronization analysis of delayed complex networks with time-varying couplings, Physica A 387, No. 14, 3729-3737 (2008)
[30] Lou, X.; Cui, B.: Synchronization of neural networks based on parameter identification and via output or state coupling, Journal of computational and applied mathematics 222, No. 2, 440-457 (2008) · Zbl 1168.65041 · doi:10.1016/j.cam.2007.11.015
[31] Lu, J.; Cao, J.: Adaptive synchronization of uncertain dynamical networks with delayed coupling, Nonlinear dynamics 53, No. 1--2, 107-115 (2008) · Zbl 1182.92007 · doi:10.1007/s11071-007-9299-x
[32] Xiang, L. Y.; Liu, Z. X.; Chen, Z. Q.; Chen, F.; Yuan, Z. Z.: Pinning control of complex dynamical networks with general topology, Physica A 379, No. 1, 298-306 (2007)
[33] Chen, G.; Ueta, T.: Yet another chaotic attractor, International journal of bifurcation and chaos 9, No. 7, 1465-1466 (1999) · Zbl 0962.37013 · doi:10.1142/S0218127499001024