Abstract.
Complexity of dynamical networks can arise not only from the complexity of the topological structure but also from the time evolution of the topology. In this paper, we study the synchronous motion of coupled maps in time-varying complex networks both analytically and numerically. The temporal variation is rather general and formalized as being driven by a metric dynamical system. Four network models are discussed in detail in which the interconnections between vertices vary through time randomly. These models are: 1) i.i.d. sequences of random graphs with fixed wiring probability, 2) groups of graphs with random switches between the individual graphs, 3) graphs with temporary random failures of nodes, and 4) the meet-for-dinner model where the vertices are randomly grouped. We show that the temporal variation and randomness of the connection topology can enhance synchronizability in many cases; however, there are also instances where they reduce synchronizability. In analytical terms, the Hajnal diameter of the coupling matrix sequence is presented as a measure for the synchronizability of the graph topology. In topological terms, the decisive criterion for synchronization of coupled chaotic maps is that the union of the time-varying graphs contains a spanning tree.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Jost, M.P. Joy, Phys. Rev. E 65, 016201 (2001); Y.H. Chen, G. Rangarajan, M. Ding, Phys. Rev. E 67, 026209 (2003); W. Lu, T. Chen, Physica D 198, 148 (2004); R.E. Amritkar, S. Jalan, C.-K. Hu, Phys. Rev. E 72, 016212 (2005)
K. Kaneko, Prog. Theor. Phys. 74, 1033 (1985); T. Bohr, O.B. Christensen, Phys. Rev. Lett. 63, 2161 (1989); K. Kaneko, Theory and Applications of Coupled Map Lattices (Wiley, New York, 1993)
A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A universal concept in nonlinear sciences (Cambridge University Press, 2001)
S. Morita, Phys. Rev. E 58, 4401 (1998); A.M. Batista, S.E. de. S. Pinto, R.L. Viana, S.R. Lopes, Phys. Rev. E 65, 056209 (2002)
T. Shinbrot, Phys. Rev. E 50, 3230 (1994); Y. Jiang, P. Parmananda, Phys. Rev. E 57, 4135 (1998); P.M. Gade, C.-K. Hu, Phys. Rev. E 65, 6409 (2000)
L.M. Pecora, T.L. Caroll, Phys. Rev. Lett. 80, 2109 (1998); G. Rangarajan, M. Ding, Phys. Lett. A 296, 204 (2002); F.M. Atay, T. Biyikoglu, J. Jost, Physica D 224, 35 (2006)
C.W. Wu, Nonlinearity 18 1057 (2005); W. Lu, T. Chen, IEEE T. CAS:-II 54:2, 136 (2007)
R. Olfati-Saber, R.M. Murray, IEEE T. Autom. Control 49:9, 1520 (2004)
Y. Hatano, M. Mesbahi, IEEE Trans. Autom. Control 50:11, 1867 (2004)
T. Vicsek, A. Czirok, E.B. Jacob, I. Cohen, O. Schochet, Phys. Rev. Lett. 75, 1226 (1995); A. Jadbabaie, J. Lin, A.S. Morse, IEEE T. Autom. Control 48, 988 (2003).
L. Moreau, IEEE Trans. Autom. Control 50:2, 169 (2005)
J.H. Lü, G. Chen, IEEE Trans. Autom. Control 50, 841 (2005); I.V. Belykh, V.N. Belykh, M. Hasler, Physica D 195, 159 (2004); D.J. Stilwell, E.M. Bollt, D.G. Roberson, SIAM J. Appl. Dyn. Syst. 5:1, 140 (2006); S. Boccaletti, D.-U. Hwang, M. Chavez, A. Amann, J. Kurths, L.M. Pecora, Phys. Rev. E 74, 016102 (2006)
J. Hajnal, Proc. Camb. Phil. Soc. 54, 233 (1958); J. Hajnal, Proc. Camb. Phil. Soc. 52, 67 (1956)
W. Lu, F.M. Atay, J. Jost, SIAM Journal on Math. Anal. 39 4, 1231 (2007)
W. Lu, F.M. Atay, J. Jost, Synchronization in coupled map networks with uncertain topologies (in preparation)
L. Arnold, Random Dynamical Systems (Springer-Verlag, Heidelberg, 1998); M. Dunford, J.T. Schwartz, Linear Operators (Interscience, New York, 1958)
P. Ashwin, J. Buescu, I. Stewart, Nonlinearity 9, 703 (1996)
J. Milnor, Commun. Math. Phys. 99, 177 (1985)
L. Arnold, J. Diff. Equ. 177, 235 (2001)
J. Shen, Wavelet Anal. Multi. Meth. LNPAM 212, 341 (2000)
Y. Fang, Ph.D. thesis (Case Western Reserve University, 1994)
P. Erdös, A. Rényi, Publ. Math. Debrecen 6, 290 (1959)
R. Albert, H. Jeong, A.-L. Barabási, Nature 406:27, 378 (2000)
A.-L. Barabási, R. Albert, Science 286, 509 (1999)
H. Hong, B.J. Kim, M.Y. Choi, H. Park, Phys. Rev. E 69, 067105 (2004)
W. Ren, R.W. Beard, T.W. McLain, Springer-Verlag Series: LNCIS 309, 171 (2004)
F.M. Atay, T. Biyikoglu, Phys. Rev. E 72, 016217 (2005); F.M. Atay, T. Biyikoglu, J. Jost, IEEE Trans. Circ. Syst.-I 53:1, 92 (2006); F.M. Atay, T. Biyikoglu, J. Jost, Physica D 224, 35 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Lu, W., Atay, F. & Jost, J. Chaos synchronization in networks of coupled maps with time-varying topologies. Eur. Phys. J. B 63, 399–406 (2008). https://doi.org/10.1140/epjb/e2008-00023-3
Received:
Published:
Issue date:
DOI: https://doi.org/10.1140/epjb/e2008-00023-3