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)
Emergence of synchronization in complex networks of interacting dynamical systems. (English) Zbl 1130.37347
Summary: We study the emergence of coherence in large complex networks of interacting heterogeneous dynamical systems. We show that for a large class of dynamical systems and network topologies there is a critical coupling strength at which the systems undergo a transition from incoherent to coherent behavior. We find that the critical coupling strength at which this transition takes place is $k_{c}=(Z\lambda )^{ - 1}$, where Z depends only on the uncoupled dynamics of the individual systems on each node, while $\lambda$ is the largest eigenvalue of the network adjacency matrix. Thus we achieve a separation of the problem into two parts, one depending solely on the node dynamics, and one depending solely on network topology.

37C99Smooth dynamical systems
94C99Circuits, networks
Full Text: DOI
[1] Newman, M. E. J.: SIAM rev.. 45, 167 (2003)
[2] Barabási, A. -L.; Albert, R.: Rev. modern phys.. 74, 47 (2002)
[3] Pikovsky, A.; Rosenblum, M. G.; Kurths, J.: Synchronization: A universal concept in nonlinear sciences. (2001) · Zbl 0993.37002
[4] Mosekilde, E.; Maistrenko, Y.; Postnov, D.: Chaotic synchronization: applications to living systems. (2002) · Zbl 0999.37022
[5] Kuramoto, Y.: Chemical oscillations, waves, and turbulence. (1984) · Zbl 0558.76051
[6] Ott, E.: Chaos in dynamical systems. (2002) · Zbl 1006.37001
[7] Restrepo, J. G.; Ott, E.; Hunt, B. R.: Phys. rev. E. 71, 036151 (2005)
[8] Restrepo, J. G.; Ott, E.; Hunt, B. R.: Chaos. 16, 015107 (2006)
[9] Ichinomiya, T.: Phys. rev. E. 70, 026116 (2004)
[10] Lee, D. -S.: Phys. rev. E. 72, 026208 (2005)
[11] Ichinomiya, T.: Phys. rev. E. 72, 016109 (2005)
[12] A. Jadbabaie, N. Motee, M. Barahona, Proceedings of the American Control Conference, ACC 2004
[13] Moreno, Y.; Pacheco, A. E.: Europhys. lett.. 68, 603 (2004)
[14] Pikovsky, A.; Rosenblum, M. G.; Kurths, J.: Europhys. lett.. 34, 165 (1996)
[15] Sakaguchi, H.: Phys. rev. E. 61, 7212 (2000)
[16] Topaj, D.; Kye, W. -H.; Pikovsky, A.: Phys. rev. Lett.. 87, 074101 (2001)
[17] Baek, S. -J.; Ott, E.: Phys. rev. E. 69, 066210 (2004)
[18] Ott, E.; So, P.; Barreto, E.; Antonsen, T.: Physica D. 173, 29 (2002)
[19] J.G. Restrepo, E. Ott, B.R. Hunt, cond-mat/0601639 eprint
[20] Maccluer, C. R.: SIAM rev.. 42, 487 (2000)
[21] Pecora, L. M.; Carroll, T. L.: Phys. rev. Lett.. 80, 2109 (1998)
[22] Chung, F.; Lu, L.; Vu, V.: Proc. natl. Acad. sci.. 100, 6313 (2003)
[23] Y. Wang, et al., Epidemic spreading in real networks: An eigenvalue viewpoint, in: 22nd Symposium On Reliable Distributed Computing, SRDS2003, 6--8 October, Florence, Italy, 2003
[24] Boguñá, M.; Serrano, M. A.: Phys. rev. E. 72, 016106 (2005)
[25] M. Brede, S. Sinha, cond-mat/0507710 eprint
[26] Cvetkovic, D.; Rowlinson, P.: Linear multilinear algebra. 28, 3 (1990)