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SYNCHRONIZATION AND GRAPH TOPOLOGY

Laboratory of Nonlinear Systems, Swiss Federal Institute of Technology Lausanne (EPFL), EPFL-IC-ISC-LANOS, Station 14, 1015 Lausanne, Switzerland

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MARTIN HASLER

Laboratory of Nonlinear Systems, Swiss Federal Institute of Technology Lausanne (EPFL), EPFL-IC-ISC-LANOS, Station 14, 1015 Lausanne, Switzerland

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MENNO LAURET

Dynamics and Control Group, Department of Mechanical Engineering, Eindhoven University of Technology, PO Box 513 5600 MB Eindhoven, The Netherlands

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, and

HENK NIJMEIJER

Dynamics and Control Group, Department of Mechanical Engineering, Eindhoven University of Technology, PO Box 513 5600 MB Eindhoven, The Netherlands

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https://doi.org/10.1142/S0218127405014143Cited by:133 (Source: Crossref)

Abstract

This paper clarifies the relation between synchronization and graph topology. Applying the Connection Graph Stability method developed by Belykh et al. [2004a] to the study of synchronization in networks of coupled oscillators, we show which graph properties matter for synchronization. In particular, while we explicitly link the stability of synchronization with the average path length for a wide class of coupling graphs, we prove by a simple argument that the average path length is not always the crucial quantity for synchronization. We also show that synchronization in scale-free networks can be described by means of regular networks with a star-like coupling structure. Finally, by considering an example of coupled Hindmarsh–Rose neuron models, we demonstrate how global stability of synchronization depends on the parameters of the individual oscillator.

Dedicated to Leonid P. Shilnikov on the Occasion of his 70th Birthday.

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References

V. S. Afraimovich and L. P. Shilnikov, Dokl. Acad. Nauk USSR 219, 1981 (1974). Google Scholar
V. S. Afraimovich and L. P. Shilnikov, Prikladnaja Matematika i Mehanika 41, 618 (1977). Google Scholar
V. S. Afraimovich, N. N. Verichev and M. I. Rabinovich, Radiophys. Quant. Electron. 29, 795 (1986). Crossref, Google Scholar
V. S. Afraimovich and L. P. Shilnikov, Amer. Math. Soc. Transl. 149, 201 (1991). Google Scholar
A. A. Andronov and A. A. Vitt, Archiv für Elektrotechnik XXIV, 99 (1930). Google Scholar
A. L. Barabási and R. Albert, Science 286, 509 (1999). Crossref, Web of Science, Google Scholar
M. Barahona and L. M. Pecora, Phys. Rev. Lett. 89, 054101 (2002). Crossref, Web of Science, Google Scholar
V. N. Belykhet al., Chua's Circuit: A Paradigm for Chaos, ed. R. N. Madan (World Scientific, Singapore, 1993) pp. 325–335. Link, Google Scholar
V. N. Belykh, I. V. Belykh and M. Hasler, Phys. Rev. E 62, 6332 (2000). Crossref, Web of Science, Google Scholar
I. V. Belykhet al., Chaos 13, 165 (2003). Crossref, Web of Science, Google Scholar
V. N. Belykh, I. V. Belykh and M. Hasler, Physica D 195, 159 (2004). Crossref, Web of Science, Google Scholar
I. V. Belykh, V. N. Belykh and M. Hasler, Physica D 195, 188 (2004). Crossref, Web of Science, Google Scholar
M. L. Cartwright and J. E. Littlewood, J. Lond. Math. Soc. 20, 180 (1945). Google Scholar
H. Fujisaka and T. Yamada, Progr. Theor. Phys. 69, 32 (1983). Crossref, Web of Science, Google Scholar
S. Guattery and G. L. Miller, SIAM J. Matrix Anal. Appl. 21, 703 (2000). Crossref, Web of Science, Google Scholar
J. L. Hindmarsh and R. M. Rose, Proc. Roy. Soc. London B 221, 87 (1984). Crossref, Web of Science, Google Scholar
R. Huertaet al., Phys. Rev. E 55, 2108 (1997). Crossref, Web of Science, Google Scholar
J. Kurthset al. (eds.), Chaos 13, (2003). Google Scholar
A. G. Maier, Tech. Phys. USSR 2(5), 1 (1935). Web of Science, Google Scholar
T. Nishikawaet al., Phys. Rev. Lett. 91, 014101 (2003). Crossref, Web of Science, Google Scholar
L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 64, 821 (1990). Crossref, Web of Science, Google Scholar
L. M. Pecoraet al., Chaos 7, 520 (1997). Crossref, Web of Science, Google Scholar
L. M. Pecora, Phys. Rev. E 58, 347 (1998). Crossref, Web of Science, Google Scholar
L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 80, 2109 (1998). Crossref, Web of Science, Google Scholar
A. Pikovsky , M. Rosenblum and J. Kurths , Synchronization: A Universal Concept in Nonlinear Science ( Cambridge University Press , Cambridge , 2001 ) . Crossref, Google Scholar
A. Yu. Pogromsky and H. Nijmeijer, IEEE Trans. Circuits Syst.-I 48, 152 (2001). Crossref, Web of Science, Google Scholar
L. P. Shilnikov et al. , Methods of Qualitative Theory in Nonlinear Dynamics. (Part I) , World Scientific Series on Nonlinear Science, Series A 4 ( 1998 ) . Crossref, Google Scholar
L. P. Shilnikov et al. , Methods of Qualitative Theory in Nonlinear Dynamics. (Part II) , World Scientific Series on Nonlinear Science, Series A 5 ( 2001 ) . Crossref, Google Scholar
A. L. Shilnikov, L. P. Shilnikov and D. V. Turaev, Int. J. Bifurcation and Chaos 14, 2143 (2004). Link, Web of Science, Google Scholar
S. H. Strogatz, Nature 410, 268 (2001). Crossref, Web of Science, Google Scholar
B. Van der Pol, London, Edinburgh and Dublin Phil. Mag. 3, 65 (1927). Crossref, Google Scholar
X. F. Wang and G. Chen, IEEE Trans. Circuits Syst.-I 49, 54 (2002). Crossref, Web of Science, Google Scholar
C. W. Wu and L. O. Chua, IEEE Trans. Circuits Syst.-I 43, 161 (1996). Web of Science, Google Scholar
C. W. Wu , Synchronization in Coupled Chaotic Circuits and Systems , World Scientific Series on Nonlinear Science, Series A 41 ( 2002 ) . Crossref, Google Scholar