Belykh, Igor; Hasler, Martin; Lauret, Menno; Nijmeijer, Henk Synchronization and graph topology. (English) Zbl 1107.34047 Int. J. Bifurcation Chaos Appl. Sci. Eng. 15, No. 11, 3423-3433 (2005). The authors consider a network of \(n\) identical oscillators that are linearly coupled \[ \dot x_i = F(x_i) + \sum_{j=1}^{n} \varepsilon_{ij}(t) P x_j, \quad j=1,\dots,n, \] where \(x_i\) is the vector containing the coordinates of the \(i\)th oscillator, the matrix \(P\) determines which variables couple the oscillators, and \((\varepsilon_{ij})\) is an \(n\times n\) symmetric coupling matrix.The main goal of the paper is to clarify the relation between synchronization and the topology of the considered network. In particular, the authors explicitly link the stability of the synchronization with the average path length of the graph, which corresponds to the network. Finally, the example of coupled Hindmarsh-Rose neuron models is considered. Reviewer: Sergiy Yanchuk (Berlin) Cited in 44 Documents MSC: 34D05 Asymptotic properties of solutions to ordinary differential equations 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 92B20 Neural networks for/in biological studies, artificial life and related topics 94C15 Applications of graph theory to circuits and networks 34D20 Stability of solutions to ordinary differential equations Keywords:synchronization; graph topology; connection graph stability method; Hindmarsh-Rose model; network PDF BibTeX XML Cite \textit{I. Belykh} et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 15, No. 11, 3423--3433 (2005; Zbl 1107.34047) Full Text: DOI References: [1] Afraimovich V. S., Dokl. Acad. Nauk USSR 219 pp 1981– [2] Afraimovich V. S., Prikladnaja Matematika i Mehanika 41 pp 618– [3] DOI: 10.1007/BF01034476 [4] Afraimovich V. S., Amer. Math. Soc. Transl. 149 pp 201– · Zbl 0751.58024 [5] Andronov A. A., Archiv für Elektrotechnik pp 99– [6] Barabási A. L., Science 286 pp 509– · Zbl 1353.94001 [7] DOI: 10.1103/PhysRevLett.89.054101 [8] DOI: 10.1142/9789812798855_0014 [9] DOI: 10.1103/PhysRevE.62.6332 [10] DOI: 10.1063/1.1514202 · Zbl 1080.37525 [11] DOI: 10.1016/j.physd.2004.03.012 · Zbl 1098.82622 [12] DOI: 10.1016/j.physd.2004.03.013 · Zbl 1098.82621 [13] Cartwright M. L., J. Lond. Math. Soc. 20 pp 180– [14] DOI: 10.1143/PTP.69.32 · Zbl 1171.70306 [15] DOI: 10.1137/S0895479897329825 · Zbl 0942.05040 [16] DOI: 10.1098/rspb.1984.0024 [17] DOI: 10.1103/PhysRevE.55.R2108 [18] Kurths J., Chaos 13 [19] Maier A. G., Tech. Phys. USSR 2 pp 1– [20] DOI: 10.1103/PhysRevLett.91.014101 [21] DOI: 10.1103/PhysRevLett.64.821 · Zbl 0938.37019 [22] DOI: 10.1063/1.166278 · Zbl 0933.37030 [23] DOI: 10.1103/PhysRevE.58.347 [24] DOI: 10.1103/PhysRevLett.80.2109 [25] DOI: 10.1017/CBO9780511755743 [26] DOI: 10.1109/81.904879 · Zbl 0994.82065 [27] DOI: 10.1142/9789812798596 [28] DOI: 10.1142/9789812798558 [29] DOI: 10.1142/S0218127404010539 · Zbl 1077.37509 [30] DOI: 10.1038/35065725 · Zbl 1370.90052 [31] DOI: 10.1080/14786440108564176 [32] DOI: 10.1109/81.974874 · Zbl 1368.93576 [33] Wu C. W., IEEE Trans. Circuits Syst.-I 43 pp 161– [34] DOI: 10.1142/9789812778420 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.