Gilli, Marco; Bonnin, Michele; Corinto, Fernando On global dynamic behavior of weakly connected oscillatory networks. (English) Zbl 1089.34033 Int. J. Bifurcation Chaos Appl. Sci. Eng. 15, No. 4, 1377-1393 (2005). Summary: The paper considers weakly coupled oscillatory networks \[ \dot X = F_i(X_i)+\varepsilon G_i ({\mathbf X}), \quad {\mathbf X}=[X_1^T,\dots,X_n^T]^T, \] where \(F_i:\mathbb{R}^m\to \mathbb{R}^m,\) \(G_i: \mathbb{R}^{m\times n}\to \mathbb{R}^m\), \(1\leq i \leq n\) and \(\varepsilon\) is a small coupling parameter.Using Malkin’s theorem and a first harmonic approximation, the authors derive analytically equations for the phase dynamics. Analysing the phase equations, the total number of limit cycles and their stability are estimated. Although, the paper is focused on the one-dimensional array of coupled Chua oscillators, the proposed technique is general and can be applied to others complex networks. Reviewer: Sergiy Yanchuk (Berlin) Cited in 5 Documents MSC: 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 34C25 Periodic solutions to ordinary differential equations 34D20 Stability of solutions to ordinary differential equations 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms 92B20 Neural networks for/in biological studies, artificial life and related topics Keywords:weakly coupled networks; Malkin’s theorem; phase equations; Kuramoto system; Chua’s system PDFBibTeX XMLCite \textit{M. Gilli} et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 15, No. 4, 1377--1393 (2005; Zbl 1089.34033) Full Text: DOI References: [1] DOI: 10.1109/31.7600 · Zbl 0663.94022 · doi:10.1109/31.7600 [2] DOI: 10.1109/31.7601 · doi:10.1109/31.7601 [3] DOI: 10.1109/81.222795 · Zbl 0800.92041 · doi:10.1109/81.222795 [4] Chua L. O., IEEE Trans. Circuits Syst.-I 40 pp 163– [5] DOI: 10.1109/81.473564 · doi:10.1109/81.473564 [6] DOI: 10.1109/81.473564 · doi:10.1109/81.473564 [7] DOI: 10.1017/CBO9780511754494 · doi:10.1017/CBO9780511754494 [8] DOI: 10.1002/(SICI)1097-007X(199601/02)24:1<37::AID-CTA902>3.0.CO;2-S · doi:10.1002/(SICI)1097-007X(199601/02)24:1<37::AID-CTA902>3.0.CO;2-S [9] DOI: 10.1109/81.473566 · doi:10.1109/81.473566 [10] DOI: 10.1137/1.9781611971446 · doi:10.1137/1.9781611971446 [11] DOI: 10.1142/S0218126693000125 · doi:10.1142/S0218126693000125 [12] DOI: 10.1109/81.473589 · doi:10.1109/81.473589 [13] DOI: 10.1002/(SICI)1097-007X(199707/08)25:4<279::AID-CTA968>3.0.CO;2-Q · doi:10.1002/(SICI)1097-007X(199707/08)25:4<279::AID-CTA968>3.0.CO;2-Q [14] DOI: 10.1109/TCSI.2004.827627 · Zbl 1374.82022 · doi:10.1109/TCSI.2004.827627 [15] G. Golub and C. Van Loan, Matrix Computations, 3rd edn. (The Johns Hopkins University Press, Baltimore, 1996) p. 198. · Zbl 1118.65316 [16] DOI: 10.1109/81.473568 · doi:10.1109/81.473568 [17] DOI: 10.1007/978-1-4612-1828-9 · doi:10.1007/978-1-4612-1828-9 [18] DOI: 10.1103/PhysRevLett.82.2983 · doi:10.1103/PhysRevLett.82.2983 [19] Khibnik A. I., J. Circuits Syst. Comput. 2 pp 145– [20] DOI: 10.1007/978-3-642-69689-3 · doi:10.1007/978-3-642-69689-3 [21] DOI: 10.1142/9789812798855_0004 · doi:10.1142/9789812798855_0004 [22] Mees A. I., Dynamics of Feedback Systems (1981) · Zbl 0454.93003 [23] DOI: 10.1109/81.473576 · doi:10.1109/81.473576 [24] DOI: 10.1109/81.704820 · Zbl 0951.92002 · doi:10.1109/81.704820 [25] DOI: 10.1109/81.704819 · Zbl 0951.92001 · doi:10.1109/81.704819 [26] A. R. Vàzquez, Towards the Visual Microprocessor – VLSI Design and Use of Cellular Network Universal Machines, eds. T. Roska and A. Rodriguez-Vazquez (J. Wiley, Chichester, 2000) pp. 87–131. 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.