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On global dynamic behavior of weakly connected oscillatory networks. (English) Zbl 1089.34033

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.

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
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