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Discontinuity, Nonlinearity, and Complexity 3(3) (2014) 347--365 | DOI:10.5890/DNC.2014.09.009

Julio Rodriguez

tecData AG, Bahnhofstrasse 108/114, 9240 Uzwil, Switzerland

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Abstract

We consider a network of interacting phase oscillators endowed with adaptive mechanisms, leading the collective motion to a consensual dynamical state. Specifically, for a given network topology (i.e. an adjacency matrix) governing the mutual interactions, the adaptive mechanisms enable all oscillators to ultimately adopt a consensual frequency. Once reached, the consensual frequency subsists even if interactions between the oscillators are switched off. For the class of models we consider, the consensual frequency is independent of the network topology. Even though this independence might suggest that extension totime-dependent networks is straightforward, this is not true here. For time-dependent networks and spectra of the underlying Laplacian matrices, one may observe the emergence of more complex dynamics. Due to their high degree of complexity, these dynamics generally offer little hope for analytical tractability. In this paper, we focus on connected time-dependent networks with circulant adjacency matrices. The simple spectral structures and commutativity properties enjoyed by circulant matrices enable an analytical stability analysis of the consensus state. Ultimately, we are able to reduce the stability analysis to a dissipative harmonic oscillator with parametric pumping.

Acknowledgments

The author warmly thanks Dr. Martin ANDEREGG for the interesting discussions on commuting matrices. This paper was supported by the DFG-IRTG 1132 (Deutsche Forschungsgemeinschaft - International Research Training Group) under the project entitled “Internationales Graduiertenkolleg - Stochastics and Real World Models”.

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