Discontinuity, Nonlinearity, and Complexity
Random Parametric Resonance in TimeDependent Networks of Adaptive Frequency Oscillators
Discontinuity, Nonlinearity, and Complexity 3(3) (2014) 347365  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 totimedependent networks is straightforward, this is not true here. For timedependent 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 timedependent 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 DFGIRTG 1132 (Deutsche Forschungsgemeinschaft  International Research Training Group) under the project entitled “Internationales Graduiertenkolleg  Stochastics and Real World Models”.
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