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Phys Rev E Stat Nonlin Soft Matter Phys. 2007 Feb;75(2 Pt 2):027104. Epub 2007 Feb 20.

Analysis of nonlinear synchronization dynamics of oscillator networks by Laplacian spectral methods.

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Department of Chemistry, University of Toronto, Toronto, Ontario, Canada M5S 3H6.


We analyze the synchronization dynamics of phase oscillators far from the synchronization manifold, including the onset of synchronization on scale-free networks with low and high clustering coefficients. We use normal coordinates and corresponding time-averaged velocities derived from the Laplacian matrix, which reflects the network's topology. In terms of these coordinates, synchronization manifests itself as a contraction of the dynamics onto progressively lower-dimensional submanifolds of phase space spanned by Laplacian eigenvectors with lower eigenvalues. Differences between high and low clustering networks can be correlated with features of the Laplacian spectrum. For example, the inhibition of full synchoronization at high clustering is associated with a group of low-lying modes that fail to lock even at strong coupling, while the advanced partial synchronization at low coupling noted elsewhere is associated with high-eigenvalue modes.

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