Coarse Graining the Dynamics of Heterogeneous Oscillators in Networks with Spectral Gaps

We present a computer-assisted approach to coarse-graining the evolutionary dynamics of a system of nonidentical oscillators coupled through a (fixed) network structure. The existence of a spectral gap for the coupling network graph Laplacian suggests that the graph dynamics may quickly become low-dimensional. Our first choice of coarse variables consists of the components of the oscillator states -their (complex) phase angles- along the leading eigenvectors of this Laplacian. We then use the equation-free framework [1], circumventing the derivation of explicit coarse-grained equations, to perform computational tasks such as coarse projective integration, coarse fixed point and coarse limit cycle computations. In a second step, we explore an approach to incorporating oscillator heterogeneity in the coarse-graining process. The approach is based on the observation of fastdeveloping correlations between oscillator state and oscillator intrinsic properties, and establishes a connection with tools developed in the context of uncertainty quantification.