An equation-free approach to coupled oscillator dynamics: The Kuramoto model example

Abstract:

We present an equation-free multi-scale approach to the computational study of the collective dynamics of the Kuramoto model [Chemical Oscillations, Waves, and Turbulence}, Springer-Verlag (1984)], a prototype model for coupled oscillator populations. Our study takes place in a reduced phase space of coarse-grained ``observables" of the system: the first few moments of the oscillator phase angle distribution. We circumvent the derivation of explicit dynamical equations (approximately) governing the evolution of these coarse-grained macroscopic variables; instead we use the equation-free framework [Kevrekidis et al., Comm. Math. Sci. 1 (4), 715 (2003)] to computationally solve these equations without obtaining them in closed form. In this approach, the numerical tasks for the conceptually existing but unavailable coarse-grained equations are implemented through short bursts of appropriately initialized simulations of the ``fine-scale", detailed coupled oscillator model. Coarse projective integration and coarse fixed point computations are illustrated.

International Journal of Bifurcations and Chaos, accepted for publication, in press (2005); http://arxiv.org/abs/nlin.AO/0502016


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