An equation-Free Approach to Coupled oscillator Dynamics: the Kuramoto Model Example

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 equationfree 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. Author to whom correspondence should be directed