Numerical Multiscale Methods for Coupled Oscillators

A multiscale method for computing the effective slow behavior of a system of weakly coupled general oscillators is presented. The oscillators may be either in the form of a periodic solution or a stable limit cycle. Furthermore, the oscillators may be in resonance with one another and thereby generate some hidden slow dynamics. The proposed method relies on correctly tracking a set of slow variables that is sufficient to approximate any variable and functional that are slow under the dynamics of the ODE. The technique is more efficient than existing methods and its advantages are demonstrated with examples. The algorithm follows the framework of the heterogeneous multiscale method.