Young Measure Approach to Computing Slowly Advancing Fast Oscillations

We offer a multiscale and averaging strategy to compute the solution of a singularly perturbed system when the fast dynamics oscillates rapidly; namely, the fast dynamics, rather than settling on a manifold of smaller order, forms cycle-like limits which advance along with the slow dynamics. We describe the limit as a Young measure with values being supported on the limit cycles, averaging with respect to which induces the equation for the slow dynamics. In particular, computing the tube of limit cycles establishes a good approximation for arbitrarily small singular parameters. Possible algorithms are displayed and concrete numerical examples are exhibited.