General asymptotic methods on
various time scales are developed for periodic systems of ordinary differential
equations in order to treat global motion in multi-oscillatory systems. Moreover, we
show that bifurcations of an attractive and essentially nonperiodic nature can arise in
systems that also possess several (often unstable) Hopf bifurcations. Such attractor
bifurcations frequently dominate the long term system behavior. In addition, the
methods here can be used to determine the flow on a center manifold in cases where
center manifold theory indicates an instability at the origin of that manifold and
little else about the flow. Finally, various examples of mixed scale motion are
treated.
