  Multiscale phenomena that evolve on multiple distinct timescales are
prevalent throughout the sciences. It is often the case that the governing
equations of the persistent and approximately periodic fast scales are
prescribed, while the emergent slow scale evolution is unknown. Yet the
course-grained, slow scale dynamics is often of greatest interest in practice.
In this work we present an accurate and efficient method for extracting the
slow timescale dynamics from signals exhibiting multiple timescales that are
amenable to averaging. The method relies on tracking the signal at
evenly-spaced intervals with length given by the period of the fast timescale,
which is discovered using clustering techniques in conjunction with the dynamic
mode decomposition. Sparse regression techniques are then used to discover a
mapping which describes iterations from one data point to the next. We show
that for sufficiently disparate timescales this discovered mapping can be used
to discover the continuous-time slow dynamics, thus providing a novel tool for
extracting dynamics on multiple timescales.
