  A major challenge in the study of dynamical systems is that of model
discovery: turning data into models that are not just predictive, but provide
insight into the nature of the underlying dynamical system that generated the
data. This problem is made more difficult by the fact that many systems of
interest exhibit diverse behaviors across multiple time scales. We introduce a
number of data-driven strategies for discovering nonlinear multiscale dynamical
systems and their embeddings from data. We consider two canonical cases: (i)
systems for which we have full measurements of the governing variables, and
(ii) systems for which we have incomplete measurements. For systems with full
state measurements, we show that the recent sparse identification of nonlinear
dynamical systems (SINDy) method can discover governing equations with
relatively little data and introduce a sampling method that allows SINDy to
scale efficiently to problems with multiple time scales. Specifically, we can
discover distinct governing equations at slow and fast scales. For systems with
incomplete observations, we show that the Hankel alternative view of Koopman
(HAVOK) method, based on time-delay embedding coordinates, can be used to
obtain a linear model and Koopman invariant measurement system that nearly
perfectly captures the dynamics of nonlinear quasiperiodic systems. We
introduce two strategies for using HAVOK on systems with multiple time scales.
Together, our approaches provide a suite of mathematical strategies for
reducing the data required to discover and model nonlinear multiscale systems.
