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A major challenge in the study of dynamical systems is that of modeldiscovery: turning data into models that are not just predictive, but provideinsight into the nature of the underlying dynamical system that generated thedata. This problem is made more difficult by the fact that many systems ofinterest exhibit diverse behaviors across multiple time scales. We introduce anumber of data-driven strategies for discovering nonlinear multiscale dynamicalsystems 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 fullstate measurements, we show that the recent sparse identification of nonlineardynamical systems (SINDy) method can discover governing equations withrelatively little data and introduce a sampling method that allows SINDy toscale efficiently to problems with multiple time scales. Specifically, we candiscover distinct governing equations at slow and fast scales. For systems withincomplete observations, we show that the Hankel alternative view of Koopman(HAVOK) method, based on time-delay embedding coordinates, can be used toobtain a linear model and Koopman invariant measurement system that nearlyperfectly captures the dynamics of nonlinear quasiperiodic systems. Weintroduce two strategies for using HAVOK on systems with multiple time scales.Together, our approaches provide a suite of mathematical strategies forreducing the data required to discover and model nonlinear multiscale systems.

Discovery of Nonlinear Multiscale Systems: Sampling Strategies and Embeddings