Dynamic mode decomposition, Koopman operators, diffusion maps, equations free modeling, Lagrangian coherent systems, finite time Lyapunov exponents — are some of the new methods that have been introduced in recent decades to analyze dynamical systems. Three lectures will survey these methods along with earlier ones of “nonlinear time series analysis.” The goal will be to describe theoretical principles and algorithmic approaches suitable for working with empirical data and computer defined systems. Attempts will be made to characterize overlaps and features of dynamical problems that distinguish them from other areas of machine learning and data science. Don’t expect new results!

This will be the first lecture of a mini-course. The first lecture will review invariant measures for chaotic systems together with three ways of quantifying their complexity: dimension, Lyapunov exponents and entropy.