Nonlinear Time Series Analysis and Manifold Learning

Over the last 50 years or so nonlinear dynamics and chaos has revolutionized our understanding of the complexity of the natural world. However, the vast majority of this understanding, has come from the study of toy models that captured the flavor of real problems but were not models of real systems. As such dynamical systems theory has become more an area of mathematics than natural experimental science. Much of this is because experiments were difficult to perform and we lived in a data limited world. Now with the advent of big data and increases in computing performance in the 3rd decade of the 21st century we are no longer data-limited and can take full advantage of the theory of dynamical systems for the analysis of observed data. This data science course is focused on the analysis of dynamical systems, mostly from nonlinear time series, to find hidden properties and causal relationships in complex systems. The course teaches the students what can practically be done using the intellectual framework of chaos theory and dynamical systems using a data driven approach to maximally extract information from time series and their mathematical properties hidden in the geometry of their embeddings.