Dynamical systems theory discovers universal patterns of how things change in time. Its perspective is abstract, but its ability to unite seemingly unrelated phenomena is astounding. This lecture will give examples from biomechanics, neuroscience and climate science that illustrate how mathematics helps us analyze empirical data.

In human locomotion, the body resembles an upright, unstable pendulum. We can neither explain this stability nor build legged robots with agility comparable to our own. Data driven mathematical strategies give insight about locomotion while avoiding the complexity of our coupled nervous and musculoskeletal systems.

To illustrate how similar dynamical phenomena appear in unrelated systems, I shall compare dynamics in two systems from different disciplines: a model of mixed mode oscillations in chemical reactors and a reduced model for El Nino, a climate phenomenon in the tropical Pacific that affects weather globally. Multiple time scales are key to explaining why the same dynamical patterns are observed in these unrelated systems.