Lyapunov and Auxiliary Functions - Data-Driven Dyanmics | Lecture 12

Many important statements in dynamical systems can be posed in terms of finding scalar functions that satisfy certain pointwise inequalities that imply the desired result. Such functions are called auxiliary functions, with a classical example being a Lyapunov function. In this lecture we review the relevant theory and show that auxiliary function methods primarily rely on the Lie derivative. The general framework of this video lecture shows that Lyapunov functions can be found for both stochastic and deterministic dynamical systems in both continuous and discrete time. We further demonstrate another example of bounding time averages, which is a problem coming from ergodic theory. Learn more about Lyapunov functions and Dulac's criterion: https://www.youtube.com/watch?v=Gh2BtRjUJDg Learn more about ergodic theory: https://www.youtube.com/watch?v=LX-CyxMrrHs Learn more about the Lorenz equations: https://www.youtube.com/watch?v=0Rpy-xSsAo8 Get the book here: https://epubs.siam.org/doi/10.1137/1.9781611978162 Scripts and notebooks to reproduce all examples: https://github.com/jbramburger/DataDrivenDynSyst This book provides readers with: - methods not found in other texts as well as novel ones developed just for this book; - an example-driven presentation that provides background material and descriptions of methods without getting bogged down in technicalities; - examples that demonstrate the applicability of a method and introduce the features and drawbacks of their application; and - a code repository in the online supplementary material that can be used to reproduce every example and that can be repurposed to fit a variety of applications not found in the book. More information on the instructor: https://hybrid.concordia.ca/jbrambur/ Follow @jbramburger7 on Twitter for updates.