Defense Event

Aspects of Invariant Manifold Theory and Applications

Matthew Kvalheim

Monday, September 10, 2018
10:00am - 12:00pm
EECS 3316

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About the Event

Abstract: Recent years have seen a surge of interest in “data-driven” approaches to determine the equations governing complex systems. Yet in spite of modern computing advances, the high dimensionality of many systems — such as those occurring in biology and robotics — renders direct machine learning approaches infeasible. This dissertation develops tools for the experimental study of complex systems, based on mathematical concepts from dynamical systems theory. Our approach uses the fact that parsimonious assumptions often lead to strong insights from dynamical systems theory; such insights can be leveraged in learning algorithms to mitigate the “curse of dimensionality” and make these algorithms practical. We use dynamical insights from the theory of normally hyperbolic invariant manifolds (NHIMs) which come in two flavors: existence of reduced-order models, and existence of “normal forms” for the equations of motion. Our contributions include a data-driven phase estimation algorithm for nonlinear oscillators, and a data-driven algorithm — utilizing geometric mechanics and NHIM theory — to estimate the governing equations for a class of biolocomotion systems. These algorithms have several applications to the sciences and engineering. We also make fundamental contributions to NHIM theory, including global linearization results generalizing the celebrated Hartman-Grobman theorem.

Additional Information

Sponsor(s): Professor Shai Revzen

Open to: Public