Dynamics and Linear Algebra

This course provides an introduction to the interplay between linear algebra and di¤erential equations/dynamical systems in continuous time. We …rst introduce linear di¤erential equations in Euclidian space R and on Grassmannian and ‡ag manifolds induced by a single matrix A, with emphasis on characterizations of the constant matrix A from a dynamics point of view. We then introduce linear skew product ‡ows as a way to model time varying linear systems _ x = A(t)x with, e.g., periodic, measurable ergodic, and continuous chain transitive time dependencies. We develop generalizations of (real parts of) eigenvalues and eigenspaces as a starting point for a linear algebra for classes of time varying linear systems, namely periodic, random, and robust systems. Finally we present some basic ideas to study genuinely nonlinear systems via linearization, emphasizing invariant manifolds and Grobman-Hartman type results that compare nonlinear behavior locally to the behavior of associated linear systems. We will conclude with some engineering applications of this circle of ideas. AMS classi…cation: 15A21, 34C, 34D08, 37B25, 37C75, 37H15, 93E15