xii Introduction dynamical system with certain recurrence properties. In this way, the mul- tiplicative ergodic theorem of Oseledets from 1968 [109] resolves the basic issues for measurable linear systems with stationary time dependencies, and the Morse spectrum together with Selgrade’s theorem [124] goes a long way in describing the situation for continuous linear systems with chain transitive time dependencies. A third important area of interplay between dynamics and linear algebra arises in the linearization of nonlinear systems about fixed points or arbitrary trajectories. Linearization of a differential equation ˙ = f(y) in Rd about a fixed point y0 Rd results in the linear differential equation ˙ = f (y0)x and theorems of the type Grobman-Hartman (see, e.g., Bronstein and Kopan- skii [21]) resolve the behavior of the flow of the nonlinear equation locally around y0 up to conjugacy, with similar results for dynamical systems over a stochastic or chain recurrent base. These observations have important applications in the natural sciences and in engineering design and analysis of systems. Specifically, they are the basis for stochastic bifurcation theory (see, e.g., Arnold [6]), and robust stability and stabilizability (see, e.g., Colonius and Kliemann [29]). Stabil- ity radii (see, e.g., Hinrichsen and Pritchard [68]) describe the amount of perturbation the operating point of a system can sustain while remaining stable, and stochastic stability characterizes the limits of acceptable noise in a system, e.g., an electric power system with a substantial component of wind or wave based generation. Goal This book provides an introduction to the interplay between linear alge- bra and dynamical systems in continuous time and in discrete time. There are a number of other books emphasizing these relations. In particular, we would like to mention the book [69] by M.W. Hirsch and S. Smale, which always has been a great source of inspiration for us. However, this book restricts attention to autonomous equations. The same is true for other books like M. Golubitsky and M. Dellnitz [54] or F. Lowenthal [96], which is designed to serve as a text for a first course in linear algebra, and the relations to linear autonomous differential equations are established on an elementary level only. Our goal is to review the autonomous case for one d × d matrix A via induced dynamical systems in Rd and on Grassmannians, and to present the main nonautonomous approaches for which the time dependency A(t) is given via skew-product flows using periodicity, or topological (chain re- currence) or ergodic properties (invariant measures). We develop general- izations of (real parts of) eigenvalues and eigenspaces as a starting point
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