Introduction xiii for a linear algebra for classes of time-varying linear systems, namely peri- odic, random, and perturbed (or controlled) systems. Several examples of (low-dimensional) systems that play a role in engineering and science are presented throughout the text. Originally, we had also planned to include some basic concepts for the study of genuinely nonlinear systems via linearization, emphasizing invari- ant manifolds and Grobman-Hartman type results that compare nonlinear behavior locally to the behavior of associated linear systems. We decided to skip this discussion, since it would increase the length of this book consider- ably and, more importantly, there are excellent treatises of these problems available in the literature, e.g., Robinson [117] for linearization at fixed points, or the work of Bronstein and Kopanskii [21] for more general lin- earized systems. Another omission is the rich interplay with the theory of Lie groups and semigroups where many concepts have natural counterparts. The mono- graph [48] by R. Feres provides an excellent introduction. We also do not treat nonautonomous differential equations via pullback or other fiberwise constructions see, e.g., Crauel and Flandoli [37], Schmalfuß [123], and Ras- mussen [116] our emphasis is on the treatment of families of nonautonomous equations. Further references are given at the end of the chapters. Finally, it should be mentioned that all concepts and results in this book can be formulated in continuous and in discrete time. However, sometimes results in discrete time may be easier to state and to prove than their ana- logues in continuous time, or vice versa. At times, we have taken the liberty to pick one convenient setting, if the ideas of a result and its proof are par- ticularly intuitive in the corresponding setup. For example, the results in Chapter 5 on induced systems on Grassmannians are formulated and derived only in continuous time. More importantly, the proof of the multiplicative ergodic theorem in Chapter 11 is given only in discrete time (the formula- tion and some discussion are also given in continuous time). In contrast, Selgrade’s Theorem for topological linear dynamical systems in Chapter 9 and the results on Morse decompositions in Chapter 8, which are used for its proof, are given only in continuous time. Our aim when writing this text was to make ‘time-varying linear alge- bra’ in its periodic, topological and ergodic contexts available to beginning graduate students by providing complete proofs of the major results in at least one typical situation. In particular, the results on the Morse spectrum in Chapter 9 and on multiplicative ergodic theory in Chapter 11 have de- tailed proofs that, to the best of our knowledge, do not exist in the current literature.
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