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Dynamical Systems and Linear Algebra

Fritz Colonius and Wolfgang Kliemann
American Mathematical Society
Publication Date: 
Number of Pages: 
Graduate Studies in Mathematics 158
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Felipe Zaldivar
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Linear algebra and differential equations are, arguably, two of the most beautiful areas of mathematics: both are sophisticated enough, dealing with fundamentally important subjects, and at the same time both are down to earth, allowing non-trivial computations either as illustrations or, more importantly, well-established and important applications. As luck will have it, the interplay between these two areas is a classical and ever-evolving fertile ground of mathematical research.

This is, essentially, the subject of the book under review, and what a subject: the intertwining of linear algebra and dynamical systems, with differential equations for continuous time or difference equations in discrete time.

The classical treatments include Hirsch and Smale’s Differential Equations, Dynamical Systems, and Linear Algebra (Academic Press, 1974) or its reincarnation, Differential Equations, Dynamical Systems and an Introduction to Chaos (Academic, 2004 and 2012) with a new catchy title, a new coauthor, and, of course, the addition of some well-chosen new topics. This book is still one of the best introductions to the subject, with the understanding that it is essentially devoted to the case of autonomous equations, that is, systems of differential or difference equations, determined by a fixed matrix, in other words, with constant coefficients.

The book under review considers in its first part, a little bit more than a third of the book, the same case of autonomous differential or difference equations. Here the authors take a quick and efficient approach, dealing with linear dynamical systems in real space \({\mathbb R}^d\), but also considering linear dynamical systems in real projective space and even in (real) Grassmannians. The overlap with Hirsch and Smale is minimal, and each book treats or leaves out some topics, somehow complementing each other.

The second part of the book under review considers the case of non-autonomous (with time-varying coefficients or matrices \(A(t)\)) differential or difference equations. The analysis of these non-autonomous systems starts with the case of periodic coefficients (Floquet theory), followed by a topological analysis of linear flows for which the main tool is the Morse decomposition, and culminating with the analysis of the time-dependency of \(A(t)\) via ergodic properties. Perhaps the major contribution of the book under review is its unified approach, using Lyapunov exponents, for the proofs of the properties needed for the analysis of the time-dependency of \(A(t)\). I should also add that all topological or ergodic properties are carefully developed, making the book attractive for a graduate course or self-study.

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Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is