Review of "Ordinary Differential Equations and Dynamical Systems" by T. Sideris

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Ordinary Differential Equations and Dynamical Systems

Ordinary Differential Equations and Dynamical Systems
by Thomas C. Sideris
Atlantis Studies in Differential Equations
Atlantis Press
225 pp. (2013)
ISBN: : 978-94-6239-021-8
Reviewed by: Douglas S.Shafer
Department of Mathematics
University of North Carolina at Charlotte
Email: dsshafer (at) uncc.edu

This book is intended as a textbook for use in beginning graduate courses on ordinary differential equations and as with many if not most such books it arose from the author’s experience teaching such a course at UC Santa Barbara. The words that come to mind after reading through it carefully are “terse,” “self-contained,” and “thorough” in what it covers. It is about half the length of the books of Carmen Chicone and Larry Perko and seems to move through the material very quickly. Virtually every result, from lemma to theorem, is fully proved (in rare cases the proof is relegated to the exercises). The topics that are selected for coverage are generally treated in some depth. The prerequisites for the students are stated as a “solid background in analysis and linear algebra.” My personal impression is that many students will find the book more than a little demanding. For example, the concepts of limit inferior and upper and lower semi-continuity are used or referred to without comment.

A list of the chapter titles and their lengths gives a fairly good idea of the material covered:

  • Chapter1 Introduction (3 pages),
  • Chapter2 Linear Systems (15 pages, 9 exercises),
  • Chapter3 Existence Theory (31 pages, 16 exercises),
  • Chapter4 Nonautonomous Linear Systems (20 pages, 5 exercises),
  • Chapter5 Results from Functional Analysis (15 pages, 9 exercises),
  • Chapter6 Dependence on Initial Conditions and Parameters (6 pages, 2 exercises),
  • Chapter7 Linearization and Invariant Manifolds (24 pages, 11 exercises),
  • Chapter8 Periodic Solutions (38 pages, 7 exercises),
  • Chapter9 Center Manifolds and Bifurcation Theory (44 pages, 4 exercises),
  • Chapter 10 The Birkhoff Smale Homoclinic Theorem (19 pages, 1 exercise).

There is also a two-page appendix that contains statements of the Contraction Mapping Theorem, The Implicit Function Theorem, and the Inverse Function Theorem.

The first chapter gives a well-written overview of what an ordinary differential equation is, what a solution is, why in most cases we can restrict to first order equations, and why linear systems are important. Chapter 2 is confined to systems x' = Ax, A constant; the more general case is deferred to Chapter 4. Chapter 3 includes the definition of a Lyapunov function and the Lyapunov Stability Theorem. The unusual Chapter 5 contains the definition of a Banach space, the Open Mapping Theorem, the Fréchet derivative, the Contraction Mapping Theorem and the Implicit Function Theorem in this context, and the Lyapunov-Schmidt technique, everything completely proved, except for the Open Mapping Theorem. Chapter 7 includes the Flowbox Theorem. Chapter 8 treats T-periodic perturbations of autonomous systems, stability of periodic solutions of periodic nonautonomous systems, stable and unstable manifolds of period solutions, and the Poincaré-Bendixson Theorem. Chapter 9 includes a section on normal forms.

It is apparent that all the fundamental results for a course of this type are present, exactly as one would expect. Two things that stood out  were the treatment of nonautonomous systems that in my experience is much more complete than in most comparable textbooks and the very sparse treatment of two-dimensional systems. The only strictly two-dimensional result is the Poincaré-Bendixson Theorem, and a family as important as the Liénard systems appears only in an example illustrating Lyapunov stability.

The book is structured in theorem-proof format with very little additional discussion, and in particular with very few examples; Chapter 3 has fourteen, but none of the rest have more than five examples and a couple have none at all. It is difficult to know how this book would work in a course without actually trying it out, but my personal impression is that for many this book would function as a kind of framework around which to organize a course. I believe that if the students were to come away with any real understanding of what is going on then the instructor would have to provide a great deal of supplementary explanation in the form of concrete examples and practical exercises.

The author, who as far as I can tell is a specialist in PDEs, states in his preface that he could almost subtitle the work “ODE, as told by an analyst,” and this book certainly has that feel to it. It seems almost completely non-geometric; the concept of a phase portrait hardly appears at all.

On the whole, the book is very well written from the point of view of organization and language usage, and the mathematical standards are high. Typesetting is up to today's standards and typographical and other such errors are very few. The pricing is more than competitive. To the extent that the choice of topics matches an individual instructor's views on what should be covered in a beginning course in ordinary differential equations, and for a course populated by sufficiently well-prepared students, this could be a top choice as a text for such a course.

 

 

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