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Differential dynamical systems (Monographs on Mathematical Modeling and Computation)
Meiss J., Society for Industrial and Applied Mathematics, Philadelphia, PA, 2007. 434 pp.  Type: Book (9780898716351)
Date Reviewed: Sep 4 2008

This well-written book is aimed at advanced undergraduate and graduate students who have taken a traditional course on ordinary differential equations (ODEs) and linear algebra, and have been exposed to advanced calculus.

The text addresses traditional ODE material, combined with the modern theory of dynamical systems viewed as evolution rules that describe smooth evolution in time. It proceeds to a classification of equilibria that leads to a theory of invariant manifolds, nonhyperbolic orbits, and approaches to the theory of bifurcations and the phenomena of chaos. The text represents a very valuable collection of important theorems and their proofs related to differential dynamical systems, providing all of the necessary concepts. It also represents a successful illustration of the theory, by providing many well-chosen examples taken from engineering, biology, physics, and chemistry applications.

Chapter 1 provides an introduction, with a wealth of examples from one- and two-dimensional dynamics and a nice introduction to the Lorenz chaos model. More examples are presented as exercises resulting from a long didactic experience. Chapter 2 presents a review of techniques for solving linear systems of ODEs. I particularly liked the presentation of nonautonomous linear systems and Flouqet theory. Chapter 3 addresses the existence and uniqueness of solutions of ODEs leading to the contraction-mapping theorem (the Picard-Lindelof theorem). In chapter 4, the author introduces the concept of flow, aimed at providing a qualitative perspective of differential dynamical systems. Topological concepts of conjugacy and equivalence of dynamical systems are introduced. This is followed by attractor sets and basins, along with Poincare maps.

Chapter 5 discusses the theory of invariant manifolds, and the way they give rise to a mechanism for chaos. Stable manifolds and global stable manifolds are discussed, along with center manifolds leading later to bifurcation theory. The chapter ends with the statement of the center manifold theorem that is used in chapter 8, dedicated to bifurcations. Nonhyperbolic equilibria, along with a method to obtain global phase portraits specific to two dimensions, are discussed in chapter 6. This is followed by discussions of the Poincare Bendixxon theorem, Lienard systems, and the behavior at infinity for an ODE on the Poincare sphere.

Chapter 7 introduces a wealth of concepts necessary for understanding chaos, such as the Lyapunov exponents, strange attractors, the Hausdorff dimension, and strange nonchaotic attractors, illustrated by well-chosen examples. Chapter 8 discusses bifurcation theory, introducing basic ideas of normal forms and addressing co-dimension-1 and 2 bifurcations, illustrated by examples. Relatively modern results are presented, such as Shilnikov bifurcation and Melnikov’s method for the onset of chaos. A wealth of exercises completes the chapter.

Chapter 9 seems to discuss the preferred research topic of the author: Hamiltonian dynamics, emphasizing geometrical aspects. The variational foundation of Hamiltonian dynamics is also addressed, along with their spectral properties, and the advanced Kolmogorov, Arnold, and Moser (KAM) theory.

A short appendix addresses the use of mathematical software, providing examples in MATLAB, Mathematica, and Maple for basic implementations. The bibliography is relevant and up to date.

This very useful book constitutes a most welcome addition to the shelves of teachers and researchers on the topic of ODEs, with a comprehensive introductory perspective on the modern theory of dynamical systems. I hope that in follow-up editions, an instructor’s manual of solutions to chosen exercises will be made available along with the excellent text.

Reviewer:  I. M. Navon Review #: CR136026 (0908-0725)
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Ordinary Differential Equations (G.1.7 )
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Mathematical Software (G.4 )
Partial Differential Equations (G.1.8 )
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