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Zbl 1144.34001
Meiss, James D.
Differential dynamical systems. (English)
[B] Mathematical Modeling and Computation 14. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). xxii, 412~p. \$~79.00 (2007). ISBN 978-0-898716-35-1/pbk

The book under review is an introduction to ordinary differential equations. It consists of 9 chapters, appendix, and bibliography. Chapter 1 is an introduction. The author describes here what is a dynamical system and an ordinary differential equation. One-dimensional, two-dimensional systems, and simplest chaotic systems are considered. In Chapter 2, the author reviews and extends the standard techniques for solving linear systems of ordinary differential equations (ODEs). He considers the splitting of a matrix into its diagonalizable (semisimple) and nilpotent parts, as well as the treatment of linear, time-periodic systems (Floquet theory). The goal of Chapter 3 is to prove the fundamental theorems of existence and uniqueness for solutions of ODEs. Theorems on the smooth dependence on initial conditions and on the continuous dependence on parameters are also considered. In Chapter 4, the author develops a classification of the qualitative properties of dynamics and investigates the asymptotic behavior -- what happens as $t\to\infty$. The first part of this study concerns the trajectories of a dynamical system in a local neighborhood. The goals are to classify equilibria by their stability, invariant manifolds, and topological types. Chapter 5 is devoted to invariant manifolds of ODEs. Two-dimensional ODEs are studied in Chapter 6. Methods to obtain global phase portraits are described here. Chaotic dynamics and bifurcations are considered in Chapters 7 and 8, respectively. Finally, Chapter 9 investigates Hamiltonian systems. Some mechanical examples, Poisson dynamics, the action principle are also considered here. There are a lot of exercises at the end of each chapter. The book under review is an undergraduate-level text. It is a good starting point for scientists and students that would like to move into the field of studying ODEs.
[Alexander O. Ignatyev (Donetsk)]

MSC 2000:: *34-01 Textbooks (ordinary differential equations)
34A30 Linear ODE and systems
34A34 Nonlinear ODE and systems, general
37-01 Instructional exposition (Dynamical systems and ergodic theory)
34Cxx Qualitative theory of solutions of ODE
34Dxx Stability theory of ODE

Keywords: ordinary differential equation; stability; chaos; bifurcation; fractals

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