Berechenbares Chaos in Dynamischen Systemen

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Berechenbares Chaos in Dynamischen Systemen

R. Stoop and W.-H. Steeb
Birkhäuser (2006), 264 pp., Price: $39.95 (paperback)
ISBN 978-3-7643-7550-8
Reviewer: Charles Morgan
DSWeb Book Reviews editor
Department of Mathematics
Lock Haven University of Pennsylvania
Lock Haven, Pennsylvania, U.S.A.

Although available only in German, this is one of the best textbooks introducing dynamical systems to the advanced undergraduate or graduate student. The text begins with a thorough discussion of one-dimensional systems. In this discussion, the authors introduce essential notions such as Lyapunov exponents, invariant densities, the Perron-Frobenius operator, and conjugacy. The text also uses Monte Carlo methods to calculate Lyapunov exponents for Bernoulli-shift maps, circle maps, and logistic maps.

The second chapter begins with a minimal but complete introduction to manifolds and hyperbolic fixed points. Then the authors give brief but excellent discussions of Hénon attractors, a class of maps known as Takens-Ruelle-Newhouse maps, Lozi attractors, and Baker transformations.

The third chapter begins with the Grobman-Hartman Theorem before introducing stable and unstable manifolds and stable and unstable bundles. It is curious, though, that the Smale horseshoe map and the Smale solenoid are not used in the introduction to stable and unstable manifolds. The chapter continues with a further discussion of Lyapunov exponents for higher-dimensional systems and the Poincaré map. The fourth chapter concentrates mostly on flows, the fifth chapter on numerous excellent examples of dissipative systems encountered in physics, chemistry, etc. The next three chapters engage the reader in a thorough discussion of Hamiltonian systems, chaos control in physical systems, the Painlevé Property and complex dynamics; and the final chapter is a very brief discussion of fractal dimension and Hausdorff dimension.

This is the perfect textbook for both the pure mathematics student and the applied mathematics student interested in dynamical systems. I wish that I had such a book when I started graduate school, but I had already graduated three years before the book was published. Almost every page of this book has excellent examples and exercises which fit perfectly into the discussion, and many of the exercises and examples use Mathematica. Many chapters contain biographies of some of the most important contributors to dynamical systems, although a biography of Stephen Smale is absent.

Even if one's students do not speak German, any faculty member teaching an introductory course in dynamical systems should require this textbook. The text would probably require two semesters for an undergraduate course, but it certainly could be completed in one semester in a graduate course.

Table of Contents

  1. Grundparadigmen dynamischer Systeme in 1-d
  2. Zweidimensionale diskrete Abbildungen
  3. Kontinuerliche dissipative Systeme
  4. Vertiefungen
  5. Wichtige dissipative Systeme
  6. Hamiltonische Systeme
  7. Fortsetzung ins Komplexe
  8. Chaoskontrolle
  9. Fraktale Dimensionen
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