An Introduction to Dynamical Systems: Continuous and Discrete Clark Robinson Prentice Hall (2004), 672 pp., price $80.- ISBN: 0131431404
Reviewer: Todd Young, Department of Mathematics, Ohio University, Athens OH, USA

The book is an introduction to dynamical systems, primarily intended as a textbook for an advanced undergraduate course. As the title suggests, the book covers both continuous and discrete dynamical systems.

It is surprising to me that discrete and continuous parts of the book are completely segregated. The first part is about nonlinear differential equations and the second part covers iteration of functions. This is in contrast to other introductory dynamics books, such as [1,2,4,5,6,7], that treat discrete and continuous systems in a more integrated fashion. Other textbooks such as [3,8] treat only discrete or continuous dynamical systems: this is the first to the reviewer's knowledge that treats both in a fully separate manner. The author states in the Preface that the parts can be covered in either order. The biggest problem I see with this approach is that the connections between the two parts might not be as clear as they should be.

The second thing that strikes me about the book after the arrangement of topics, is the thoroughness of the treatment. The book not only covers a wide variety of topics, but for each of these topics the author presents concepts, methods, applications and proofs. In fact, if the book has a fault it might be that it covers too much; those interested in a mathematical approach might be put off by the extensive treatment of applications and the introduction of concepts through examples, while those interested in an applied approach might be equally dissatisfied with the large number of definitions, theorems and proofs in the book.
The author attempts to avoid the later problem by placing the harder proofs in separate sections titled ``Theory and Proofs'' at the end of each chapter. However, the main text is still rich in rigorous mathematics. Given this, the book may become the choice for those who want their students to be well-grounded in both worlds.

The book begins with a historical prologue and an introduction to the ``Geometric Approach to Differential Equations'', which is in fact a very brief overview of Part I of the book. Part I consists of the following chapters:

2. Linear Systems
3. The Flow: Solutions of Nonlinear Equations
4. Phase Portraits with an Emphasis on Fixed Points
5. Phase Portraits Using Energy and Other Test Functions
6. Periodic Orbits
7. Chaotic Attractors

The second part of the book begins with an introduction to discrete dynamical systems. Chapter titles include:

 9. One-Dimensional Maps
10. Itineraries for One-Dimensional Maps,
11. Invariant Sets
12. Higher Dimensional Maps
13. Invariant Sets for Higher Dimensional Maps
14. Fractals

For example, the third chapter begins with the logistic equation followed by an explanation that we might not be able to find explicit solutions for nonlinear equations. It proceeds with the traditional existence, uniqueness and differential dependence on initial conditions theorem, the proof of which is postponed until the ``Theory and Proof'' section. Examples of nonexistence and non-uniqueness are presented. It then covers the definition of the flow defined by the equation, the group property of the flow, the definition of periodic orbits and invariant sets and the first variation equation. Then the author introduces the Euler, Improved Euler (Heun), and Runga-Kutta methods, followed by a treatment of local and global errors. This is followed by the example of the Euler method applied to the harmonic oscillator and a brief discussion of more advanced numerical methods. The Theory and Proofs section of this chapter introduces the Lipschitz condition, the Picard iteration scheme, Gronwall's inequality, complete proofs of the claims about existence, uniqueness and continuous dependence on initial conditions and an explanation of how the first variation equation gives differential dependence. The section concludes with theorems and proofs about the order of the local and global errors of the Euler and Heun methods. All the chapters contain a similar mixture of example, theory and application. At the end of each chapter are a wide variety of well written exercises, including proofs, derivations, numerical simulations and applications.

In summary I think that this book would be an excellent text for a one year sequence for advanced undergraduates in math or beginning graduate students in engineering or the sciences. Personally, I would advise the instructor to be careful to make connections between discrete and continuous parts.
 

1. K. Alligood, T. Sauer, and J. Yorke, Chaos: An Introduction to Dynamical Systems, Texts in Mathematical Sciences, Springer-Verlag, New York, 1997.

2. D. Arrowsmith and C. Place, An Introduction to Dynamical Systems, Cambridge University Press, 1990.

3. R. Devaney, An Introduction to Chaotic Dynamical Systems, Westview Press, 2nd edition, 2003.

4. J. Hale and J. Kocak, Dynamics and Bifurcations, Texts in Applied Mathematics, vol. 3, Springer-Verlag, New York, 1991.

5. B. Hasselblatt and A. Katok, A First Course in Dynamics: with a Panorama of Recent Developments, Cambridge University Press, 2003.

6. R. Hilborn, Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers, Oxford University Press; 2nd edition, 2001.

7. D. Kaplan and L. Glass, Understanding Nonlinear Dynamics, Springer-Verlag, New York, 1995.

8. L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, vol. 7, Springer-Verlag, New York; 3rd edition, 2001.