A good introduction to dynamical systems, which is accessible to the advanced
undergraduate or beginning graduate student, can be very difficult to find. In order to get a good
introduction to the subject, the beginning graduate student must track down a number of books
or articles, each of which addresses a fairly narrow topic: chaos theory, complex dynamics, ergodic theory, symbolic dynamics, topological properties, random walks, mechanics, etc. Dynamical systems is compartmentalized early in the graduate program, and the result is an army of new Ph.D.s who are quite narrowly focused in their research. Finding one approachable book which gives a brief discussion of all of the major areas and shows the conceptual and mathematical connections among them is just about impossible.
Until now. Shlomo Sternberg's book Dynamical Systems is that
excellent introduction which many of us sought when we were first-year graduate students, who
became intrigued by our differential equations class but who had at that time no idea how
diverse the field truly has become.
In just 265 pages, Sternberg has managed to address almost all the major topics which form the basis of modern research in dynamical systems. The chapter titles alone are sufficiently titillating to make one buy the book.
- Iteration and fixed points
(Includes a discussion of attractors, repellers, and basins of attraction)
(Includes a discussion of the logistic family and Newton's Method)
- Sarkovsky's Theorem, Singer's Theorem, Intermittency
(Includes chaos, sensitivity to initial conditions, symbolic dynamics and the shift map)
- Space and time averages
- The contraction fixed point theorem
- The Hausdorff metric and Hutchinson's Theorem
(Includes fractals and fractal dimension)
(Includes a nice discussion of invariant manifolds and the graph transform method)
- The Perron-Frobenius Theorem
- Some topics in ordinary differential equations
(Includes an excellent discussion of entropy)
- Symbolic dynamics
(Includes topological entropy and the Henon map)
The selling point of Sternberg's book, which is a considerable expansion of his lecture notes from a course he taught at Harvard, is that he does an excellent job of showing the important ideas without getting the reader bogged down in a plethora of corollaries and propositions which are subordinate to the real discussion. We mathematicians are almost always guilty of getting bogged down in theorems and propositions which are interesting but tangential to the primary topic of a discussion, but Sternberg stays on topic on every page of Dynamical Systems. He presents the main thesis of each chapter clearly and concisely without skipping the necessary proofs! He presents the essential results and leaves the important but tangential discussions for further reading.
Another selling point of this book is that it is readable. Sternberg has made this work an interesting read. The language is pleasant.
Sternberg has written a perfect introduction to dynamical systems. This is the ideal book to whet any graduate student's appetite for the subject. I will go further: every graduate student should read this book. This book should be included in every new student's welcome package to graduate school.
The only deficiency I see in Dynamical Systems is the lack of a bibliography at the end of each chapter. When the reader finishes this book, it would be expected that he or she would then like to return to some favorite chapters and delve into the deeper ideas which follow. There is a bibliography of books in the appendix. They are all excellent books which cover the range of topics Sternberg introduces, but
since each chapter addresses a more-or-less self-contained topic, I would strongly recommend a list of accessible journal articles at the end of each chapter.
Finally, it is good to see Dover Publications getting into the new book market instead of dealing only with reprints. Graduate students need textbooks and reference books which don't require a major loan to purchase them. By publishing new books, Dover is putting the big publishers on notice that it is possible to charge a reasonable price for a book, make a profit, and have a successful business. If we could just convince Dover to get into the business of publishing academic journals, then authors might be able to avoid having to pay $300 to have their own articles published.