Differential Equations and Dynamical Systems

Rating:
Summary: Very good graduate ODEs text, with some flaws
Review: Perko's book is one of the best books that gives an advanced introduction to dynamical systems from the point of differential equations. Many other good books tread the same ground, without emphasizing the connection to ODEs. Perko's text is particularly strong in several respects. First, the dynamical systems it considers are almost always expressed in terms of underlying differential equations. Second, it gives proofs or outlines of proofs of most major theorems used in this field. Third, it covers the most important topics, including: local theory of hyperbolic equilibria, invariant manifolds, Hamiltonian systems, flows on R^2, stability theory, and elementary bifurcations. Also reviewed are the results from linear systems theory, in a particularly well-written and easy to follow introductory chapter. Another great feature of this book is its solid coverage of center manifold theory, which is an important and somewhat difficult topic.

There are a couple of problems with this book. The proofs to some of the major theorems are occasionally abstruse or poorly derived. Perko seems to bend over backwards to give analytical proofs, when algebraic or topological proofs might be easier. Many of the problems reuse the same elementary example equations. This is OK insofar as it allows the reader to see how different techniques can be used to analyze the same systems, but it limits the reader's exposure to the full variety of interesting dynamical systems that can arise in practice. The author also tends to emphasize polynomial vector fields, which is a potential limitation. Occasionally the problems are significantly more difficult than the examples worked in the text.

Overall, Perko's text is a very solid introduction to advanced ODEs and continuous dynamics. It is especially well-suited for scientists and engineers who want a readable introduction to the qualitative theory of ODEs.

Rating:
Summary: A Book on Advanced Dynamical Systems
Review: This book is a useful textbook for advanced courses on differential equations and dynamical systems for senior undergraduate students or first year graduate students.

The book presents a systematic study of the qualitative and geometric theory of nonlinear differential equations and dynamical systems.

The book has a sketch of the proof of the Hartman-Grobman Theorem which was useful for my second undergraduate course on dynamical systems and nonlinear differential equations.

I liked the book and I am quite sure it will become a classic textbook on this very useful branch of Math that has so many old and new applications in Physics, Economics and Finance.