Differential Equations and Dynamical Systems [Hardcover]

Lawrence Perko (Author)

Book Description

Publication Date: April 1, 2006 | ISBN-10: 0387951164 | ISBN-13: 978-0387951164 | Edition: 3rd

This textbook presents a systematic study of the qualitative and geometric theory of nonlinear differential equations and dynamical systems. Although the main topic of the book is the local and global behavior of nonlinear systems and their bifurcations, a thorough treatment of linear systems is given at the beginning of the text. All the material necessary for a clear understanding of the qualitative behavior of dynamical systems is contained in this textbook, including an outline of the proof and examples illustrating the proof of the Hartman-Grobman theorem. In addition to minor corrections and updates throughout, this new edition includes materials on higher order Melnikov theory and the bifurcation of limit cycles for planar systems of differential equations.

Editorial Reviews

Review

Reviews from the first edition: “...The text succeeds admiraby ... Examples abound, figures are used to advantage, and a reasonable balance is maintained between what is proved in detail and what is asserted with supporting references ... Each section closes with a set of problems, many of which are quite interesting and round out the text material ... this book is to be highly recommended both for use as a text, and for professionals in other fields wanting to gain insight into modern aspects of the geometric theory of continuous (i.e., not discrete) dynamical systems.” MATHEMATICAL REVIEWS

From the Back Cover

  This textbook presents a systematic study of the qualitative and geometric theory of nonlinear differential equations and dynamical systems. Although the main topic of the book is the local and global behavior of nonlinear systems and their bifurcations, a thorough treatment of linear systems is given at the beginning of the text. All the material necessary for a clear understanding of the qualitative behavior of dynamical systems is contained in this textbook, including an outline of the proof and examples illustrating the proof of the Hartman-Grobman theorem, the use of the Poincare map in the theory of limit cycles, the theory of rotated vector fields and its use in the study of limit cycles and homoclinic loops, and a description of the behavior and termination of one-parameter families of limit cycles. In addition to minor corrections and updates throughout, this new edition contains materials on higher order Melnikov functions and the bifurcation of limit cycles for planar systems of differential equations, including new sections on Francoise's algorithm for higher order Melnikov functions and on the finite codimension bifurcations that occur in the class of bounded quadratic systems.

Product Details

• Hardcover: 568 pages
• Publisher: Springer; 3rd edition (April 1, 2006)
• Language: English
• ISBN-10: 0387951164
• ISBN-13: 978-0387951164
• Product Dimensions: 9.3 x 6.4 x 1.4 inches
• Shipping Weight: 2.2 pounds (View shipping rates and policies)
• Average Customer Review: 3.2 out of 5 stars  See all reviews (5 customer reviews)
• Amazon Best Sellers Rank: #509,511 in Books (See Top 100 in Books)
  • #70 in Books > Professional & Technical > Professional Science > Physics > Dynamics > Fluid Dynamics

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Customer Reviews

Most Helpful Customer Reviews

24 of 25 people found the following review helpful:

4.0 out of 5 stars Very good graduate ODEs text, with some flaws, December 11, 2003

By 

David Elder "elddm" (Boston, Ma United States) - See all my reviews
(REAL NAME)   

This review is from: Differential Equations and Dynamical Systems (Hardcover)

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.

12 of 17 people found the following review helpful:

4.0 out of 5 stars A Book on Advanced Dynamical Systems, April 7, 2000

By 

Juan P Medina-Mora (Mexico City, Mexico) - See all my reviews

This review is from: Differential Equations and Dynamical Systems (Texts in Applied Mathematics, No 7) (Hardcover)

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.

1 of 1 people found the following review helpful:

3.0 out of 5 stars it could be so much better, November 8, 2011

By 

Peter Hinow - See all my reviews

Amazon Verified Purchase

This review is from: Differential Equations and Dynamical Systems (Hardcover)

I used this book the last two times I taught ordinary differential equations at the graduate level. It's a decent book and it covers a lot of topics, but it has an amazing amount of errors, given that it is already in its 3rd edition. Some proof techniques I find archaic (for example, the insistence on Picard iterations, where a single call to Banach's fixed point theorem would be much clearer). A good point are the many exercises, but again, they can contain typos.