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Publisher:

Oxford University Press

Publication Date:

2007

Number of Pages:

531

Format:

Paperback

Edition:

4

Price:

60.00

ISBN:

9780199208258

Category:

Textbook

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by , on ]

Henry Ricardo

07/6/2008

The authors view this book as an introduction to dynamical systems via ordinary differential equations. The treatment of topics, considered by the authors to be suitable for senior undergraduate or master’s degree courses in the UK, is broad but not deep. The emphasis is on qualitative methods and applications for scientists and engineers, and this well written book presents the material in a simple but elegant way. Proofs are given, but the authors also rely on calculations produced by *Mathematica* and some excellent figures to explain the theory. Software is not necessary for using this book, although it is to be hoped that a student has access to a CAS such as *Maple* or *Mathematica* to supplement his or her understanding.

I’ve had the previous edition of this book on my shelves almost since its publication. Although I’ve never used it for a course, I have found the first three chapters valuable as a reference for teaching nonlinear equations and systems in an undergraduate ODE course and the rest of the book wonderful reading for “self-tuition,” as the authors might put it. Revised portions of this edition include an extended explanation of Mathieu’s equation, an expanded treatment of the exponential matrix, and a detailed account of Liapunov exponents for both difference and differential equations.

There are 124 fully worked examples in the text, and the book contains over 500 end-of-chapter problems, many containing significant applications and developments of the theory in the chapter. This edition has 88 new ‘Exercises’ set off in grey boxes, routine problems with selected answers but no full solutions. In total, there are now over 750 examples and problems in the Jordan-Smith text. The book ends with five appendices containing useful background information and two pages of references and suggestions for further reading. For the first time, there is a complete solutions manual, available as a separately sold volume: Nonlinear Ordinary Differential Equations: Problems and Solutions (Oxford University Press, 2007). This handbook has 584 pages and contains more than 500 fully solved problems, including 272 diagrams.

The large number of citations of this work in the research literature seems to indicate that NODE has transcended its use as a text and has come to be regarded as a source of techniques for a wide range of applied problems. Any reader combining this new edition with Strogatz’s Nonlinear Dynamics and Chaos (Addison-Wesley, Reading, MA, 1994) will have an excellent view of nonlinear differential equations, especially their applications.

Henry Ricardo (henry@mec.cuny.edu) is Professor of Mathematics at Medgar Evers College of The City University of New York and serves as Secretary (until May, 2009) and Governor of the Metropolitan NY Section of the MAA. He is the author of A Modern Introduction to Differential Equations, which has just been translated into Spanish; and he is currently writing a linear algebra text.

Preface

1. Second-order differential equations in the phase plane

2. Plane autonomous systems and linearization

3. Geometrical aspects of plane autonomous systems

4. Periodic solutions; averaging methods

5. Perturbation methods

6. Singular perturbation methods

7. Forced oscillations: harmonic and subharmonic response, stability, and entrainment

8. Stability

9. Stability by solution perturbation: Mathieu's equation

10. Liapurnov methods for determining stability of the zero solution

11. The existence of periodic solutions

12. Bifurcations and manifolds

13. Poincaré sequences, homoclinic bifurcation, and chaos

Answers to the exercises

Appendices

A. Existence and uniqueness theorems

B. Topographic systems

C. Norms for vectors and matrices

D. A contour integral

E. Useful identities

References and further reading

Index

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