Ordinary Differential Equations with Applications (Texts in Applied Mathematics) - Hardcover

Carmen Chicone

Hardcover

ISBN 10: 0387985352 ISBN 13: 9780387985350

Publisher: Springer, 1999

Synopsis

This book is based on a year long course taught by the author to graduate students at the University of Missouri over several years. The main objective is to provide, in the first semester, a student friendly account of some of the standard topics encountered in an introductory course of ordinary differential equations, and then, in the second semester, to expand on these ideas by introducing more advanced concepts and applications. A central theme in the book is the use of Implicit Function Theorem. A proof of the existence and uniqueness results for solutions of differential equations based on the Implicit Function Theorem in Banach spaces is presented. Then in the latter sections of the book, the basic ideas of perturbation theory are introduced as applications of the Implicit Function Theorem (with the Lyapunov-Schmidt reduction technique): continuation of subharmonics and the existence of periodic solutions via the averaging method. Finally, local bifurcations-saddle node and Hopf-are studied as applications of the Lyapunov-Schmidt reduction. The book also contains material that is different from standard treatments. For example, the Fiber Contraction Principle is introduced as a useful tool for proving the smoothness of functions that are obtained as fixed points of contractions. This idea is used to give an alternate proof of the smoothness of the flow of a differential equation. Later, the Fiber Contraction Principle appears in the nontrivial proof of the smoothness of invariant manifolds at a rest point. The proof for this existence and smoothness of the stable center manifolds is obtained as a corollary of a more general existence theorem for invariant manifolds at a rest point in the presence of a "spectral gap". The ideas introduced in this section can be extended to infinite dimensions.

About this title

From the Back Cover

This book developed over 20 years of the author teaching the course at his own university. It serves as a text for a graduate level course in the theory of ordinary differential equations, written from a dynamical systems point of view. It contains both theory and applications, with the applications interwoven with the theory throughout the text. The author also links ordinary differential equations with advanced mathematical topics such as differential geometry, Lie group theory, analysis in infinite-dimensional spaces and even abstract algebra.

The second edition incorporates corrections and improvements of the original text. New material includes a proof of the Grobman-Hartman theorem for flows based on the Lie derivative, more extensive treatment of the Euler-Lagrange equation and its applications, a proof of Noether's theorem on the existence of first integrals in the presence of symmetries and a new section on dynamic bifurcation with a proof of Pontryagin's formula.  The impressive array of existing exercises has been more than doubled in size and further enhanced in scope, providing mathematics, physical science and engineering graduate students with a thorough introduction to the theory and application of ordinary differential equations.

Reviews of the first edition:

``As an applied mathematics text on linear and nonlinear equations, the book by Chicone is written with stimulating enthusiasm. It will certainly appeal to many students and researchers.'' -- F. Verhulst, SIAM Review

``The author writes lucidly and in an engaging conversational style. His book is wide-ranging in its subject matter, thorough in its presentation, and written at a generally high level of generality, detail, and rigor.'' -- D. S. Shafer, Mathematical Reviews

Edition details

• Publisher: Springer
• Publication date: 1999
• Language: English
• ISBN 10: 0387985352
• ISBN 13: 9780387985350
• Binding: Hardcover
• Edition number: 1
• Number of pages: 584