Differential Equations and Dynamical Systems: v. 7 (Texts in Applied Mathematics) - Hardcover

Perko, L.

Hardcover

ISBN 10: 3540974431 ISBN 13: 9783540974437

Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K, 1991

Synopsis

This book contains a systematic study of autonomous systems of ordinary differential equations and dynamical systems. It begins with a thorough treatment of linear systems; however, the main topic of the book is local and global behaviour of nonlinear systems. The main purpose of the book is to introduce students to the qualitative and geometric theory of ordinary differential equations originated at the end of the 19th century. It is also intended as a reference book for mathematicians doing research on dynamical systems. There are several new features in this book such as the simplified proof of the HartmanGrobman Theorem and examples illustrating the proof, map in the theory of limit cycles, an efficient method for obtaining the global phase portrait of two-dimensional systems, and the description of the behaviour of a one-parameter family of limit cycles. Readers of this book will find that, except for certain topics of current mathematical research such as the number of limit cycles and the nature of attracting sets of dynamical systems, the global qualitative theory of a nonlinear dynamical system leads to an understanding of the solution set of the nonlinear system that rivals the understanding what we have of linear flows. This textbook on control/systems theory and ordinary differential equations is intended for mathematicians and physicists.

About this title

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.

Edition details

• Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
• Publication date: 1991
• Language: English
• ISBN 10: 3540974431
• ISBN 13: 9783540974437
• Binding: Hardcover
• Number of pages: 415