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CLASSICS IN MATHEMATICS Nam Parshad Bhatia· Giorgio P. Szego Stability Theory of Dynamical Systems Springer Classics in Mathematics N.P. Bhatia • G.P. Szego Stability Theory of Dynamical Systems Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Tokyo N. P. Bhatia · G. P. SzegO Stability Theory of Dynamical Systems Springer-Verlag Berlin· Heidelberg· New York 1970 Professor Nam Parshad Bhatia University of Maryland Division of Mathematics, Baltimore and Institute for Fluid Dynamics and Applied Mathematics, College Park Maryland, U.S.A. Professor George Philip Szego University of Milan Milan, Italy Geschaftsfiihrende H erausgeber: Professor Dr. B. Eckmann Eidgenossische Technische Hochschule Zurich Professor Dr. B. L. van der Waerden Mathematisches Institut der Universitat Zurich This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broad casting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. @ by Springer-Verlag, Berlin ·Heidelberg 1970. Printed in Germany Library of Congress Catalog Card Number 70-126892 Title No. 5144 To Sushiela and Emilia Preface This book contains a systematic exposition of the elements of the theory of dynamical systems in metric spaces with emphasis on the stab ility theory and its application and extension for ordinary autonomous differential equations. In our opinion, the book should serve as a suitable text for courses and seminars in the theory of dynamical systems at the advanced under graduate and beginning graduate level, in mathematics, physics and engineering. It was never our intention to write a treatise containing all known results on the subject; but we have endeavored to include most of the important new results and developments of the past 20 years. The extensive bibliography at the end should enhance the usefulness of the book to those interested in the further exploration of the subject. Students should have completed an elementary course in ordinary differential equations and have some knowledge of metric space theory, which is usually covered in undergraduate courses in analysis and topo logy. Each author strongly feels that any mistakes left in the book are attributable to the other author, but each would appreciate receiving any comments from the scientific community. We are obliged to Professor AARON STRAUSS for reading the entire typescript and pointing out several corrections. We would also like to thank Doctors FLORENCIO CASTILLO, LAWRENCE FRANKLIN, CESI.AW OLECH and GIULIO TRECCANI for help in proofreading the galleys. June 1970 N. P. BHATIA · G. P. SzEGO Contents Notation . . . x Introduction 1 I. D y n am i c a 1 Sy st e m s . 5 1. Definition and Related Notation . . 5 2. Examples of Dynamical Systems . . . 6 Notes and References . . . . 10 II. Elementary Concepts 12 1. Invariant Sets and Trajectories. . 12 2. Critical Points and Periodic Points . . 15 3. Trajectory Closures and Limit Sets 19 4. The First Prolongation and the Prolongational Limit Set . . 24 Notes and References . . . . . . . . . . . . 30 III. R e c u r s i v e C o n c e p t s . . 31 1. Definition of Recursiveness . . . . . . . . . . 31 2. Poisson Stable and Non-wandering Points . 31 3. Minimal Sets and Recurrent Points . . . . . . . 36 4. Lagrange Stability and Existence of Minimal Sets 41 Notes and References . . . . . . . . . 42 IV. D i s p e r s i v e C o n c e pt s 43 1. Unstable and Dispersive Dynamical Systems 43 2. Parallelizable Dynamical Systems 48 Notes and References . . . . . . . 55 V. Stability Theory. 56 1. Stability and Attraction for Compact Sets . . . . . . . . . . 56 2. Liapunov Functions: Characterization of Asymptotic Stability . 66 3. Topological Properties of Regions of Attraction 79 4. Stability and Asymptotic Stability of Closed Sets . 84 5. Relative Stability Properties . . . . . . . . . . . . . . 99 6. Stability of a Motion and Almost Periodic Motions 106 Notes and References . . . . . . . . . 111 V. Flow near a Compact Invariant Set ... . 114 1. Description of Flow near a Comi:act Invariant Set . 114 2. Flow near a Compact Invariant Set (Continued) . 116 Notes and References . . . . . . . . . . . . . . . . 117