ERGEBNISSE DER MATHEMATIK UND IHRER GRENZGEBIETE

UNTER MITI1I'IRKUNG DER SCHRIFTLEITUNG DES "ZENTRALBLATT FUR MATHEMATIK"

HERAUSGEGEBEN VON

L. V. AHLFORS . R. BAER . F. L. BAUER . R. COURANT . A. DOLD J. L. DOOB· S. EILENBERG· M. KNESER· T. NAKAYAMA

H. RADEMACHER . F. K. SCHMIDT . B. SEGRE . E. SPERNER

REDAKTION P. R. HALMOS

====== NEUE FOLGE· BAND 16

ASYMPTOTIC BERA VIOR AND STABILITY PROBLEMS IN ORDINARY

DIFFERENTIAL EQUATIONS

BY

LAMBERTO CE SARI

SECOND EDITION

WITH 37 FIGURES

Springer-Verlag Berlin Heidelberg GmbH

Die Betieher des "Zentt'alblatt fu1' Mathematik" erhalten die ,,Ergebnisse der Mathematik"

zu einem gegenube1' defn Ladenp1'eis Ufn 10% erfnăjJigten V01'zugsp1'eis

ISBN 978-3-662-00107-3 ISBN 978-3-662-00105-9 (eBook) DOI 10.1007/978-3-662-00105-9

Alle Rechte, insbesondere das der Obersetzung in fremde Sprachen,

vorbehaIten

Ohne ausdruckliche Genehmigung des Verlages ist es auch nieht gestattet, dieses Buch oder Teile daraus

auf photomechanischem Wege (Photokop1e, Mikrokopie) oder auf andere ArI zu vervielfaItigen

© by Springer-Verlag Berlin Heidelberg 1963

Originally published by Springer-Verlag OHG. Berlin' Gottingen. Heidelberg in 1959 and 1963

Softeover reprint of the hardeover 2nd edition 1963

Library of Congress Catalog Card Number 63-12926

Preface to the Second Edition

This second edition, which has become necessary within so short a time, presents no major changes.

However new results in the line of work of the author and of J. K. HaIe have made it advisable to rewrite seetion (8.5). Also, some references to most recent work have been added.

June 1962

LAMBERTO CESARI

University of Michigan Ann Arbor

Preface to the First Edition In the last few decades the theory of ordinary differential equations

has grown rapidly under the action of forces which have been working both from within and without: from within, as a development and deepen­ing of the concepts and of the topological and analytical methods brought about by LYAPUNOV, POINCARE, BENDIXSON, and a few others at the turn of the century; from without, in the wake of the technological development, particularly in communications, servomechanisms, auto­matie controls, and electronics. The early research of the authors just mentioned lay in challenging problems of astronomy, but the line of thought thus produced found the most impressive applications in the new fields.

The body of research now accumulated is overwhelming, and many books and reports have appeared on one or another of the multiple aspects of the new line of research which some authors call .. qualitative theory of differential equations"

The purpose of the present volume is to present many of the view­points and questions in a readable short report for which completeness is not claimed. The bibliographical notes in each section are intended to be a guide to more detailed expositions and to the original papers.

Some traditional topics such as the Sturm comparison theory have been omitted. Also excluded were all those papers, dealing with special differential equations motivated by and intended for the applications. General theorems have been emphasized wherever possible. Not all proofs are given but only the typical ones for each seetion and some are just outlined.

I wish to thank the colleagues who have read parts of the manuscript and have made suggestions: W.R FULLER, RA. GAMBILL, M. GOLOMB, J.K.HALE, N.D.KAZARINOFF, C.RPUTNAM, and E.SILVERMAN. I am indebted to A. W. RANSOM and W.E. THOMPSON for helping with the proofs. Finally, I want to express my appreciation to the Springer Verlag for its accomplished and disceming handling of the manuscript.

Lafayette, Ind. June 1958 L. CESARI

Contents

Preiace . III/V

Chapter I. The concept 01 stability and systems with constant coelficients

§ 1. Some remarks on the concept of stability

1.1. Existence, uniqueness, continuity,' p. 1. - 1.2. Stability in the sense of LYAPUNOV, p.4. - 1.3. Examples, p.6. - 1.4. Bounded­ness, p. 7. - 1.5. Other types oi requirements and comments, p. 8. -1.6. Stability of equilibrium, p. 9. - 1. 7. Variation al systems, p. 10. -1.8. Orbital stability, p. 12. - 1.9. Stability and change of coordi­nates, p, 12. - 1.10. Stability of the m-th order in the sense of G.D.BIRKHOFF, p. 13. - 1.11. A general remark and bibliographical notes, p. 14.

§ 2. Linear systems with constant coefficients. . . . . . . . . . . . . 14

2.1. Matrix notations, p. 14. - 2.2. First applications to differential systems, p. 18. - 2.3. Systems with constant coefficients, p. 19. -2.4. The RouTH-HuRWITZ and other criteria, p. 21. - 2.5. Systems of order 2, p. 24. - 2.6. Nonhomogeneous systems, p. 26. 2.7. Linear resonance, p.27. - 2.8. Servomechanisms, p.28. -2·9· Bibliographical notes, p. 33.

Chapter 11. Genetallinear systems . . . 34

§ 3. Linear systems with variable coefficients 34

3.1. A theorem of LYAPUNOV, p. 34. - 3.2. A proof of (3.1. i), p. 35· -3·3· Boundedness oi the solutions, p. 36. - 3.4. Further conditions for boundedness, p. 37. - 3.5. The reduction to L-diagonal form and an outline of the proofs of (3.4. iii) and (3.4. iv), p. 39. - 3.6. Other conditions, p. 41. - 3.7. Asymptotic behavior, p. 41. - 3.8. Linear asymptotic equilibrium, p. 42. - 3.9. Systems with variable coeffi­cients, p. 44. - 3.10. Matrix conditions, p. 48. - 3.11. Nonhomo­geneous systems, p. 49. - 3.12. LYAPUNOV's type numbers, p. 50. -3.13. First application of type numbers to differential equations, p. 51. 3.14. Normal systems of solutions, p. 52. - 3.15. Regular differential systems, p. 53. - 3.16. A relation between type numbers and generalized characteristic roots, p. 54. - 3.17. Bibliographical notes, p. 55.

§ 4. Linear systems with periodic coefficients . . . . . . . . . . . .. 55

4.1. Floquet theory, p. 55. - 4.2. So me important applica­tions, p. 59. - 4.3. Further results concerning equation (4.2.1) and extensions, p. 61. - 4.4. Mathieu equation, p. 65. - 4.5 SmaU periodic perturbations, p. 66. - 4.6. Bibliographical notes, p. 79.

§ 5. The second order linear differential equation and generalizations 80

5·1. Oscillatory and non-oscillatory solutions, p. 80. - 5.2. FUBINI'S theorems, p. 81. - 5.3. So me transformations, p. 84. - 5.4. BELLMAN'S and PRODI'S theorems, p. 84. - 5.5. The case I{t) ~ + 00, p. 85. -

VIII Contents

5.6. Solutions of c1ass L2, p. 86. - 5.7. Parseval relation for functions of c1ass L2, p. 88. - 5.8. Some properties of the spectrum S, p. 89. 5.9. Bibliographical notes, p. 89.

Chaptel' IlI. Nonlinear systems 91

§ 6. Some basic theorems on nonlinear systems and the first method of LYAPUNOV . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.1. General considerations, p. 91. - 6.2. A theorem of existence and uniqueness, p. 91. - 6.3. Periodic solutions of periodic systems, p. 96. - 6.4. Periodic solutions of autonomous system~, p. 98. -6.5. A method of successive approximations and the first method ofLYA­PUNOV, p. 99. - 6.6. Some results of BYLOV and VINOG~AD, p. 101.-6.7. The theorems of BELLMAN, p. 102. - 6.8. Invariant measure, p. 103. - 6.9. Differential equations on a torus, p. 106. - 6.10. Biblio­graphical notes, p. 107.

§ 7. Thc second method of LYAPUNOV ................ 107 7.1. The function V of LYAPUNOV, p. 107. - 7.2. The theorems of

LYAPUNOV, p. 109. - 7.3. More recent results, p. 111. - 7.4. A par­ticular partial differential equations, p. 113. - 7.5. Autonomous systems, p. 114. - 7·6. Bibliographical notes, p. 114.

§ 8. Analytical methods . . . . . . . . . . . . . . . . . . . . . . 115 8.1. Introductory considerations, p. 115. - 8.2. Method of

LINDSTEDT, p. 116. - 8.3. Method of POINCARE, p. 118. - 8.4. Method of KRYLOV and BOGOLYUBOV, and of VAN DER POL, p. 120. -8.5. A convergent method for periodic solutions and existence theo­rems, p.123. - 8.6. The perturbation method, p. 136. - 8.7. The Lienard equation and its periodic solutions, p. 139. - 8.8. An oscilla­tion theorem for equation (8.7.1). p. 143. - 8.9. Existence of a perio­dic solution of equation (8.7.1.), p. 145.. - 8.10. Nonlinear free oscilla­tions, p. 145. - 8.11. Invariant surfaces, p. 148. - 8.12. Bibliographi­cal notes, p. 150. - 8.13. Nonlinear resonance, p. 150. - 8.14. Prime movers, p. 151. - 8.15. Relaxation oscillations, p. 155.

§ 9. Analytic-topological methods . . . . . . . . . . . . . . . . . . 156 9·1. Poincare theory of the critical points, p. 156. - 9.2. Poincare­

Bendixson theory, p. 163. - 9.3. Indices, p.167. - 9.4. A configuration concerning LIENARD'S equation, p. 170. - 9.5. Another existence theorem for the Lienard equation, p. 174. - 9.6. The method of the fixed point, p. 176. - 9.7. The method of M. L. CARTWRIGHT, p. 177. - 9.8. The method of T. WAZEWSKI, p. 179.

Chapter IV. Asymptotic developments . . . . . . . 182

§ 10. Asymptotic developments in general. . . . 182 10.1. POINCARE'S concept of asymptotic development, p. 182. -

10.2. Ordinary, regular and irregular singular points, p. 184. -10.3: Asymptotic expansions for an irregular singular point of finite type, p. 186. - 10.4. Asymptotic developments deduced from Taylor expansions, p. 187. - 10.5. Equations containing a large parameter, p. 189. - 10.6. Turningpoints and the theoryof R. E. LANGER, p.192. -10.7. Singular perturbation, p. 195.

Bibliogl'aphy

Index

197

267