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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann 702 Yuri N. Bibikov Local Theory of Nonlinear Analytic Ordinary Differential Equations Springer-Verlag Berlin Heidelberg New York 1979 Author Yuri N. Bibikov Department of Mathematics Mechanics University Leningrad Leningrad USSR AMS Subject Classifications (1970): 34A25, 34A45, 34C05, 34C20, 34C25, 34C30, 34 D10, 3 4 D 2 0 ISBN 3-540-09114-9 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-09114-9 Springer-Verlag NewYork Heidelberg Berlin 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, broadcasting, 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 1979 Printed in Germany PREFACE These notes p r e s e n t a u t h o r at the D i v i s i o n during the second a course of l e c t u r e s of A p p l i e d M a t h e m a t i c s , semester They are b a s e d on a course, the m o t i o n , matics g i v e n by the of the a c a d e m i c year Brown University 1975-1976. on the t h e o r y of the s t a b i l i t y of w h i c h the a u t h o r gave at the D e p a r t m e n t and M e c h a n i c s last several years, at the U n i v e r s i t y of M a t h e - of L e n i n g r a d during and on some r e c e n t p u b l i c a t i o n s by the the author. The a u t h o r members is very g r a t e f u l of the D i v i s i o n discussions. is also g r a t e f u l proofreading June, for their help and Miss S a n d r a S p i n a c c i The author 1976 J a c k K. Hale, for t h e i r w a r m h o s p i t a l i t y The a u t h o r thanks Messrs. K. Lyons and N. A l i k a k o s script, to P r o f e s s o r and and u s e f u l R. M a l e k - M a d a n i , in p r e p a r i n g for her m e t i c u l o u s to R. M a l e k - M a d a n i the m a n u typing. for his c a r e f u l of the material. Yuri N. B i b i k o v P r o v i d e n c e , R. I. TABLE OF CONTENTS Page Basic Notation §0. ,,.., . . . . . . . . . . ,..., . . . . . . . . . . . . . . . . . ,.. Introduction Chapter I. Analytic §i. Auxiliary §2. Normal Form Form §3. Normal LiapunovVs §5. Analytic §6. Special §7. Bifurcation Chapter II. ................................... 1 Families 3 Lemma §4° 6 on an Invariant Method Liapunov's Surface ............ ........................ of P e r i o d i c Solutions .......... .................................. Equation Stability §9. ............ 3 First by Solutions ................................ Family Stability of .................................... Cases §8. 48 Approximation Method ......................