Local Theory of Nonlinear Analytic Ordinary Differential Equations
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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 §8.
II.
1
Families
3
............
3 6
on
an Invariant
Family
Method
Surface
............
........................
of P e r i o d i c
Solutions
..........
..................................
Equation
Stability
Liapunov's
Solutions
................................
First
by
of
....................................
Cases
Stability
§9.
...................................
Lemma
§4°
48
Approximation
Method
.......................
Reduction
§12.
Critical
§13.
Critical Case of One Pair of Pure Imaginary Eigenvalues ..........................
§14.
Data Loading...
