Asymptotic Behavior of Monodromy Singularly Perturbed Differential E
This book concerns the question of how the solution of a system of ODE's varies when the differential equation varies. The goal is to give nonzero asymptotic expansions for the solution in terms of a parameter expressing how some coefficients go to infini
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B. Eckmann. Zurich F. Takens. Groningen
1502
Carlos Simpson
Asymptotic Behavior of Monodromy Singularly Perturbed Differential Equations on a Riemann Surface
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Author Carlos Simpson Laboratoire de Topologie et Geometrie, URA Universite Paul Sabatier, U. F. R. M. 1.G. 31062 Toulouse-Cedex, France
Mathematics Subject Classification (1991): 34015, 14E20, 30E15, 34B25, 41A60
ISBN 3-540-55009-7 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-55009-7 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991 Printed in Germany Typesetting: Camera ready by author Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 46/3140-543210- Printed on acid-free paper
CONTENTS
Introduction (I-XII)
1
1
Ordinary differential equations on a Riemann surface
12
2
Laplace transform, asymptotic expansions, and the method of stationary phase
17
3
Construction of flows
31
4
Moving relative homology chains
41
5
The main lemma
54
6
Finiteness lemmas
60
7
Sizes of cells
68
8
Moving the cycle of integration
84
9
Bounds on multiplicities
93
10
Regularity of individual terms
101
11
Complements and examples
III
12
The Sturm-Liouville problem
127
References
135
Index
138
INTRODUCTION
We will study the question of how the solution behaves when an algebraic system of linear ordinary differential equations varies. We will consider a family of ordinary differential equations indexed linearly by a parameter t, different from the parameter of differentiation z, Fix initial conditions at a point z = P; then each equation uniquely determines its solution at all other points. The solutions depend on the equations, so they depend on the parameter t. Evaluating the solutions at a point z = Q, we get a function m( Q, t). We will investigate the behavior of this family of solutions, as t 00 in such a way that some coefficients of the equations go to infinity. The simplest example of this situation is the family of equations
dm
dz
tm =
a.
With initial conditions mea, t) 1, the solution is m(z, t) = et z . For a fixed z = Q, this behaves exponentially in t. We will look at some families of equations generalizing this basic example, allowing r x r matrix-valued functions for the coefficients and solutions, and considering equations which are holornorphic and defined globally on a compact Riemann surf
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