Asymptotic Behavior of Monodromy: Singularly Perturbed Differential Equations on a Riemann Surface

Carlos Simpson (auth.)9780387550091, 0-387-55009-7, 3540550097-:

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 infinity. A particular classof families of equations is considered, where the answer exhibits a new kind of behavior not seen in most work known until now. The techniques include Laplace transform and the method of stationary phase, and a combinatorial technique for estimating the contributions of terms in an infinite series expansion for the solution. Addressed primarily to researchers inalgebraic geometry, ordinary differential equations and complex analysis, the book will also be of interest to applied mathematicians working on asymptotics of singular perturbations and numerical solution of ODE’s.

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Table of contents :
Introduction….Pages 1-11
Ordinary differential equations on a Riemann surface….Pages 12-16
Laplace transform, asymptotic expansions, and the method of stationary phase….Pages 17-30
Construction of flows….Pages 31-40
Moving relative homology chains….Pages 41-53
The main lemma….Pages 54-59
Finiteness lemmas….Pages 60-67
Sizes of cells….Pages 68-83
Moving the cycle of integration….Pages 84-92
Bounds on multiplicities….Pages 93-100
Regularity of individual terms….Pages 101-110
Complements and examples….Pages 111-126
The Sturm-Liouville problem….Pages 127-134