Topics in Stability and Bifurcation Theory
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309 David H. Sattinger University of Minnesota, Minneapolis, MN/USA
Topics in Stability and Bifurcation Theory
Springer-Verlag Berlin · Heidelberg · New York 1973
AMS Subject Classifications (1970): 35-02, 35B35, 35G20, 35)60, 35K55, 35Q10, 46-xx, 76D05, 76E99
ISBN 3-540-06133-9 Springer-Verlag Berlin · Heidelberg· New York ISBN 0-387-06133-9 Springer-Verlag New York · 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 1973. Library of Congress Catalog Card Number 72-96728. Printed in Germany. Offsetdruck: Julius Beltz, Hems bach /Bergstr.
Preface In analyzing the dynamics of a physical system governed by nonlinear equations the following questions present themselves: Are there equilibrium states of the system? Are they stable or unstable? varied?
How many are there?
What happens as external parameters are
As the parameters are varied, a given equilibrium may lose
its stability (although it may continue to exist as a mathematical solution of the problem) and other equilibria or time periodic oscillations may branch off.
Thus, bifurcation is a phenomenon
closely related to the loss of stability in nonlinear physical systems. The subjects of bifurcation and stability have always attracted the interest of pure mathematicians, beginning at least with Poincar$ and Lyapounov.
In the past decade an increasing amount of attention has
focused on problems in partial differential equations.
The purpose of
these notes is to present some of the basic mathematical methods which have developed during this period.
They are primarily mathematical in
their approach, but it is hoped they will be of value to those applied mathematicians and engineers interested in learning the mathematical techniques of the subject. In a number of sections we have explained the basic mathematical tools needed for the development of the subject of bifurcation theory, for example elements of the theory of elliptic boundary value problems,
IV the Riesz-Schauder
theory of compact operators~ the Leray-Schauder
topological degree theory~ and the implicit function theorem in a B~uach space.
Nevertheless, the reader will have to have a certain
amotmt of background in mathematical analysis, particularly in the areas of partial differential equations and functional analysis, in order to benefit from these notes. These notes were the basis of a course given at the University of Minnesota during the academic year 1971-1972.
The author would
like to express his thanks to Professors D. G. Aronson, Gene Fabes, Charles McCarthy, and Daniel Joseph for their live
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