Topological Degree Approach to Bifurcation Problems
Topological bifurcation theory is one of the most essential topics in mathematics. This book contains original bifurcation results for the existence of oscillations and chaotic behaviour of differential equations and discrete dynamical systems under varia
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Topological Fixed Point Theory and Its Applications VOLUME 5
For other titles published in this series, go to www.springer.com/series/6622
Michal Feˇckan
Topological Degree Approach to Bifurcation Problems
Michal Feˇckan Department of Mathematical Analysis and Numerical Mathematics Faculty of Mathematics, Physics and Informatics Comenius University Mlynská dolina 842 48 Bratislava Slovakia
ISBN 978-1-4020-8723-3
e-ISBN 978-1-4020-8724-0
Library of Congress Control Number: 2008931004 All Rights Reserved c 2008 Springer Science + Business Media B.V. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com
To my beloved family
Contents 1 Introduction 1.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 An Illustrative Perturbed Problem . . . . . . . . . . . . . . . . . 1.3 A Brief Summary of the Book . . . . . . . . . . . . . . . . . . . . 2 Theoretical Background 2.1 Linear Functional Analysis . . . . . . . . . . 2.2 Nonlinear Functional Analysis . . . . . . . . . 2.2.1 Implicit Function Theorem . . . . . . 2.2.2 Lyapunov-Schmidt Method . . . . . . 2.2.3 Leray-Schauder Degree . . . . . . . . . 2.3 Differential Topology . . . . . . . . . . . . . . 2.3.1 Differentiable Manifolds . . . . . . . . 2.3.2 Symplectic Surfaces . . . . . . . . . . 2.3.3 Intersection Numbers of Manifolds . . 2.3.4 Brouwer Degree on Manifolds . . . . . 2.3.5 Vector Bundles . . . . . . . . . . . . . 2.3.6 Euler Characteristic . . . . . . . . . . 2.4 Multivalued Mappings . . . . . . . . . . . . . 2.4.1 Upper Semicontinuity . . . . . . . . . 2.4.2 Measurable Selections . . . . . . . . . 2.4.3 Degree Theory for Set-Valued Maps . 2.5 Dynamical Systems . . . . . . . . . . . . . . . 2.5.1 Exponential Dichotomies . . . . . . . 2.5.2 Chaos in Discrete Dynamical Systems 2.5.3 Periodic O.D.Eqns . . . . . . . . . . . 2.5.4 Vector Fields . . . . . . . . . . . . . . 2.6 Center Manifolds for Infinite Dimensions . . .
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3 Bifurcation of Periodic Solutions 3.1 Bifurcation of Periodics from Homoclinics I . . . . . . . . 3.1.1 Discontinuous O.D.Eqns . . . . . . . . . . . . . . . 3.1.2 The Linearized Equation . . . . . . . . . . . . . . 3.1.3 Subharmonics for Regular Periodic Perturbations . 3.1.4 Subharmonics for Singular Periodic Perturbations
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