Qualitative Theory of ODEs. An Introduction to Dynamical Systems Theory PDF
Preview Qualitative Theory of ODEs. An Introduction to Dynamical Systems Theory
Qualitative Theory of ODEs An Introduction to Dynamical Systems Theory TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk Qualitative Theory of ODEs An Introduction to Dynamical Systems Theory Henryk Żołądek University of Warsaw, Poland Raul Murillo Complutense University of Madrid, Spain World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO Published by World Scientific Publishing Europe Ltd. 57 Shelton Street, Covent Garden, London WC2H 9HE Head office: 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 Library of Congress Cataloging-in-Publication Data Names: Żołądek, Henryk, 1953– author. | Murillo, Raul, author. Title: Qualitative theory of ODEs : an introduction to dynamical systems theory / Henryk Żołądek (University of Warsaw, Poland), Raul Murillo (Complutense University of Madrid, Spain). Description: New Jersey : World Scientific, [2023] | Includes bibliographical references and index. Identifiers: LCCN 2022021010 | ISBN 9781800612686 (hardcover) | ISBN 9781800612693 (ebook) | ISBN 9781800612709 (ebook other) Subjects: LCSH: Differential equations--Qualitative theory. | Dynamics--Mathematical models. Classification: LCC QA372 .Z643 2023 | DDC 515/.352--dc23/eng20220723 LC record available at https://lccn.loc.gov/2022021010 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2023 by World Scientific Publishing Europe Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/Q0374#t=suppl Desk Editors: Nimal Koliyat/Adam Binnie/Shi Ying Koe Typeset by Stallion Press Email: [email protected] Printed in Singapore Preface The Qualitative Theory of Ordinary Differential Equations occu- pies a rather special position both in Applied Mathematics and TheoreticalMathematics.Ontheonehand,itisacontinuation ofthe standard course on Ordinary Differential Equations (ODEs). On the other hand,it is an introduction to Dynamical Systems, which is one of the main mathematical disciplines in recent decades. Moreover, it will turn out to be very useful for graduates when they encounter differential equations at their work; usually such equations are very complicated and cannot be solved by standard methods. Boasting aside, the first lecture on the Qualitative Theory of ODEs at the Faculty of Mathematics Informatics and Mechanics of Warsaw Uni- versity was delivered by Henryk Z˙o(cid:3)la¸dek in 1985; as you can see, the idea turned out to be successful. The main idea of the qualitative analysis of differential equations is to be able to say something about the behavior of solutions of equations without solving them explicitly. Therefore,inthefirstinstance,suchpropertieslikethestabilityof solutions stand out. It is the stability with respect to changes in the initial conditions of the problem. Note that even with the numerical approach to differential equations, all calculations are subject to cer- tain inevitable errors. Therefore, it is desirable that the asymptotic behavior of the solutions is insensitive to perturbations of the initial state. Chapter 1 of these lecture notes roughly focuses on this point. v vi Qualitative Theory of ODEs Another important concept of this theory is structural stability. This is the stability of the entire system, e.g., the phase portrait, with respect to changes of parameters, which usually appear (and in large quantities) on the right side of the equations. In the absence of structural stability, we deal with bifurcations. Methods of the Qual- itative Theory allow for precise and accurate investigation of such bifurcations. We describe them in Chapter 3. In the case of two-dimensional autonomous systems, phase por- traits areconceptually simple—theyconsistofsingularpoints,their separatrices and limit cycles; one should add to this the resolution of singularities and behavior at infinity. It is worth mentioning that the problem of limit cycles for polynomial vector fields is the still unresolved Hilbert’s 16th problem. These topics are discussed in Chapter 2. Chapter4isdedicatedtoseveralissuesinwhichasmallparameter appears (in different contexts). It includes averaging (in different situations),theKAMtheoryandthetheoryofrelaxationoscillations. In multi-dimensional systems, new phenomena appear: transitiv- ity, even distribution of trajectories, ergodicity, mixing and chaos. In Chapter 5, we discuss the ergodicity of translations on tori and dynamics of orientation-preserving diffeomorphisms of a circle. The most elementary example of a chaotic system is the famous Smale horseshoe map, defined for a single transformation. In Chapter5,weshowhowtheSmalehorseshoeappearsinsuchelemen- tary systems like a swing moved by periodic external force. We will also give other examples of chaotic behaviors, like Anosov systems and attractors. In the Appendix (Chapter 6), the reader will find collected main facts from the course on ODEs. The last section, about Hamiltonian systems, is more advanced and rather minimal compared to the others; the reader must forgive us for this. Each chapter contains a series of problems (with varying degrees of difficulty) and a self-respecting student should try to solve them. This book is based on Raul Murillo’s translation of Henryk Z˙o(cid:3)la¸dek’s lecturenotes,which wereinPolish andedited intheportal Matematyka Stosowana (Applied Mathematics) in the University of Warsaw. Preface vii At the end of the introduction, we would like to thank Zbigniew Peradzyn´ski, who carefully read the Polish version of the manuscript and gave a list of comments and errors, and to Maciej Borodzik, whose useful remarks have helped to improve the text. Henryk Z˙o(cid:3)la¸dek was supported by the Polish NCN OPUS grant No. 2017/25/B/ST1/00931. TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk About the Authors Henryk Z˙o(cid:2)la¸dek completed his education at the University of Warsaw in 1978. In 1983, he obtained his PhD degree from Moscow State University. He is working permanently at the University of Warsaw. He is also a coeditor (with G. Filipuk, A. Lastra, S. Michalik and Y. Takei) of the proceedings Complex Differ- ential and Difference Equations, de Gruyter Proceedings in Mathematics, 2020. He is spe- cializing in: Ordinary Differential Equations (limit cycles, normal forms), Mathematical Physics (classical and quantum mechanics) and Algebraic Geometry (algebraic curves). He is a member of the editorial board of the journals: Topological Meth- ods in Nonlinear Analysis,Qualitative Theory of Dynamical Systems, Journal of the Belarussian State University: Mathematics and Infor- matics. He is also an author of nearly a hundred papers and the monograph The Monodromy Group (Birkha¨user, 2006). He has pub- lishedhisworksinjournalssuchasJournal of Differential Equations, Journal of Statistical Physics, Israel Journal of Mathematics, and Lecture Notes in Mathematics. ix
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