Table Of ContentQualitative
Theory of ODEs
An Introduction to
Dynamical Systems Theory
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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
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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
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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.
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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.
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