Introduction to applied nonlinear dynamical systems and chaos

864 Pages·2003·5.013 MB·English

by  Wiggins S.

2
Texts in Applied Mathematics
Editors
J.E. Marsden
L. Sirovich
S.S. Antman
Advisors
G. Iooss
P. Holmes
D. Barkley
M. Dellnitz
P. Newton
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Stephen Wiggins
Introduction to
Applied Nonlinear Dynamical
Systems and Chaos
Second Edition
With 250 Figures
StephenWiggins
SchoolofMathematics
UniversityofBristol
Clifton,BristolBS81TW
UK
[email protected]
SeriesEditors
J.E.Marsden L.Sirovich
ControlandDynamicalSystems,107–81 DivisionofAppliedMathematics
CaliforniaInstituteofTechnology BrownUniversity
Pasadena,CA91125 Providence,RI02912
USA USA
[email protected] [email protected]
S.S.Antman
DepartmentofMathematics
and
InstituteforPhysicalScience
andTechnology
UniversityofMaryland
CollegePark,MD20742-4015
USA
[email protected]
MathematicsSubjectClassification(2000):58Fxx,34Cxx,70Kxx
LibraryofCongressCataloging-in-PublicationData
Wiggins,Stephen.
Introductiontoappliednonlineardynamicalsystemsandchaos/StephenWiggins.—2nded.
p.cm.—(Textsinappliedmathematics;2)
Includesbibliographicalreferencesandindex.
ISBN0-387-00177-8(alk.paper)
1.Differentiabledynamicalsystems. 2.Nonlineartheories. 3.Chaoticbehaviorin
systems. I.Title. II.Textsinappliedmathematics;2.
QA614.8.W5442003
003′.85—dc21 2002042742
ISBN0-387-00177-8 Printedonacid-freepaper.
2003,1990Springer-VerlagNewYork,Inc.
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Series Preface
Mathematics is playing an ever more important role in the physical and
biological sciences, provoking a blurring of boundaries between scientific
disciplinesandaresurgenceofinterestinthemodernaswellastheclassical
techniques of applied mathematics. This renewal of interest, both in re-
search and teaching, has led to the establishment of the series Texts in
Applied Mathematics (TAM).
The development of new courses isa natural consequence of a high level
ofexcitementontheresearchfrontierasnewertechniques,suchasnumeri-
cal and symbolic computer systems, dynamical systems, and chaos, mix
with and reinforce the traditional methods of applied mathematics. Thus,
the purpose of this textbook series is to meet the current and future needs
of these advances and to encourage the teaching of new courses.
TAM will publish textbooks suitable for use in advanced undergraduate
and beginning graduate courses, and will complement the Applied Mathe-
maticalSciences(AMS)series,whichwillfocusonadvancedtextbooksand
research-level monographs.
Pasadena, California J.E. Marsden
Providence, Rhode Island L. Sirovich
College Park, Maryland S.S. Antman
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Preface to the Second Edition
This edition contains a significant amount of new material. The main rea-
son for this is that the subject of applied dynamical systems theory has
seen explosive growth and expansion throughout the 1990s. Consequently,
a student needs a much larger toolbox today in order to begin research on
significant problems.
I also try to emphasize a broader and more unified point of view. My
goal is to treat dissipative and conservative dynamics, discrete and con-
tinuous time systems, and local and global behavior, as much as possible,
on the same footing. Many textbooks tend to treat most of these issues
separately(e.g.,dissipative,discretetime,localdynamics;globaldynamics
of continuous time conservative systems, etc.). However, in research one
generally needs to have an understanding of each of these areas, and their
inter-relations. For example, in studying a conservative continuous time
system, one might study periodic orbits and their stability by passing to a
Poincar´emap(discretetime).Thequestionofhowstabilitymaybeaffected
by dissipative perturbations may naturally arise. Passage to the Poincar´e
maprendersthestudyofperiodicorbitsalocalproblem(i.e.,theyarefixed
pointsofthePoincar´emap),buttheirmanifestationinthecontinuoustime
problem may have global implications. An ability to put together a “big
picture” from many (seemingly) disparate pieces of information is crucial
for the successful analysis of nonlinear dynamical systems.
This edition has seen a major restructuring with respect to the first
edition in terms of the organization of the chapters into smaller units with
a single, common theme, and the exercises relevant to each chapter now
being given at the end of the respective chapter.
The bulk of the material in this book can be covered in three ten week
terms. This is an ambitious program, and requires relegating some of the
materialtobackgroundreading(describedbelow).Mygoalwastohavethe
necessary background material side-by-side with the material that I would
lecture on. This tends to be more demanding on the student, but with the
right guidance, it also tends to be more rewarding and lead to a deeper
understanding and appreciation of the subject.
The mathematical prerequisites for the course are really not great; ele-
mentary analysis, multivariable calculus, and linear algebra are sufficient.
In reality, this may not be enough on its own. A successful understanding
of applied dynamical systems theory requires the students to have an inte-
viii Preface to the Second Edition
grated knowledge of these prerequisites in the sense that they can fluidly
manipulate and use the ideas between the subjects. This means they must
possess the quality often referred to as “mathematical maturity.” A study
of dynamical systems theory can be a good way to obtain this. In addi-
tion, an ordinary differential equations course from the geometric point of
view (e.g., the material in the books of Arnold [1973] or Hirsch and Smale
[1974]) would be ideal.
Chapters 1-17 form the core of the first term material. It provides stu-
dentswiththebasicconceptsandtoolsforthestudyofdynamicalsystems
theory. I tend to cover chapters 7, 11 and 12 at a brisk pace. The main
point there is the ideas and main results. The details can be grasped over
time,andinothersettings.Chapters13-17couldbeviewedasbelongingto
the common theme of “dynamical systems with special structure.” Chap-
ter14isthemostimportantofthesechapters.Therelation,andcontrasts,
between Hamiltonian and reversible systems is useful to understand, and
is the reason for including chapter 16. I often just assign selected back-
ground reading from chapter 13, but knowledge of the relation between
Lagrangian and Hamiltonian dynamical systems is of growing importance
in applications. Gradient dynamical systems arise in numerous applica-
tions (e.g., in biologically related areas) and knowledge of the nature of
their dynamics, and how it contrasts with, e.g., Hamiltonian dynamics, is
important.Chapter17isshort,butIhavealwaysfeltthatstudentsshould
be aware of these results because there are numerous examples of systems
arising in applications that experience a “transient temporal disturbance.”
ThroughouttheearlychaptersIdiscussanumberofresultsandtheoretical
frameworks for general nonautonomous vector fields (i.e., time-dependent
vector fields whose time dependence is not periodic). This area tradition-
ally has not been a part of dynamical systems from a geometric point of
view, but this situation is changing rapidly, and I believe it will play an
increasingly important role in applications in the near future.
Chapters 18-22 are covered in the second term. The subject is “local
bifurcation theory.” The two key tools for the local analysis of dynami-
cal systems are center manifold theory and normal form theory, covered in
chapters18and19.Thechapteronnormalformtheoryisgreatlyexpanded
from the first edition. The main new material is the normal form work of
Elphick, Tirapegui, Brachet, Coullet and Iooss, a discussion of Hamilto-
nian normal form theory (following Churchill, Kummer, and Rod), and
some material on symmetries (whose possible existence, and implications,
shouldbeconsideredinthecourseofstudyofanydynamicalsystem).Pos-
sibly sections 19.1-19.3 could have been omitted in this edition; however
it has been my experience that students understand the later (and more
difficult) material more easily once they have been exposed to this more
pedestrian introduction. In chapters 20 and 21 I tend not to cover in much
detail the material related to the codimension of a bifurcation and versal
deformations. This is a standard language used in discussing the subject
Preface to the Second Edition ix
and it is important that the students have all the details available to them
for background reading and see it in the context of the material I lecture
on. New material on Hamiltonian bifurcations and circle maps is included.
The inclusion of introductory material on Hamiltonian bifurcations is an
example of the effort to have a broader and more unified point of view as
discussed earlier. For example, we first describe the “generic saddle-node
bifurcation at a single zero eigenvalue.” It is then natural to ask about the
saddle-node bifurcation in a Hamiltonian system, which turns out to be
rather different. Chapter 22 mainly serves as a warning that the way in
which bifurcation phenomena are discussed in applications may not agree
with the mathematical reality, and appropriate pointers to the literature
are given.
Chapters 23-33 are covered in the third term. The subject is “global
dynamics, bifurcations, and chaos.” There is a sprinkling of new material
throughout these chapters (e.g., a proof of a simple version of the lambda
lemmaandaproofoftheshadowinglemma),butthestructureisbasically
the same as the first edition.
There is not a great deal of overlap between the material in the individ-
ual terms, and with the appropriate prerequisites, each of these one term
courses could be viewed as an independent course in itself. The textbook
provides the necessary background for the students to make this a possi-
bility.
Some material has been left out of this edition; in particular, material
onaveraging,thesubharmonicMelnikovfunction,andlobedynamics.The
reason is that over time I have begun to cover averaging and the subhar-
monicMelnikovfunctionastopicsinacoursesolelydevotedtoperturbation
methods. I cover lobe dynamics in a course devoted to transport phenom-
ena in dynamical systems, which has developed in the last ten years to the
point that it now justifies an independent course of its own, with applica-
tions taken from many diverse disciplines.
It has been my experience over time that a significant obstacle for stu-
dents in their study of the subject is the sheer amount of (initially) unfa-
miliar jargon. In order to make this a bit easier to deal with I have now
included a glossary of frequently used terms. The bibliography has also
been updated and greatly expanded.
I would also like to take this opportunity to express my gratitude to
the National Science Foundation and to Dr. Wen Masters and Dr. Reza
Malek-Madani of the Office of Naval Research for their generous support
of my research over the years. Research and teaching are two sides of the
same coin, and it is only through an active and fruiful research program
that the teaching becomes alive and relevant.
Bristol, England Stephen Wiggins
2003