Chaos - An Introduction for Applied Mathematicians

311 Pages·2020·12.975 MB·English

by  Fowler, Andrew, McGuinness, Mark

Andrew Fowler
Mark McGuinness
Chaos
An Introduction
for Applied Mathematicians
Chaos
Andrew Fowler Mark McGuinness
(cid:129)
Chaos
An Introduction for Applied Mathematicians
123
AndrewFowler Mark McGuinness
MACSI Schoolof Mathematics andStatistics
University of Limerick Victoria University ofWellington
Limerick, Ireland Wellington, NewZealand
ISBN978-3-030-32537-4 ISBN978-3-030-32538-1 (eBook)
https://doi.org/10.1007/978-3-030-32538-1
Mathematics Subject Classification (2010): 34A34, 34C23, 34C28, 34C37, 37B10, 37C29, 37D45,
37E05,37J40
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We dedicate this book to Stephen O’Brien, who
has variously brought both chaos and order, in
differing measures, to each of us. Stephen is a
leader in applied mathematics in Ireland at the
present time, and is a figurehead in the effort to
bring the subject there to a level where it
seriously engages with industry, science and
nature. Under his leadership, the Mathematics
Applications Consortium for Science and
Industry (MACSI) at the University of Limerick
hasbecomeaflagshipforappliedmathematicsin
Ireland.
Preface
The modern theory of chaos arguably stems from the publication of Ed Lorenz’s
remarkablepaperin1963ondeterministicnonperiodicflow.Ittookawhileforitto
be noticed outside the meteorological literature, but by the 1970s, the subject was
gainingmomentum,courseswerecreatedandinthe1980sbooksstartedtoappear.
At first few in number, but eventually they too became a roaring current. The
subject is now well established, and as well as plentiful literature, there are a
number of dedicated journals.
So thefirst thing anauthor needstodo istoprovidean excuse for why another
book should be added to the list. There are two reasons, both of which provided a
motivationfortheoriginaldraftingofthismaterial,andbothofwhichremainvalid
in my opinion.
Thefirstisthatmostofthebooksondynamicalsystemstakeaparticularpointof
view,whereasactuallythesubjectisacompositeofanumberofdifferenthistorical
strands.Forexample,Poincaré’s(1892)earlyinvestigationsweremotivatedbythe
problem of predictability in celestial mechanics, particularly with regard to the
stabilityofthesolarsystem,andthatworkledinonewaytotheKAMtheoremand
muchofthematerialconcerningstochasticityinHamiltoniansystems;butequallyit
led(viathePoincaré–Lindstedtmethod)tomodernmethodsofperturbationtheory
and bifurcation theory. This homogeneity of the subject has largely been lost in
books on chaos, which mostly focus on one subject area or another. The intention
here is to provide a view which accommodates all the different directions the
subject has taken, whether it be Hamiltonian systems, maps or bifurcation theory.
The second reason is found in the book’s title; I am an applied mathematician,
andthismeansIchartamiddlecoursebetweenthatofthephysicistandthatofthe
purer mathematician. But most of the books on chaos are by people who reside in
oneorother ofthese camps,andtheflavour oftheappliedmathematicianisrather
different.
vii
viii Preface
Applied mathematicians are less interested in theorems and proofs,1 because
most of what they do is too hard to prove anything about. That is not always true,
but it mostly is. They are interested in practicalities, and often this means
approximations. An applied mathematician is less interested in a proof that, for
example, the Lorenz equations have chaotic solutions: he2 is willing to take it on
trust since there is no reason not to think it so. On the other hand, he is a mathe-
matician, and will therefore be worried by the minutiae of (for example) dealing
withhomoclinicbifurcations,whereitisnecessaryforpeaceofmindtopursuethe
details and mechanism of proof: these would not be items of much concern for
physicists, who are yet more interested in observations, to the exclusion of some
of the technical detail.
For both these reasons, I, or rather we, feel that the present text will fill a small
gap in the literature. We provide in chapter one a semi-historical overview of the
development and context of the subject, and then in the following chapters we tell
the story of the four horsemen: maps, Hopf and homoclinic bifurcations and
Hamiltoniansystems.Thefinalchapterrepresentsanexplorationofothertopicsof
more recent and practical or simply intellectual interest. The treatment is succinct,
but in places we push it to a more challenging level. Such excursions provide
anotherdeviationfromothertexts.Inordertopreservewhatreadabilitythereis,we
havenotclutteredthetextwithreferences,andwecertainlydonotaimtobeoverly
extensive in our bibliography. The ‘Notes and references’ sections at the end of
eachchapterprovidebothcitationsandalsofurthercommentaryon,orelaboration
of, some of the earlier text.
Thematerialmaybesuitableforfinal-yearundergraduateorgraduatecourses;it
assumes a working knowledge of applied mathematics curricular subjects such as
differentialequations,mechanicsandnonlineardynamics.Inplaces,itmayassume
a familiarity with other likely staples of the curriculum: fluid mechanics and
mathematicalbiology,forexample.Butitalsodabbleswithmaterialwhichmaybe
less familiar to the intended audience, though still essential to the text: point set
topology, and some elements offunctional analysis.
In preparing this book for publication, I have taken on as co-author my
long-standing colleague and good friend Mark McGuinness, whose personal
entwinement with my career dates back 40 years. Mark and I met as postdoctoral
research fellows in Dublin in 1979, and we proceeded to write several papers on
chaos in the Lorenz equations, before going our separate ways: he to CalTech and
eventually Wellington, I to M.I.T. and eventually Oxford and subsequently
Limerick. And, eventually,we met up again, andhave since beenfrequentvisitors
toeachother'sinstitutions.Theplantofinishthisbookcoincidedwithavisitbyme
1But,Iwillconfess,youwillfindelementsofbothhere.Itissimplynecessarytothenarrative.
2Orshe:butforprosaicconvenienceIwilldeemthepronountoconnotepersonsofeithergender,
whenitisnotspecific.
Preface ix
to Victoria University, Wellington, where Mark is Professor in the School of
Mathematics and Statistics, and I have a position of Adjunct Professor, and I am
gratefulforthisaward,withoutwhichthisbookmightneverhaveseenthelightof
day!
Kilkee, Co. Clare, Ireland Andrew Fowler
February 2019
I am delighted to be involved in writing a book on chaos with my colleague,
co-author and friend Andrew Fowler. It is apposite that this book began during a
visit to Victoria University of Wellington from Andrew in his capacity as Adjunct
Professor, and was completed during my first visit as Adjunct Professor to the
University of Limerick.
Myfirstforaysintochaoticdynamicswerein1979withAndrewandwithJohn
Gibbon,inDublin,Ireland.WewereallnewlyintownandfreshoutofourPh.D.s,
andfreeandwillingtotalkaboutmathematicsinIrishpubsandovermealsateach
other’s homes. It was classic stuff, scribbling on napkins, finding connections
between soliton equations and the Lorenz equations, sharpening cusps and using
asymptotic methods to make the connection between differential equations and
one-dimensional maps. It was not until much later that I was to realise that what
happenedcollaborativelyandinstinctively inIrelandwasveryspecialindeed.This
bookisinmanywaysacelebrationofthosegoodtimes,andoftheveryproductive
years that followed.
Wellington, New Zealand Mark McGuinness
February 2019
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Bifurcation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Hopf Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Strange Attractors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Turbulence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Stochasticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 The Lorenz Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.7 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.7.1 Poincaré and the Swedish Prize . . . . . . . . . . . . . . . . . . . . 14
1.7.2 Weather Predictability . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 One-Dimensional Maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Chaos in Maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Bifurcations in Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Period-Doubling and the Feigenbaum Conjectures . . . . . . . . . . . . 28
2.3.1 Bifurcation Trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Symbolic Dynamics: The Kneading Sequence . . . . . . . . . . . . . . . 38
2.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.5.1 The Feigenbaum Conjectures . . . . . . . . . . . . . . . . . . . . . . 42
2.5.2 Kneading Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.5.3 Period-Doubling in Experiments. . . . . . . . . . . . . . . . . . . . 43
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3 Hopf Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1 Hopf Bifurcation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1.1 Implicit Function Theorem. . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.2 Proof of Hopf Bifurcation Theorem . . . . . . . . . . . . . . . . . 55
3.1.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
xi
xii Contents
3.2 Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.3 Centre Manifold Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3.1 Formal Power Series Expansion . . . . . . . . . . . . . . . . . . . . 69
3.3.2 Application to Hopf Bifurcation . . . . . . . . . . . . . . . . . . . . 72
3.4 Secondary Hopf Bifurcations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.4.1 Stability of Periodic Orbits. . . . . . . . . . . . . . . . . . . . . . . . 74
3.4.2 Floquet Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.4.3 Normal Forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.4.4 Weak and Strong Resonance . . . . . . . . . . . . . . . . . . . . . . 80
3.4.5 Circle Maps, Arnold Tongues, Frequency Locking . . . . . . 81
3.4.6 Structural Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.5 Tertiary Hopf Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.6.1 Many Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.6.2 Multiple Scale Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.6.3 Circle Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.6.4 Tertiary Hopf Bifurcation. . . . . . . . . . . . . . . . . . . . . . . . . 92
3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4 Homoclinic Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.1 Lorenz Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.1.1 Homoclinic Bifurcations. . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.1.2 One-Dimensional Map. . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.2 Symbolic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.2.1 The Smale Horseshoe . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.2.2 Strange Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.3 Shilnikov Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.3.1 Approximation and Proof. . . . . . . . . . . . . . . . . . . . . . . . . 119
4.4 Matched Asymptotic Expansions for n-dimensional
Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.4.1 Strange Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.4.2 Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . 129
4.4.3 Transition to Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.5.1 The Lorenz Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.5.2 Shilnikov Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.5.3 Infinite Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.5.4 Turbulence and Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5 Hamiltonian Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.1 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.1.1 Hamilton’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.1.2 Hamilton’s Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 147