168 pages on 40 lb Velocity Spine: 3/16" Print Cover on 9pt
CarolinaISSN 0002-9920 Volume 57, Number 9 Notices of the American
Mathematical Society Volume 57, Number 9, Pages 10731240, October
2010of the American Mathematical SocietyTrim: 8.25" x 10.75"October
2010Topological Methods for Nonlinear Oscillationspage 1080Tilings,
Scaling Functions, and a Markov Processpage 1094Reminiscences of
Grothendieck and His Schoolpage 1106Speaking with the Natives:
Reectionson Mathematical Communicationpage 1121New Orleans
Meetingpage 1192 What if Newton saw an apple as just an apple?Take
a closer look at TI-Nspire software and get started for free. Visit
education.ti.com/us/learnandearn Mac is a registered trademark of
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page 1151) ,!4%8 ,!4%8,!4%8c=LL vcn ===Lic=+iceHow to appIy
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atwww.sciencesmaths-paris.frand, for PGSM,
atwww.sciencesmaths-paris.fr/pgsmrcLc=+ic eciccce M=+icm=+icLce cc
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re& ==nie ccccx ce1cL. : + & ee c - r=x : + & ee
crcLccne ecic+ivc ==n+cne ==n+cneThe Foundation Sciences
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entire list of titles on sale.AMERICAN MATHEMATICAL SOCIETYFeatures
Communications 1116 Alice Turner Schafer (19152009):
RemembrancesGeorgia Benkart, Bhama Srinivasan, Mary Gray, Ellen
Maycock, Linda Rothschild, edited by Anne Leggett 1125 WHAT IS... a
Linear Algebraic Group?Skip Garibaldi 1127 Doceamus: The Doctor Is
InRobert Borrelli 1132 Meta-Morphism: From Graduate Student to
Networked MathematicianAndrew Schultz 1137 Lovsz Receives Kyoto
Prize Commentary 1079 Opinion: Agenda for a Mathematical
RenaissanceTeodora-Liliana R adulescu and Vicentiu R adulescu 1129
The Cult of Statistical SignificanceA Book ReviewReviewed by Olle
HggstrmNoticesof the American Mathematical SocietyThe feature
articles this month exhibit the dynamic and diversity of modern
mathematics. The article by Chris Byrnes considers applications of
modern topology to questions of physics. The article by Dick Gundy
shows how wavelets can be used to understand probability theory.
And the article by Jerry Folland considers questions of
communication that will resonate with us all. The interview with
Luc Illusie aboutAlexander Grothendieck will give us all pause for
thought.Steven G. KrantzEditor 1080 Topological Methods for
Nonlinear Oscillations Christopher I. Byrnes 1094 Tilings, Scaling
Functions, and a Markov Process Richard F. Gundy 1106 Reminiscences
of Grothendieck andHis School Luc Illusie, with Alexander
Beilinson, Spencer Bloch, Vladimir Drinfeld, et al. 1121 Speaking
with the Natives: Reflections on Mathematical Communication Gerald
B. FollandOctober 20101106 11291094 1116From the AMS SecretaryAMS
Officers and Committee Members . . . . . . . . . . . . . . . . . .
. 1152Statistics on Women Compiled by the AMS . . . . . . . . . . .
. . . . 1163EDITOR: Steven G. KrantzASSOCIATE EDITORS: Krishnaswami
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durability.DepartmentsAbout the Cover . . . . . . . . . . . . . . .
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People . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 1138Nagel and Wainger Receive 2007-2008 Bergman
Prize, AMS Menger Awards at the 2010 ISEF, Gupta and
Grattan-Guinness Awarded May Prize, Buchweitz Receives Humboldt
Research Award, Prizes of the Royal Society, SIAM Prizes Awarded,
Prizes of the London Mathematical Society, Prizes of the Canadian
Mathematical Society, 2010 International Mathematical Olympiad,
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. . . . . . . . . . . . . . 1239Noticesof the American Mathematical
SocietyOpinions expressed in signed Notices articles are those of
the authors and do not necessarily reflect opinions of the editors
or policies of the American Mathematical Society.I thank Randi D.
Ruden for her splendid editorial work, and for helping to assemble
this issue. She is essential to everything that I do.Steven G.
KrantzEditorGOOD MATH.ITS WEBASSIGN, ONLY BETTER.Only WebAssign is
supported by every publisher and supports every course in the
curriculum from developmental math to calculus and beyond. Thats
good. Now weve added the latest homework tools to bring you the
ultimate learning environment for math. Thats better. Welcome to a
better WebAssign. There are new tools including number line, that
lets students plot data graphically, then automatically assesses
their work. There are more books, over 170 titles from every major
publisher. And theres more server capacity with an expanded support
team just waiting to serve you and over half a million loyal
WebAssign users each term. Its the culmination of a team-wide
effort to build the best WebAssign ever. The ingenuity of WebAssign
for math. Its available now and its free to faculty at
WebAssign.net. Now what could be better than that?Visit
www.webassign.net to sign up for your free faculty account
today.Students use number line to plot data graphically with
automatic assessment.Terence Tao Professor of Mathematics,
University of California, Los AngelesThe American Mathematical
Society presentsThe AMS Einstein Public Lecture in MathematicsThe
Cosmic Distance LadderHow do we know the distances from the earth
to the sun and moon, from the sun to the other planets, and from
the sun to other stars and distant galaxies? Clearly we cannot
measure these directly. Nevertheless there are many indirect
methods of measurement, combined with basic mathematics, which can
give quite convincing and accurate results without the need for
advanced technology (for instance, even the ancient Greeks could
compute the distances from the earth to the sun and moon to
moderate accuracy). These methods rely on climbing a cosmic
distance ladder, using measurements of nearby distances to deduce
estimates on distances slightly farther away. In this lecture, Tao
will discuss several of the rungs in this ladder.Saturday, October
9, 2010 6:15 p.m. with a reception to follow Schoenberg Hall on the
UCLA campusSponsored by the American Mathematical Society Hosted by
the UCLA Department of Mathematics This event is part of the AMS
2010 Fall Western Sectional Meeting, October
9-10www.ams.org/meetings/einstein-lect.html Mars:
NASA/JPL-Caltech/University of ArizonaSaturn: NASA/ESA/Erich
Karkoschka (University of Arizona)Jupiter: NASA/JPLGalaxy-1:
NASA/JPL-Caltech/STSclGalaxy-2 (background): MPIA/NASA/Calar Alto
ObservatoryOCTOBER 2010 NOTICES OF THE AMS 1079OpinionAgenda for a
Mathematical RenaissanceMost of us have wondered at some point in
our careers how to motivate the interest of students in mathematics
and encourage young talents to learn mathematics. In 1975 Paul
Halmos [3] said: The best way to learn is to do; the worst way to
teach is to talk. The best way to teach is to make students ask,
and do. Dont preach factsstimulate acts. Halmos was a wise and
respected scholar and there is general agreement that what he said
is correct and important, but change is so difficult that perhaps
we need more than words; we need a new agenda.One may begin with a
recent analysis developed for The Wall Street Journal [4] that
evaluates 200 professions. Ac-cording to the study mathematician is
the top job in the U.S., placing first in terms of good
environment, income, employ-ment outlook, physical demands, and low
stress. Actuary and statistician, two related professions, rank
second and third, respectively. The sociological analysis provides
answers to the young students question Why do mathematics?
Math-ematicians are in demand in terms of job prospects. Also,
should anyone wonder if interest in mathematics has slowed down,
the study shows that the role of mathematicians, both pure and
applied, in the development of our society is as important as
ever.Science and technology developed in an impressive way before,
during, and immediately after World War II, which at-tracted
post-war students to careers in research. This trend received a
powerful stimulus in the 1960s and 1970s with the advent of the
space age: launch of the Soviet satellite Sputnik (1957), Gagarins
first human travel into space (1961), and the first manned mission
to land on the Moon (Neil A. Armstrong and Edwin E. Aldrin, Apollo
11, 1969). The impact of these advances was huge. Science was
recognized for the political and economic power it could generate.
In those golden years, research became as important to society as
it was fascinating to practitioners.Decades later, societys
interest in many areas of research has diminished in most countries
around the world. With exceptions in biomedical areas, genetics, or
software engineer-ing, the importance of mathematics and science in
society is less often recognized. Mathematics and the sciences are
no longer perceived as offering desirable career opportuni-ties.
Both in developed and developing countries, brilliant mathematics
students who could have chosen careers in mathematics are not doing
so.It is strange that students interest is low at a time when
career opportunities for professional mathematicians are greater
than ever [2]. This is true both for the applied fields where
demand for mathematicians will continue to grow rap-idly in the
next decades and for traditional areas that are rich with new
developmentssee the seven Millennium Problems, cf. [1]. Yet today
we notice a fundamental lack of appreciation for the richness and
relevance of mathematics itself.It is possible for mathematics and
mathematicians to regain social stature. The scientific enterprise
can function at full potential if there is a fast flow of knowledge
between the creators and users of mathematics. This is something
mathematics education can and should facilitate, especially since
mathematics is currently so active and vital both in research and
applications.The culture of this millennium shows itself to be
highly in-teractive and collaborative. It is an opportunity for
mathemati-cians to work with scientists in other fields and also to
reach out to the community at large. Mathematicians are uniquely
qualified to articulate the value of mathematics in catalyz-ing
major advances in science, health, business, economics, biomedical
engineering, genetics, software engineering, and, more generally,
in proving the patterns and the truths of the universe in which we
live.The trend toward interactivity is an important feature of the
sciences in our time. Unfortunately, some institutions have been
slow to adapt to this reality. Mathematics loses a lot when it is
isolated or fragmented according to various paradigms. Universities
around the world, as well as many in-dustries and government
agencies, will benefit from removing barriers to collaboration. In
particular, powerful and diverse interactions between academic and
industrial mathematicians should be enhanced. While the primary
missions of academia and industries are different, the two cultures
have much to learn from one another.In short, while mathematician
is a top job in the U.S. today, it is no longer possible for a
mathematician to remain aloof from the passing needs of the world
or to continue work-ing in an ivory tower. As funds get scarce, the
future of our profession is at stake. It is time for mathematicians
to bring the vitality and usefulness of modern mathematics to the
classrooms, to demonstrate its social impact, and to support this
centurys mathematical renaissance.References[1]
http://www.claymath.org/millennium.[2] P. A. Griffiths, Mathematics
at the turn of the millennium, Current Science Online 77 (1999),
750758.[3] P. Halmos, The problem of learning to teach, Amer. Math.
Monthly 82 (1975), 466476.[4] S. E. Needleman, Doing the math to
find the good jobs. Mathematicians land top spot in new ranking of
best and worst occupations in the U.S., The Wall Street Journal,
Jan. 26, 2009, p. D2
(http://online.wsj.com/article_email/SB123119236117055127-lMyQjAxMDI5MzAxNjEwOTYyWj.html).[5]
A. N. Whitehead, The Aims of Education and Other Essays, MacMillan
Co, 1929 (reprinted in Education in the Age of Sci-ence, edited by
Brand Blanshard, New York, Basic Books, 1959).Teodora-Liliana
RadulescuDepartment of Mathematics, Fratii Buzesti National
[email protected] RadulescuInstitute of
Mathematics Simion Stoilow of the Romanian
[email protected]@math.cnrs.frTopological
Methods forNonlinear OscillationsChristopher I.
ByrnesIntroductionPeriodic phenomena play a pervasive role in
natu-ral and in man-made systems. They are exhibited,for example,
in simple mathematical models ofthe solar system and in the
observed circadianrhythms by which basic biological functions
areregulated. Electronic devices producing stable pe-riodic signals
underlie both the electrication ofthe world and wireless
communications. My inter-est in periodic orbits was heightened by
researchinto the existence of oscillations in nonlinear feed-back
systems. While these kinds of applicationsare illustrated in
Examples 2.2 and 4.2, a more de-tailed expedition into this
important applicationarea is omitted here for the sake of space
andfocus.Periodic orbits have played a prominent role inthe
mathematics of dynamical systems and its ap-plications to science
andengineering for centuries,due to both the importance of periodic
phenom-ena and the formidable intellectual challengesinvolved in
detecting or predicting periodicity.As a rst step toward addressing
this challenge,Poincar developed his method of sections, begin-ning
with the observation that, if a periodic orbit for a smooth vector
eld X exists, if x0 and ifH is a hyperplane complementary to the
tangentline Tx0() to x0 at , then on a suciently smallneighborhood
S H of x0 one can dene, by theimplicit function theorem, a (least)
positive timetx > 0 so that for each x M the solution to
theChristopher Byrnes, former dean of the School of Engi-neering
and Applied Science at Washington Universityin St. Louis, was a
distinguished visiting professor in op-timization and systems
theory at the Royal Institute ofTechnology in Stockholm when he
died unexpectedly inFebruary 2010.dierential equation dened by X
with initial con-dition x returns to H. In particular, one can
denea smooth Poincar, or rst-return, map P on S,which sends the
initial condition x to the solutionof the dierential equation at
time tx. Moreover,the dynamics of the iterates of P on S are
thenintimately related to the dynamics of X, near ,in positive
time. Conversely, if a local sectionS transverse to X exists for
which there exists aPoincar map P, the existence of periodic
pointsfor P implies the existence of periodic orbits forX, allowing
for the use of powerful topologicalxed point and periodic point
theorems in thestudy of nonlinear oscillations. The importance
ofPoincars method of sections led G. D. Birkhoto develop two sets
of necessary and sucientconditions [1] for the existence of a
section for adierential equation evolving in Rn. One of thesewas
formulated in terms of what Birkho calledan angular variable, and
the other involved what,in modern terminology, would be called an
angu-lar one-form. Both concepts are reviewed in thisarticle.The
existence of a section is, of course, bothone of the standard
paradigms for the existenceof nonlinear oscillations and one of the
grandtautologies of nonlinear dynamics, since to knowwhether S is
section for X is to know a lotabout the long-time behavior of the
trajectories ofthe corresponding dierential equationin whichcase
one might already know whether there areperiodic orbits.
Nonetheless, this paradigm hasactually been used with great success
in appli-cations, most notably beginning with Birkhosproof of
Poincars Last Theorem, which arosein the restricted three-body
problem in celestialmechanics. An easier paradigm is provided by
the1080 Notices of the AMS Volume 57, Number 9principle of the
torus, which has been widely usedin applications to biology,
chemistry, dynamics,engineering, and physics [13]. In this
literature, ifDNdenotes the closed unit disc in RN, a subman-ifold
M Rnthat is dieomorphic to Dn1 S1is called a toroidal region, and
the principle ofthe torus asserts that, if a smooth vector eld
Xleaves a toroidal region positively invariant andhas a section S
that is dieomorphic to Dn1, thenX has a periodic orbit in M by
Brouwers xedpoint theorem. Of course, among the limiting fea-tures
of the principle of the torus is the need notonly to nd a section
but also to have the abilityto characterize familiar topological
spaces such astoroidal regions, disks, and spheres.
Fortunately,remarkable advances in dynamics and topologysince
Poincars time now allow us to eectivelyaddress both of these
technical issues.Among the tools from dynamics that play arole in
the results described in this paper arethe properties of nonlinear
dynamical systemsthat dissipate energy. This has been developedin
two separate schools, one pioneered by Lia-punov and the other
beginning with Levinson andsignicantly developed by Hale,
Ladyzhenskaya,Sell, and others. One of the main results of
thelatter school concerns the existence of Liapunovstable global
attractors for dissipative systems.The topological methods
described here are alsoglobal, allowing one to bring techniques
such asthe classical combination of cobordism and ho-motopy theory,
as described in [14], to bear onthe study of nonlinear
oscillations. The early useof topological methods in the study of
nonlineardynamics dates back to the work of Poincar,Birkho,
Lefschetz, Morse, Krasnoselskii, Smale,and many others. The results
described here alsorely on global topological methods developed
byF. W. Wilson Jr. in his study of the topology ofLiapunov
functions for global attractors. For thecase of periodic orbits,
Wilsons results form thestarting point for the derivation of
necessary con-ditions, derived in [5], for the existence of
anasymptotically stable periodic orbit for a smoothvector eld dened
on an orientable manifold. Theproof uses a very general cobordism
theorem ofBarden, Mazur, and Stallings [10] in dimensionsbigger
than 5. In lower dimensions, crucial use isalso made of the
solution of the Poincar Con-jecture in dimensions 3 and 4 by
Perelman andFreedman and a result of Kirby and Siebenmannon
smoothings of 5-manifolds. The remarkablefact that the necessary
conditions are sucientfor the existence of a periodic orbit follows
froman explicit cobordism argument [5] involving theperiod maps of
one-forms, introduced by Abel.In this paper, I give a brief
overview of theproofs of these results and then describe howto
combine them to derive a new sucient con-dition, which replaces a
topological assumptionwith an assumption that the dynamical
systemdissipates energy. These sucient conditions areeasier to use
in practice and are illustrated inseveral ways, including examples
taken from pop-ulation dynamics and feedback control systems.This
exposition concludes with an existence the-orem that is valid for a
much more general classof smooth manifolds but that requires a more
re-strictive hypothesis. Among several applications,this result is
illustrated in the case of the exis-tence of periodic orbits for
smooth vector elds oncompact 3-manifolds, with or without
boundary.In closing, it is a pleasure to acknowledgevaluable advice
from Roger Brockett, Tom Farrell,Dave Gilliam, Moe Hirsch, John
Morgan, Ron Stern,and Shmuel Weinberger.Stability of Equilibria,
Periodic Orbits, andCompact AttractorsIn this section, some basic
results about asymp-totic stability of compact sets that are
invariantwithrespect toasmoothvector eldX arereviewedand
illustrated for a feedback design problem pre-sented in Example
2.2. Except for the last twosections of this survey, I will only
need theseresults for vector elds dened on Rn, on thetoroidal
cylinder, RnS1, or on the solid torus,DnS1. In this section, I will
conne the discussionto the case of vector elds on the toroidal
cylinder,which in this section is denoted by M. In Sections3 and 4,
vector elds on solid tori are studiedin more detail. It should be
noted, however, thatthese results, suitably formulated, do hold
forsmooth paracompact manifolds, with or withoutboundary, and with
careful modication they alsohold in innite dimensions [9].Any point
in M has coordinates (x, ), wherex Rnand S1, and therefore any
smoothvector eld X on M has the formX =
f1(x, )f2(x, )where f1 takes values in Rnand f2 takes valuesin
R. The vector space of smooth vector elds onM is denoted by
Vect(M). In particular, a smoothvector eld denes, and is dened by,
an ordinarydierential equation (ODE) x = f1(x, ) (2.1) = f2(x, )
(2.2)to which the local existence, uniqueness, andsmoothness
theorem for solutions to ODEs ap-plies, since small variations in
an initial conditionz0 = (x0, 0) take place in anopen subset of
RnR.(t, z0), dened for suciently small t, will de-note the solution
initialized at z0 at time t0 = 0.In this paper, only vector elds
for which (t, z0)October 2010 Notices of the AMS 1081is dened, for
each z0 M and for all t 0, areconsidered. Any such X denes a
semiow(2.3) : [0, ) M M.When t is xed, it is often convenient to
use thenotation t(z) := (t, z). In particular, denes asemigroup of
smooth embeddings t of M.Anequilibriumfor X is a point z0 M
satisfyingX(z0) = 0 or, equivalently, (t, z0) = z0 for allt 0. A
solution curve of (2.3) initialized at anonequilibrium point z M is
periodic providedt(z) = z for some t > 0. The minimumtime T >
0such that T(z) = z is its period and the set ofpoints in M
transcribed by a periodic solution iscalled a periodic orbit. A
subset I of M is positivelyinvariant for a vector eld X if (t, z) I
for eachz I and every t 0. I is invariant if (t, z) I foreach z I
and every t R. Equilibria and periodicorbits are invariant sets. A
compact invariantset K is a maximal compact invariant set for
thesemigroup (2.3) provided every compact invariantset of (2.3) is
contained in K.For any B M, the -limit set of B is dened[9] as(B) =
{z B| for zj B and tj +,with j +, (tj, zj) z}.(2.4)For B = {z},
this coincides with the -limit set(z) introduced by Birkho in [1].
The -limitsets, (B) and (z), are dened as in (2.4) withthe sequence
of times tj tending to . Following[9], a closed set A M is said to
attract a closedset B M provided the distance(2.5) (t(B), A) :=
supzB infyA d(t(z), y)betweenthe sets t(B) and Atends to 0 as t
+,where d is any complete metric on M.Denition 2.1 ([9]). A compact
invariant subset Kis said to(1) be stable provided that for every
neighbor-hood V of K, there exists a neighborhoodVof K, satisfying
t(V) V, for all t 0;(2) attract points locally if there exists a
neigh-borhood W of K such that K attracts eachpoint in W;(3) be
asymptotically stable if K is stable andattracts points
locally.Remark 2.1. If K is a compact invariant set, thenotion of
attracting a point or attracting a com-pact set is independent of
the choice of metric, asit should be. Moreover, since K is compact,
condi-tion (3) is equivalent to the existence of a
positivelyinvariant neighborhood K L for which K attractsL [9,
Lemma 3.3.1]. The largest open set, D, at-tracted by K is called
the domain of attraction ofthe attractor K.Denition 2.2. If a
compact subset K M satisesconditions (1) and attracts every point
of M, thenK is called a global attractor.Compact attractors exist
in many situations inwhich the dynamical system dissipates energy,
anotion that can be mathematized in several ways.There are two
formulations of dissipativity thatare very useful. In reverse
chronological order, onehas its roots in the work by Levinson on
the forcedvan der Pol oscillator and is developed in [9] forthe
case of Banach spaces. In this exposition, ittakes the following
form.Denition 2.3. X Vect(M) is point-dissipativeprovided there
exists a compact set K M thatattracts all points in M.Remark 2.2.
For any > 0, the -neighborhood,B = B(K), of a global attractor K
is a relativelycompact absorbing set; i.e., every trajectory
even-tually enters and remains in B. A system is point-dissipative
if, and only if, there exists a relativelycompact absorbing
set.Point-dissipative systems on Rnare also some-times referred to
as being ultimately boundedsystems, and their origin lies in
classical nonlinearanalysis.Example 2.1. Consider a Cperiodically
time-varying ordinary dierential equation(2.6) x = f (x, t), f (x,
t +T) = f (x, t)evolving on Rn. Historically, a central
questionconcerning periodic systems is whether thereexists an
initial condition (x0, 0) generating aperiodic solution having
period T. Such solu-tions are called harmonic solutions.
Followingthe pioneering work of Levinson on dissipativeforced
systems in the plane, V. A. Pliss formulatedthe following general
denition for periodicallytime-varying systems:Denition 2.4 ([16]).
The periodic dierentialequation (2.6) is dissipative provided there
existsan R > 0 such that(2.7) limtx(t; x0, t0) < R.In
particular, the ball B(0, R) of radius R about0 Rnis an absorbing
set for the time-varyingsystem (2.7). As noted in [16], the system
(2.6)denes a time-invariant vector eld X on thetoroidal cylinder M
via x = f (x, ) (2.8) = 1. (2.9)To say that (2.6) is dissipative on
Rnis to say that(2.8) is point-dissipative on M. For a
dissipativeperiodic system, one can dene a smooth Poincarmap P :
RnRndened via(2.10) P(x0) = (T; x0, 0).An important consequence
[16] of (2.7) is thatthere exists a closed ball 0 B Rnand an r
N1082 Notices of the AMS Volume 57, Number 9such that if x0 B, then
x(t; x0, 0) B for t rT;i.e.,(2.11) Pr: B B.ApplyingBrouwers
xedpoint theoremtoPr, Pliss[16] showed the existence of a forced
oscillation.In fact, Browders xed point theorem asserts thatany
mapsatisfying (2.11) must have a xedpoint inB so that there always
exists a harmonic solution.The theory of dissipative systems has
beenstudied by many mathematicians. Indeed, dissi-pative systems
play a central role in the workof Krasnoselskii, Hale,
Ladyzhenskaya, Sell, andothers for both nite- and
innite-dimensionalsystems. In the present context, the
principalresult is the following.Theorem 2.1. If X Vect(M) is
point-dissipative,then there exists a compact attractor Afor X on
M.Ais the maximal compact attractor and satisesA= {z M : {(t, z) :
< t < }is relatively compact}.(2.12)In particular, if B M is
relatively compact, then(B) A. Moreover, Ais connected.Remark 2.3.
For an ODE evolving on Rn, any tra-jectory with initial condition z
such that {(t, z) : < t < } is bounded is called Lagrange
stable.Example 2.2. The problem considered in this ex-ample is
referred to as a set-point control prob-lem in the systems and
control literature and iswidely used in engineering applications in
whichcertain physical variables need to be maintainedasymptotically
close to a desired constant. Impor-tant examples include
controlling the temperatureand air quality in buildings, for which
heating, ven-tilation, and air conditioning consume the
largestportion of energy costs, and controlling the alti-tude or
airspeed of aircraft, a problem of greatimportance for air trac
control. In more detail,consider the system(2.13) x1 = gx1+u1, x2 =
gx2+u2, x3 = x1u2x2u1x3,which models [2] the control of a rotor by
an ACmotor, with (x1, x2) being the components of themagnetic eld,
u1, u2 being the current throughthe armature coils, g the
resistance in the coils,x3 modeling the angular velocity of the
rotor, and the coecient of friction. The control objectivestudied
in [2] was to design a Ccontrol law(2.14) u1 = u1(x1, x2, x3, d),
u2 = u2(x1, x2, x3, d)so that the system (2.13)(2.14) has the
propertythat, in steady-state, limtx3(t) = d, for a de-sired
constant rate of rotation d > 0. In the engi-neering literature,
a system of this form, in whichexplicit control laws are fed back
into a controlsystem, such as (2.13), is called a closed-loop
sys-tem, and the control laws are referred to as feed-back laws.A
natural starting point is to determine nec-essary conditions on the
control laws in order tohave the closed-loop system(2.13)(2.14) be
point-dissipative on some positively invariant open setD R3and
solve the set-point control problemfor all initial conditions x D.
Point-dissipativitywould imply the existence of a global compact
at-tractor A D, while solving the set-point controlproblem would
consist of ensuring that x3|A =d. Since A is invariant, x3|A = 0,
which impliesx1 x2x2 x1 = d or(2.15) = dx21+x22> 0, for xA,where
(r, ) denotes polar coordinates in the(x1, x2)-plane. In
particular, the magnetic eldmust rotate in steady-state, as it
should in orderto generate torque. In fact, for conventional
ACmotors, the rotational rate of the magnetic eld ofthe AC motor
should be constant, and imposingthe condition = f > 0 yields
several additionalconclusions. For example, since any trajectoryon
A will be a closed curve in the ane plane,x3 = d, having constant
amplitude A =
d/f ,the global compact attractor A must consist ofthis single
periodic orbit. If one assumes that thecontrol laws (2.14) are
dened on all of R3andthat the rotational rate for the magnetic eld
ofthe AC motor is constant for all initial conditionsin D, one is
led to the further constraint(2.16) x1u2x2u1 = f (x21+x22)on
(2.14), which yieldsu1 = x1f x2+x1h(x1, x2)+H1(x1, x2, x3)(x3d),u2
= x2+f x2+x2h(x1, x2)+H2(x1, x2, x3)(x3d),for some R. Setting h =
0, each of thesecontrol laws produces a closed-loop system hav-ing
a periodic orbit with period T = 2/ onx3 = d, evolving as a
classical harmonic motion, x1 = f x2, x2 = f x1, on the circle
(x21+x22) = d/f .In [2], Brockett shows that the feedback lawu1 =
gx1f x2+(d x3)x1(2.17)u2 = gx2+f x1+(d x3)x2(2.18)where f , > 0
solves the set-point control prob-lem, inducing an asymptotically
stable periodic or-bit on D = R3 X3, where X3 is the x3-axis.In
fact, this system is point-dissipative on D R2S1, with as its
compact global attractor. Thisis easiest to see using Liapunov
methods, whichare described below.A very powerful way to formulate
dissipationof energy near an equilibrium was developed byLiapunov
in his 1892 thesis and has since beenextended to uniformly
attractive closed invariantOctober 2010 Notices of the AMS
1083sets. The main results for compact invariant setssuce for
point-dissipative systems.Denition 2.5. Suppose X leaves an open
subsetD M positively invariant and that K D is acompact invariant
subset. A Liapunov function Vfor X on the pair (D, K) is a
Cfunction V : D Rthat satises(1) V|K = 0 and V(z) > 0 for z
K,(2) V < 0 on DK, and(3) V tends to a constant value (possibly
)on the boundary, D, of D.Theorem 2.2. Suppose X leaves an open
subsetD M positively invariant and that K D is acompact invariant
subset. If a function V exists sat-isfying conditions (1) and (3)
of Denition 2.5 andif V 0 on D, then K is stable. If V also
satisescondition (2), then K is a global compact attractoron
D.Example 2.3 (Example 2.2 (bis)). Consider theclosed-loop vector
eld X Vect(R3) obtainedby implementing the feedback law (2.17) in
thesystem (2.13)(2.19) x1 =f x2+(d x3)x1, x2 =f x1+(d x3)x2, x3 =f
(x21+x22) x3.As noted above, (2.19) has a periodic solution with
initial condition x(0) = (
d/f , 0, d)T. Fol-lowing [2], consider the function V : D [0,
)dened byV(x1, x2, x3) = (d x3)2+f (x21+x22)d ln(x21+x22) +d(ln(d/f
) 1).(2.20)The function V satises conditions (1) and (3) ofDenition
2.5 for K = . Moreover,(2.21) V(x1, x2, x3) = 2(d x3)2 0,along
trajectories of X, so that is stable. Since Vis nonincreasing along
trajectories, it follows thatfor any x D, (x0) is an invariant
subset ofV1(0)a very useful result known as LaSallesinvariance
principle. To say V = 0 is to say thatx3 = d and, as shown in
Example 2.2, the only in-variant set on which x3 = d is . This
proves that is a global attractor on D. A typical sublevel setof V
together with a trajectory converging to isdepicted in Figure 1.In
fact, F. W. Wilson Jr. has shown that Liapunovfunctions for compact
attractors always exist. Inthis setting, his result takes the
following form.Theorem 2.3. Suppose that X Vect(M). A nec-essary
condition for a compact subset K M to bea global compact attractor
on an open positivelyinvariant domain D M is that there exist a
Lya-punov function V for X on the pair (D, K).! 1012!
101200.511.5x1x2x3Figure 1. The sublevel set V1[0, 1] V1[0, 1]
V1[0, 1] and thetrajectory of XXX for initial condition (1, .75,
1.5). (1, .75, 1.5). (1, .75, 1.5).Remark 2.4. Wilson [19] also
studied the topol-ogy of Liapunov functions in the case that K is
asmooth submanifold. For example, if K S1, as inthe case of an
asymptotically stable periodic orbit,then the domain of attraction
D of K always sat-ises D Rn1 S1, in harmony with Examples2.22.3 and
Figure 1, for which D = R3 X3 R2S1.Angular Variables and Angular
One-FormsThe purpose of this section is to recast Birkhosseminal
ideas on the existence of sections forsmooth dynamical systems in
modern terms andto delineate the extent to which these ideas
areapplicable. For the closed-loop system (2.19), animportant role
in the analysis of the nonlinearoscillator was played by the
variable , measuringthe rotation of the magnetic eld. More
formally,for the Liapunov function V dened in (2.20) andfor a xed
choice of c > 0, denote the sublevel setV1[0, c] by M and
consider the function(3.1) J : M S1, J(r, , x3) = ,which satises
J(x) = dJ(x), X(x) = f > 0. Thesign of J is irrelevant; only the
fact that J issign denite is important. The existence of suchan
angular variable also arises in the two-bodyproblem with a central
force eld, since conserva-tion of angular momentum implies that =
c, forc a constant and for (r, ) in an invariant plane ofmotion.The
importance of angular variables in thetheory of nonlinear
oscillations was elucidatedby G. D. Birkho in his 1927 book on
nonlineardynamics. In [1, pp.143145] Birkho derived twosets of
necessary and sucient conditions for theexistence of what he called
a surface of sectionfor an arbitrary dierential equation evolving
inRn. For Birkho, a smoothsectionfor X Vect(Rn)is a hypersurface S
M in some region M Rnso that, for each x M, the ow line
(trajectory)through x intersects S transversely at a (least)1084
Notices of the AMS Volume 57, Number 9forwardtime tx > 0and,
replacing t by t, at a leasttime sx > 0 in reverse time. In this
case, he denesthe map (x) = 2sx/(sx+ tx) and notes that increases
along every streamline (trajectory ofX) by 2 between successive
intersections withS. Abstracting from this construction, he
calledany map satisfying ddt((x(t))) > 0 for alltrajectories
x(t) M an angular variable for X.Moreover, if is an angular
variable, he observesthat S = 1(0) is a surface of section for X on
M.Of course, an angular variable is actually amultivalued function,
but it can be made single-valued as a map with values in S1. For
example,Birkhos construction leads to the map(3.2) J(x) =
exp(2isx/(sx+tx)).Birkhos second set of conditions is the
existenceof smooth functions ai such thatni=1aiXi > 0 and aixj=
ajxi, for i, j = 1, . . . , n,where the Xi are the coordinates of
X, abstractingthe observation that J = ni=1JxiXi is actually
awell-dened, real-valued function on M.More precisely, suppose M
Rnconsists of anopen subset together with a smooth boundary.In the
language of one-forms, one may dene=ni=1aidxi and note that Birkhos
conditionsassert that is closed, i.e., d = 0, and that thenatural
pairing of the one-form and the vectoreld X satises(3.3) , X
=ni=1aiXi > 0.It is important to note that (3.3) can be
checkedpointwise (in particular, without explicit knowl-edge of
trajectories) just as in the applicationsof Liapunov functions. In
this spirit, one mayalso formulate the existence of an angular
vari-able without reference to the trajectories x(t). IfVect+(M)
denotes the set of vector elds thatpoint inward on the boundary of
M, then M ispositively invariant under any X Vect+(M).Denition 3.1
([5]). Suppose X Vect+(M). Wesay that a map J : M S1is an angular
variablefor X if it satises(3.4) J = dJ, X > 0everywhere on M.
If is a closed one-form on M,then is an angular one-form for X
provided that(3.3) holds everywhere on M.For example, if M is the
solid torus, Dn1S1Rn, the smooth boundary of M is Sn2S1,
whereSrdenotes the r-sphere. In this case, every closedone-formcan
be written as =n1i=1 aidxi+and.In fact, there exists c R such
that(3.5) = cd +dffor a smooth function f : M R.This is most easily
seenusing the basic theory offundamental groups, e.g., the
fundamental group1(M) of M satises 1(M) Z with a generatorgiven by
a simple closed path transversing{0} S1in either direction. Choose
a xed basepoint x0 M and an arbitrary upper limit x Mfor the
integral(3.6)
xx0.Of course, this integral may depend on a choiceof path
joining x0 to x. Suppose that 1, 2 aretwo such paths and consider
the closed path constructed by traversing 1 and then 2 inthe
opposite direction. To say that
xx0 is pathindependent is to say that
= 0 for everyclosed path . If c =
, then
( cd) = 0and, since every closed path is homotopic to amultiple
of ,
(cd) = 0 for all closed paths . Therefore
xx0( cd) = f (x) is independentof path and = cd +df .If c in
(3.5) is an integer, then is calledan integral closed one-form, and
every such one-form denes a smooth period map J : M S1in the
following manner. As noted above, (3.6) isonly dened up to periods
of , i.e., up to theelements of the subgroup() =
: [] 1(M)
= (c) Z R.If c 0, then () is an innite cyclic subgroup,and
therefore the period map J of denes asmooth surjection from M to a
circle(3.7) J : M S1,dened via(3.8) J(x) =
xx0
R mod (c).Moreover, J satises(3.9) J = dJ, X > 0if and only
if is an angular one-form for X.Remark 3.1. If is an angular
one-form, then itcan be shown that c 0. Moreover, the
normaliza-tion /|c| is an integral angular one-form, wherethe
constant in (3.5) is 1. Therefore, an angu-lar variable exists.
Conversely, since S1 R/Z,a smooth map J : M S1can be regarded asa
multivalued map J : M R where the valueJ(x) is determined only up
to an integer constant.Nonetheless, = dJ is well-dened as a
one-formon M, since the derivative of a constant is zero.Moreover,
if J is an angular variable for X, then, X = dJ, X > 0, so that
is an angularone-form. It is convenient to employ a synthesisof
these two approaches.October 2010 Notices of the AMS 1085Remark
3.2. The calculations made above, such as(3.5), for the solid torus
also hold, without changeof notation, for the toroidal cylinder
RnS1.Periodic Orbits on Solid Tori and ToroidalCylindersWhile
Birkho was not specic about the ambientsubmanifold M Rnin which a
surface of sectionmight exist for X or what that might imply forthe
topology of M, his statements clearly excludethe choice M = Rnand
specically include sur-faces of section having a boundary.
Moreover, itis necessary for his construction that M be aninvariant
set. The development of angular vari-ables in [5] includes the case
of a smooth manifoldwith boundary that is only required to be
posi-tively invariant and allows for a characterizationof M
topologically when there exists a locallyasymptotically stable
orbit.Theorem 4.1 ([5]). Suppose that n > 1. If is an
asymptotically stable periodic orbit ofX Vect(Rn), then there
exists a smooth, pos-itively invariant n-dimensional submanifold M
Rn, homeomorphic to a solid n-torus,on which X has an angular
variable J : M S1.In fact, M is dieomorphic to Dn1 S1,
exceptperhaps when n = 4.Remark 4.1. The idea underlying the proof
is tobuild a positively invariant solid torus using a Li-apunov
function V on the domain of stability Dof , generalizing what was
observed about thesystem (2.19). Indeed, for c suciently
small,(4.1) Mc = V1[0, c]is a positively invariant compact subset,
con-sisting of an open subset and with a smoothboundary. More
succinctly, Mc is a compactorientable smooth manifold with
boundary. More-over, for c suciently small, Mc clearly admitsan
angular variable, since does. The rest ofthe proof uses several key
ingredients, startingwith the fact that, according to Remark 2.4,
thedomain of attraction, D, for is dieomorphicto Rn1 S1. It can
also be shown that the in-clusion Mc D induces an isomorphism
ofhomotopy groups. In other words, the inclusionis a homotopy
equivalence. On the other hand theprojection p2 : Rn1 S1 S1onto the
secondfactor, p2(x, ) = , is also a homotopy equiva-lence, since
Rnis contractible. Consequently, Mcis homotopy equivalent to S1.For
n = 2, by the classication of surfacesit then follows that Mc A,
the standard two-dimensional annulus. For n = 3, the solutionby
Perelman of the classical Poincar conjecture[15] implies that, up
to dieomorphism, D2 S1is the only 3-dimensional compact,
orientablemanifold that is homotopy equivalent to S1. Sim-ilarly,
for n = 4, the solution by Freedman ofthe 4-dimensional Poincar
conjecture [7] impliesthat, up to homeomorphism, D3 S1is the
only4-dimensional compact, orientable manifold thatis homotopy
equivalent to S1. In higher dimen-sions, there are innitely many
smooth compactorientable manifolds homotopy equivalent to S1,but Mc
is special. Adapting some constructions ofWilson [19], one can show
that Mc is homotopyequivalent to Sn1S1. This fact, along with
somehomotopy theory, allows one to use Freedmansproof of the
4-dimensional Poincar conjecture[7] to show that, for n = 5, Mc is
homeomorphic toD4 S1, while a fundamental result due to Kirbyand
Siebenmann [11] implies that Mc is dieomor-phic to D4S1. An
application of the theorem ofBarden, Mazur, and Stallings [10]
completes theproof [5] of Theorem 4.1 for n 6.Remark 4.2. It is
unknown how many dieren-tiable structures on D3S1may exist.In fact,
the necessary conditions for the exis-tence of an asymptotically
stable periodic orbitare also sucient for the existence of a
periodicorbit.Theorem 4.2 ([5]). If M Rnis a smooth subman-ifold
which is dieomorphic to Dn1S1, then anyX Vect(Rn) leaving M
positively invariant andhaving an angular variable J : M S1has a
pe-riodic orbit in M. Moreover, the homotopy class ofthis periodic
solution generates 1(M).The idea behind the proof is to rst use
alevel set S = J1(), for any regular value of an angular variable
J, as a section for Xand to next prove that, after modifying J
ifnecessary, S Dn1. In particular, one can applyBrouwers xedpoint
theoremto the Poincar mapP : S S.Briey, since J is an angular
variable, then dJ == cd +df according to (3.5) and, according
toRemark 3.1, c 0 . Without loss of generality, onecan assume c = 1
and can embed in the familyof one-forms = d + df , 0 1, whichdenes
a homotopy(4.2) J : M [0, 1] S1, J(x, ) =
xx0between the period mappings J0 = J(, 0) andJ1 = J(, 0) = J
and therefore a deformation ofJ10 () Dn1into J11 () S. The
remainder ofthe proof in [5] uses the fruitful relationship
be-tween homotopy and cobordism, as describedin [14], to show that
this deformation is adieomorphism.Remark 4.3. In [5], Theorems 4.1
and 4.2 areproven for the more general case in which Rnisreplaced
by an arbitrary orientable paracompactmanifold N of dimension n
> 1.1086 Notices of the AMS Volume 57, Number 9One corollary of
Theorem 4.1 and the proofof Theorem 4.2 is that, except perhaps
whenn = 4, the seemingly stringent hypotheses of theprinciple of
the torus are actually necessary forthe existence of a locally
asymptotically stableperiodic orbit. More importantly, a
combinationof the proofs of Theorems 4.1 and 4.2 yieldsan
amplication of Theorem 4.2 that combinestopology and dynamics in a
form that is easier toapply in practice.Theorem 4.3. Suppose that N
Rn S1. IfX Vect(N) is point-dissipative and has an an-gular
variable J, then X has a periodic orbit .Moreover, the homotopy
class determined by generates 1(N).This result was originally
proved [3, Theorem2.1] for the class of dissipative periodic
systemsdiscussed in Example 2.1, but the proof extends tothe
general case. The key idea is to use a Liapunovfunction V for the
global attractor Afor X to builda positively invariant torus for X
as in Remark 4.1and then apply Theorem 4.2. Since A is not
ingeneral smooth, this argument does require a bitmore work than
the proof of Theorem 4.1.Theorem 4.3 generalizes Example 2.1 and
givesa new proof of Browders theorem on the ex-istence of harmonic
oscillations for dissipativeperiodic systems (2.6) as well. Indeed,
any dis-sipative system (2.6) was noted to be equivalentto the
point-dissipative system (2.8) evolving onthe toroidal cylinder N =
Rn S1. Since (2.8)has as an angular variable, a periodic
solution{(t, x0, 0)} N exists. That {(t, x0, 0)} is har-monic
follows from the fact that the homotopyclass of (, x0, 0) generates
1(N) Z.As the next example shows, Theorem 4.3also applies directly
to the May-Leonard equa-tions, modeling the population dynamics of
threecompeting species with immigration [6].Example 4.1. The
May-Leonard model for threecompeting species with immigration (
> 0),N1 = N1(1 N1N2N3) + (4.3)N2 = N2(1 N1N2N3) + (4.4)N3 = N3(1
N1N2N3) +, (4.5)where 0 < < 1 < , + > 2, leaves
thepositive orthantP+= {(N1, N2, N3) : Ni > 0, i = 1, 2,
3}positively invariant. Let X Vect(P+) denote thevector eld dened
by this dierential equation.The function V(N1, N2, N3) = N1 + N2
+N3 is positive on P+and has derivative V =LXV(N1, N2, N3) =
N1+N2+N3(N21+N22+N23)(+)(N1N2+N2+N3+N1N3), which is negativefor
(N1, N2, N3) suciently large in norm, so thatB(0, R) P+is an
absorbing set, for the ball ofsome radius R. Since > 0, the
vector eld X|P+points inward, and therefore there is a
smaller,relatively compact absorbing set in P+. Hence, byRemark
2.2, X is point-dissipative.Following [6], there is a unique
equilibrium((), (), ()) = (1 + (1 + 4)1/2)/2 P+,where = 1 + + .
Immigration stabilizesthe population around this equilibrium if
>2( 3)/( +3)2, but for(4.6) < 2( 3)/( +3)2the equilibriumis
unstable with a one-dimensionalstable manifold Ws(0) = {(N, N, N)},
where Ni =N > 0 for i = 1, 2, 3.In this case, M = P+Ws R2S1is
positivelyinvariant. Moreover, the one-form(4.7)=
(N1dN2N2dN1)+(N2dN3N3dN2)+(N3dN1N1dN3)N21+N22+N23(N1N2+N2N3+N1N3)is
an angular one-form for X on M. Therefore, byTheorem 4.3, there
exists a periodic orbit P+whenever (4.6) is satised, as is
illustrated in Fig-ure 2.Figure 2. A periodic trajectory for = 1.5
= 1.5 = 1.5, = .75 = .75 = .75.Remark 4.4. The existence of
periodic orbits forthe May-Leonard equations when (4.6) is
satisedis well known, and some of our calculations wereinspired by
the analysis of these equations in [6],although our use of angular
one-forms and dissi-pativity is new and more streamlined. Indeed,
thetreatment in [6] proves the existence of a periodicsolution by
checking some comparatively very re-strictive hypotheses in an
existence theorem dueto Grasman [6]. In fact, Grasmans theorem is
acorollary of Theorem 4.3.Example 4.2. (Voltage Controlled
Oscillators withNonlinear Loop Filters.) A phase-locked loop
(PLL)is a basic electronic component used in wirelesscommunication
networks for the transmission ofstable periodic signals. A PLL
consists of threecomponents: a phase detector (PD), a
voltage-controlled oscillator (VCO), and a (low-pass) loopOctober
2010 Notices of the AMS 1087lter (LF), each of which can be
described in termsof a mathematical model. For example, in a
verysimple, commercially available form, the LF hasthe form y = y
+u, where u, y R are the inputand output of a one-dimensional
system, the VCOis an integrator, = y and the closed-loop
systemproduced by a PD that compares the phase resultsin the
feedback control u = asin(), where > a > 0. In this case the
region y > a is pos-itively invariant, and using
Poincar-Bendixsontheory for the pseudo-polar coordinates (y, )
inthis region shows that the interconnected feed-back system
results in a sustained, or self-excited,oscillation in the
steady-state response of y(t).In this example, I consider the
3-dimensional,nonlinear LF dened by x1 = 2x1x1ex2+x2(4.8) x2 =
x13x2x32+y (4.9) y = u (4.10)with the same VCO and feedback law as
above, re-sulting in the interconnected feedback system onM =
R3S1dened by x1 = 2x1x1ex2+x2(4.11) x2 = x13x2x32+y (4.12) y = y
++asin() (4.13) = y (4.14)where > a > 0. When = 0, this
system is un-coupled, consisting of a two-dimensional
globallyasymptotically stable systemon R2and the
classicvoltage-controlled oscillator. In fact, using Theo-rem4.3,
one can showthat there exists a sustainedoscillation for any 0.
First, note that since > a N = {(x1, x2, y, ) : y > 0} is a
positivelyinvariant submanifold which is dieomorphic toR3S1.
Moreover, is an angular variable for X onN since y > 0. Finally,
using the energy function(4.15) V(x1, x2, y) = x21+x22+(y )2,it is
straightforward to see that the system ispoint-dissipative on N,
provided 0. Therefore,by Theorem 4.3 there exists a periodic orbit,
as isillustrated in Figure 3.The Existence of Periodic Orbits for
VectorFields on Closed Three ManifoldsIn this and the next section,
I will presume famil-iarity with the concept of a smooth manifold.
Anexcellent introduction, and invitation, to the sub-ject is the
book [14]. As a prelude to investigatinghowgenerally the sucient
conditions in Theorem4.2 might hold, consider the case of X
Vect(M),where M is a compact orientable 3-manifold. Forexample, an
irrational constant vector eld onthe 3-torus, T3, is nowhere
vanishing and ape-riodic by Kroneckers theorem on DiophantineFigure
3. A periodic orbit in the case fora = 1, = 2 a = 1, = 2 a = 1, =
2, and = 1 = 1 = 1.approximations. Moreover, it is easy to
constructa constant coecient angular one-form for sucha vector eld.
Another class of counterexam-ples can be constructed from the
Heisenberggroup N = H3(R) of nonsingular upper-triangular3 3 real
matrices and its discrete subgroupsk = H3(kZ), where H3(kZ)
consists of the uppertriangular Heisenberg matrices with integer
en-tries all divisible by k Z with k 1. Explicitly,L. Auslander,
Hahn, and L. Markus have shownthat there exist (left invariant)
vector elds onN that descend to vector elds on the
compact3-manifolds Nk = N/k having Nk as their small-est closed
invariant set. In particular, any suchvector eld is aperiodic, and
I have constructedexamples which also possess angular one-forms[4].
Actually, these are the only counterexamplesin dimension
3.Theorem5.1 ([4]). Suppose that M is a compact ori-entable
3-manifold without boundary. Every X Vect(M) having an angular
variable has a periodicorbit except when M is a nilmanifold, i.e.,
exceptwhen either(1) M T3, or(2) M N/k.If J is an angular variable
for a complete vectoreld on a compact 3-manifold without
boundary,then S = J1(0) is a compact surface that is aglobal
section for X on M. Since M is orientableand X is transverse to S,
S is orientable and canbe shown to be connected [4]. Therefore S is
acompact orientable connected surface Sg with gholes, and there is
a Poincar map P : Sg Sg.In particular, periodic orbits will exist
provided Phas a periodic point, i.e., a xed point of Pkfork Z with
k 1.For what follows, I will alsoneedtoassume somefamiliarity with
algebraic topology, particularlyhomology or cohomology and the
notion of theEuler characteristic of a space. For example, the1088
Notices of the AMS Volume 57, Number 9surfaces Sg have Euler
characteristic (Sg) = 2 2g. At about the same time that
Birkhopublished[1], S. Lefschetz publisheda remarkable
xedpointtheorem that vastly generalized Brouwers xedpoint theorem.
As a special case of the generaltheorem, if f : M M is a continuous
mapon a smooth compact manifold, with or withoutboundary, Lefschetz
introduced an integer (f ),which can be computed in terms of f and
thehomology or cohomology vector spaces of M, andfor which (f ) 0
implies that f has a xedpoint. In 1953, one of Lefschetzs students,
F. B.Fuller, extended this result to a neat theorem thatimplies
that if P : N N is a homeomorphism ona compact manifold N, with or
without boundary,then(5.1)(N) 0 (Pk) 0, for some k = 1, 2, . . .and
hence P has a periodic orbit. In particular, ifS Sg for g 1, then
the Poincar map always hasa periodic point ,and therefore X has a
periodicorbit. Incaseg = 1, thesectionSis a2-torus andtheremainder
of the proof of Theorem 5.1 consistsof checking when (f ) 0 by hand
and what(f ) = 0 means geometrically, following [18]. Theremarkable
fact is the role played by nilmanifolds,which can also be expressed
algebraically in termsof fundamental groups.The Existence of
Nonlinear Oscillations onn-Manifolds With or Without BoundaryIn the
decade following Fullers publication of histheoremon periodic
points, there were substantialapplications of algebraic topology to
the studyof the existence of periodic orbits for dynamicalsystems
having an angular variable. If a vector eldX generates a solution
backward and forward forall time, then the solutions of the
correspondingdierential equation dene a mapping(6.1) : RM M,which
is said to be a ow. The case of ows is moretractable than semiows
and was developed quitegenerally by S. Schwartzman [17]. An
importantspecial case of his results is the following.Theorem 6.1
([17]). Let M be a compact man-ifold, with or without boundary, and
supposeX Vect(M) denes a ow on M. The followingconditions on a
closed submanifold S M areequivalent:(1) S is a cross section.(2)
The smooth map : R S M denes acovering space with an innite cyclic
groupof covering transformations.(3) The map : RS M is a surjective
localdieomorphism.(4) There exists a smooth angular variable J :M
S1.In particular, if the covering space has a non-vanishing Euler
characteristic, then X has a peri-odic orbit. On the other hand,
for systems thatdissipate energy, the objects of interest are
of-ten asymptotically stable invariant sets, positivelyinvariant
submanifolds, and semiows. In this di-rection, Fuller [8] also used
the method of angularvariables in the more dicult case of a
semiowina general setting that includes the case of a com-pact
manifold. In this case, following [8], a smoothconnected
hypersurface S is said to be a positivecross section for X Vect+(M)
provided S is alocal section for X everywhere in S and, for eachx
M, there is a time tx > 0 such that tx(x) S.Among the additional
topological challenges inthe fundamental work of Fuller on the
existence ofnonlinear oscillations for such dissipative systemsis
that, while the Poincar map is an embedding, itis typically not a
(surjective) dieomorphism, andhis theorem on periodic points does
not apply.Nonetheless, Fuller was able to prove the existenceof
periodic orbits in several interesting situations.Fortunately, a
renement of the notion of an-gular one-forms provides a general
approach tosurmounting this technical diculty.Denition 6.1. When M
, is said to bea nonsingular angular one-form provided it is
anangular one-form for X and |M is nonsingular.By Sards theoremfor
manifolds with boundary[14] and the compactness of M, it follows
thatthe period map (3.7) of a nonsingular angularform is a ber
bundle, since both J and J|Mare submersions. Moreover, using the
existenceof a nonsingular angular one-form, one can showthat P is
homotopic to a dieomorphism ofS, and therefore P is an automorphism
of theintegral homology ring H(S) of S. This key factenables us to
use a corollary of a result of Halpern,generalizing Fullers
periodic point theorem.Theorem 6.2. Suppose that M is a compact
mani-fold with boundary for which (M) 0. Any con-tinuous map f : M
M inducing an automorphismf on H(M) satises (fk) 0, for some k 1.
Inparticular, f has a periodic point.We summarize these results
concerning theexistence of periodic orbits as follows.Theorem 6.3
([4]). Suppose that M is a smooth,compact connected orientable
manifold, with orwithout boundary, and suppose X Vect+(M) hasa
nonsingular angular one-form. There exists asmooth compact,
connected and oriented subman-ifold S M having codimension one and
boundaryS = S M such that(1) S is a global positive section for X,
and(2) P : H(S) H(S) is an automorphism.Consequently, if (S) 0,
then X has a periodicorbit.October 2010 Notices of the AMS
1089Remark 6.1. All compact submanifolds S satisfy-ing conditions
(1)(2) have canonically isomorphicintegral homology rings, so that
(S) is intrinsi-cally dened. There are counterexamples due toFuller
for n 4 that show the inequality in (3.3)must be strict.Denoting as
before the annulus in two dimen-sions by A, the hollow torus, M =
AS1, is also theproduct of the torus T2and an interval and
there-fore admits nowhere vanishing aperiodic vectorelds, some of
which have a nonsingular angularone-form. This is the only source
of counterexam-ples to the existence of periodic orbits for
vectorelds on a 3-manifold with boundary having anonsingular
angular one-form.Theorem 6.4 ([4]). Suppose M is a
three-dimensional manifold with boundary. EveryX Vect+(M) that has
an angular one-form has a periodic solution whose homotopy classhas
innite order in 1(M), except when M isdieomorphic to a hollow torus
AS1.There are several other corollaries of Theorem6.3. For example,
[4] contains a general resultfor closed 5-manifolds that implies
that periodicorbits exist for vector elds with an angular one-form
on any closed 5-manifold with 1(M) Z.A similar result is proven in
[4] for vector eldsdened on a compact manifold M with boundarythat
is homotopy equivalent to S1.ConclusionsNonlinear oscillations are
fascinating and impor-tant but hard to rigorously detect or
predict. SincePoincars time, the best known and most accessi-ble
methods in applied mathematics and relatedelds rely onsmall
parameter analysis andprovidelocal existence criteria for periodic
motions havingsuciently small amplitudes. On the other hand,since
the period of nonlinear oscillations is gener-ally not knowna
fortiori, the existence of nonlinearoscillations is a global
phenomenon, and thereforeany comprehensive theory would necessarily
beglobal in nature. This article continues in the tra-dition of
Poincar, Birkho, and others in studyingcross sections for vector
elds, creating a globalapproach to developing criteria for the
existenceof periodic orbits using methods drawn from theglobal
theory of nonlinear dynamical systems thatdissipate some
mathematical form of energy andmethods drawn from algebraic and
dierentialtopology, particularly the fruitful combination
ofcobordism and homotopy theory.One of the major points of
departure for thisapproach is the ability to include motions ofa
dynamical system that leave a manifold withboundary positively
invariant, rather than invari-ant both forward and backward in
time. Thisenables one to discuss and characterize whatmust occur
topologically when a locally asymp-totically stable periodic orbit
exists, a necessarycondition that itself proves to be sucient for
theexistence of periodic orbits. Using the languageand methods of
dissipative systems formulatedby Hale, Ladyzhenskaya, and others,
this sucientcondition is reformulated into a global
sucientcondition that is fairly easy to apply in severalspecic
examples. The article concludes with aformulation of stricter
sucient conditions forthe existence of periodic orbits for vector
eldsdened, however, on general compact manifolds,with or without
boundary, that ber over a circle.Myowninterest inthis subject is
the existence ofasymptotically stable periodic motions in
nonlin-ear feedback systems, both manmade and natural,and possible
future directions of research shouldinclude the incorporation of
stability criteria, in-cluding classical tools such as Hopf
bifurcationsand describing function methods and the intrigu-ing
possibility of extending this work to the caseof invariant
tori.References[1] G. D. Birkhoff, Dynamical systems, Amer.
Math.Soc. Coll. Pub. 9 (revised ed.), Providence, 1966.[2] R. W.
Brockett, Pattern generation and the con-trol of nonlinear systems,
IEEE Trans. on AutomaticControl 48 (2003), 16991712.[3] C. I.
Byrnes, On the topology of Liapunovfunctions for dissipative
periodic processes, Emer-gent Problems in Nonlinear Systems and
Control,Springer-Verlag, to appear.[4] , On the existence of
periodic orbits for vectorelds on compact manifolds, submitted to
Trans. ofthe Amer. Math. Soc.[5] C. I. Byrnes and R. W. Brockett,
Nonlinear os-cillations and vector elds paired with a
closedone-form, submitted to J. of Dierential Equations.[6] M.
Farkas, Periodic motions, Springer-Verlag,Berlin, 1994.[7] M. H.
Freedman and F. Quinn, Topology of 4-manifolds, Princeton
University Press, Princeton,1990.[8] F. B. Fuller, On the surface
of section and periodictrajectories. Amer. J. of Math. 87 (1965),
473480.[9] J. K. Hale, Asymptotic behavior of dissipativesystems,
AMS Series: Surv Series 25, 1988.[10] M. Kervaire, Le thorm de
Barden-Mazur-Stallings, Comment. Math. Helv. 40 (1965),3142.[11] R.
C. Kirby and L. C. Siebenmann, FoundationalEssays on Topological
Manifolds, Smoothings, andTriangulations, Annals of Math. Studies,
AM-88. (re-vised ed.), Princeton University Press,
Princeton,1977.[12] O. Ladyzhenskaya, Attractors for semigroups
andevolution equations, Lezioni Lincee, CambridgeUniversity Press,
1991.[13] B. Li, Periodic orbits of autonomous ordinarydierential
equations: Theory and applications,Nonlinear Analysis TMA 5 (1981),
931958.1090 Notices of the AMS Volume 57, Number 9[14] J. W.
Milnor, Topology from a dierentiable view-point, University Press
of Virginia, Charlottesville,Virginia, 1965.[15] J. W. Morgan and
G. Tian, Ricci ow andthe Poincar conjecture, Amer. Math. Soc,
ClayMathematics Monographs, Vol. 3, 2007.[16] V. A. Pliss, Nonlocal
problems in the theory of non-linear oscillations, Academic Press,
New York andLondon, 1966.[17] S. Schwartzman, Global cross sections
of compactdynamical systems, Proc. N.A.S. 48 (1962), 786791.[18] W.
P. Thurston, Three-dimensional geometry andtopology, Vol. 1,
Princeton U. Press, 1997.[19] F. W. Wilson Jr., The structure of
the level sets ofa Lyapunov function, J. of Di. Eqns. 3 (1967),
323329.AMERI CAN MATHEMATI CAL SOCI ETYGo to
www.ams.org/bookstore-email to sign up for email notications of
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14777777Tilings, Scaling Functions,and a Markov ProcessRichard F.
GundyIntroductionWe discuss a class of Markov processes thatoccur,
somewhat unexpectedly, in the constructionof wavelet bases obtained
from multiresolutionanalyses (MRA). The processes in question
havebeen around for a long time. One of the rstreferences that
should be cited is a paper byDoeblin and Fortt [10] (1937),
entitled Sur leschanes liaisons compltes. In English, they
aresometimes called historical Markov processesand have been used
extensively to study the Isingmodel. However, their wavelet
connection does notseem to have ltered into the standard texts
ontime-scale analysis.The material for this article is drawn from
thepublications [9], [12], [13], as well as the priorcontributions
by various people who are cited inthe appropriate places. To keep
the expositionself-contained and as elementary as possible,
wediscuss the special case of the quincunx matrixin which the
proofs are somewhat simpler and, insome cases, radically dierent
from those foundin the above references.In the rst section, we
describe a remarkablecoincidence: two discoveries, the rst
concerninga gambling strategy and the second concerning awavelet
basis, both leading to the same mathemat-ics. It is remarkable that
these discoveries, each offundamental importance, were separated by
330years! The wavelet discovery was a particular classof
trigonometric polynomials within a class offunctions called
quadrature mirror lters (QMFfunctions, for short). These functions
turn out to beprobabilities; hence their connection to
gambling.Richard F. Gundy is professor of statistics and
mathemat-ics at Rutgers University. His email address is
[email protected] names associated with the gambling
strategyare Pascal and Fermat; the wavelet discovery towhich we
refer is due to Ingrid Daubechies.In the subsequent sections, we
are concernedwith the class of 1-periodic functions p(),
quad-rature mirror lters, that generate an MRA on Ror R2. Some do
so, but most do not; our problemis to nd out which is which. The
probabilitymethods presented here provide new informationon this
topic. We show that there is a one-to-onecorrespondence between two
disparate classes ofscaling functions, one dened on R, the
otherdened on R2. Second, the probability perspectiveallows us to
exhibit a large class of continuousp() that generate MRAs. When the
function p()is smooth and generates an MRA, it must satisfyknown
necessary conditions. However, these nec-essary conditions may be
violated in the extremefor MRAs in which the generator p() is
smoothexcept at a few points. Few here means as few asfour.In one
dimension, all of the MRAs we considerinvolve the dilation Z 2Z; in
two dimensions, aninteresting special case is the quincunx
dilation,described below. In the rst case, where R1,the QMF
function p() is periodic (with periodone); in the second case, when
R1, p() isdoubly periodic on the unit square. From this,one might
assume that the natural fundamentaldomains for these functions
would be (0, 1), Zor (0, 1) (0, 1), Z2. But the natural
assumptionis too nave. In one dimension, the appropriatedomain is
sometimes a disconnected set calleda C-tile, described below. In
two dimensions, thefundamental fun