The Poincaré-Bendixson Theorem : from Poincaré to the XXIst century

Brought to you by | Uniwersytet Jagiellonski Authenticated Download Date | 10/29/18 8:51 AM The Poincaré–Bendixson Theorem in its classical versionsays

equationsandinseveralapplications.However,inmanycasesitisnoteasytofindsuchasolution.Two-dimensional

reducedtoasystemoftwoequationsofthefirstorder.ThePoincaré–BendixsonTheoremgivesconditionswhichenable theexistenceofaperiodicorbitgivesalsotheexistenceofacriticalpoint.Theexistenceofaparticularkindofperiodic

❆❜ai:s❛❝i:✿ The Poincaré–Bendixson Theorem and the development of the theory are presented from the papers of Poincaré and Bendixson to modern results.

▼❙❈✿ 37E35, 34C25, 34-03, 01A60

❑❡②✇♦s❞a✿ Poincaré–Bendixson Theorem • Limit set • Flow • 2-dimensional system • Periodic trajectory • Critical point • Section

© Versita Sp. z o.o.

✷✶✶✵

Cent. Eur. J. Math. • 10(6) • 2012 • 2110-2128 DOI: 10.2478/s11533-012- 0110-y

❈❡♥ ❛❧❊✉ ♦♣❡❛♥❏♦✉ ♥❛❧♦❢▼❛ ❤❡♠❛ ✐❝

The Poincaré–Bendixson Theorem: from Poincaré to the

XXIst century

Krzysztof Ciesielski^1∗

❘❡✈✐❡✇ ❆

✐❝❧❡

1 Mathematics Institute, Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-364 Kraków,Poland

❘❡❝❡✐✈❡❞ ✼▼❛ ❝❤✷✵✶✷❀ ❛❝❝❡♣ ❡❞ ✸ ❏✉♥❡ ✷✵✶✷

1. Introduction

Theproblemofexistenceofperiodicorbitsandcriticalpointsisfundamentalintheanalysisofbehaviourofdifferentialsystemsplayhereanimporta ntrole.Oneofreasonsisthataone-dimensionalequationofthesecondordermaybe

ustoprovetheexistenceofaperiodicsolutionoftheequation.Moreover,fortwo-

dimensionalsystemsinmanycasesoTrhbeiotsr,eim.e..limitcycles,isinmanysituationsparticularlyinteresting.Frequently,itfollowsfr omthePoincaré–Bendixson

∗E-mail:Krzysztof.Ciesielski@im.uj.edu.pl

Consideraplaneautonomoussystemx=f(x),wherex∈ ,andassumethat

Example1.2.

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Theorem1.3.

Theorem1.4.

in [76]. For some applications in biology, see for example[45].Manyother textbooks also discuss the Poincaré–

From the Poincaré–Bendixson Theorem there easily follow several corollaries concerning the behaviour of solutionsand Theorem 1.1.

0 R²

(?) thesolutionsofthissystemaregivenuniquelyanddefinedforallt∈R.

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of thesystem. eover,

Let us present one of classical applications of this theorem. Consider the following example, due to [3].

tW oe

ps oh

lao rw

ct oh

oa rt

dit nh

ae tee

squ wa

etio gn

etx t

00

h− exs

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r⁰= rsin²θ

(1−3r²cos2θ

−2r²sin2θ

); θ⁰=

−1+1

2sin2θ(1−3r²cos2θ

−2r²sin2θ ):

ttW heeh

oa t

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nd0 ,f

(o 0r

;0r)=

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th2 ean

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theremustbeacriticalpoint. ounds(0 trajectory in

Severalexamplesofapplicationsofsimilarkindmaybefoundin[78].SomeapplicationsineconomyaredescribedBendixsonTheor emanditsapplications,forexample[3,22,23,34,41,57].

othrebiirtspirnoothfsectawno-

bdeimfoeunnsdiofnoarlesxyasmtepmles.inB[e3l,o7w8w].epresenttwoofthem(suitabledefinitionsaregiveninthenextsection);

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dimensionalsphereandassume(?).ThenapointpC

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ago.Now,thereisalargenumberofresultsrelatedtotheoriginaltheorems.Theresearchdevelopedinmany generalizationsforbroaderclassofspaceswerepresented.I n somecases,theymovedveryfarfromtheclassical

✷✶✶✶

Thetheorystartedwiththeanalysisofsolutionsofcertain2-

dimensionaldifferentialequationsmorethanonecenturydifferentdirections.Thebehaviourofsolutionswasdescribedmoreprecisely,severalnewphen omenawereobserved,rPeosiunlctas,rén–eBveenrtdhiexlseosns,Tthheeoyrehma.veThtehethPeoorinecmarsét–

ilBlehnadsixasroenmTahrkeaobrelemininfluietnscoeriognincoonrtehmapveoranraytumraalthceomnnaeticctaiolnrsesw eaitrhcht.he

IanndthBeepnadpixesr,onthuepretoarseompreesreencetendtrtehseulhtiss.toryandthedevelopmentofthetheoryfromtheachieveme ntsofPoincaré

Defienitionsin the sequel will be given forflows,but they are applicable in an obvious way to systems ofdifferentialequtions.

LetXbe a metric space. Aflow(dynamical system) onX(which is called aphase space) is a triplet(X;;π),where anyont∈(−∞;+∞).Thenthefunctiongivenbythesolutionsoftheequationfulfilstheconditionsrequiredfromflows.

everyT>0thereisan(ε;T;x)-chain.

R

R

thatπ(0;x)=xandπ(t;π(u;x))=π(t+u;x)foranyt;u;x.

tn→ ∞}andthe negative limit set of x(orα-limit set of x) asα(x) ={y∈X:π(tn; x)→yfor sometn→ −∞}. We callthesesetsalsothelimitsetsofthetrajectoryγ(x).

setAweaklynegativelyinvariantifforanyx∈Athereisanegativesolutionσthroughxwithσ(−∞;0]⊂A.AsetA

semit

R

+

R

+

the maximal invariant setAcontained inU. A critical point is called isolated if it is the only critical point in some its neighbourhood.

2. Basicdefinitions

Inthissectiontherearepresenteddefinitionswhichwillbeofuseinthefurtherdescriptionofresults. R

πau:tR×omXou→sdXiffiesreantcioanlteinquuoautisonfunxc^0t=ionf(xsu)cahndthaastsπum(0e;xth)a=tfxoraanndyπx(tt;h πe(uso;lxu)t)io=nπx((tt)

+thur;oxu)gfhorxainsyunt;iquu;ex.anCdonesxiidsetsrfaonrInstadofformulationoftheaboveconditionswewilljustsaythatthesystem givesaflow.

Byaatrajectory(anorbit)ofa

+

γ

−

(

x

)

=

{

π

t;

x)

:

t

≤

0

}

.

• criticalifπ(t;x)=xforeveryt≥0,

• periodicifthereexistsat>0suchthatπ(t;x)=xandxisnotcritical,

• regularif it is neither periodicn o r critical.

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hAasseatlAlthisesinevparroiapnetrtiifeπs.

(R×coAm)p=acAt.inAvasreiatnAtsiestmMiniimsaisloilfaitteidsinfothneermepetyx,isctlsosaende,iignhvbaoriuarnhtoaon

ddUnoofpMropseurchsutbhsaettMofiAsAwisthetγM+

(xis)a6⊂sUadadnledsγe−t(ixf)th⊂erUe.existsaneighbourhoodUofMsuchthateveryneighbourhoodVofMcontainsapointx

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Foragivensemiflow,wedefinecritical,periodicandregularpointsandpositivesemitrajectoriesinthesamewayas π(t;σ(u))=σ(t+u)foranyt;uwithu≤0,t≥0,t+u≤0.TheimageofanSegativesolutioniscalledanegative

∆⊂. A setAispositively invariantifπ(×A) =A. A setAisnegatively invariantifF([0;∞); A) =A.Wecallthe A pointxis said to be

oInfAthehacsaasellothfesseempiflroopwesrtwiees.defineω-

limits ets i n t h e s a me w a y a s i n t h e c a s e of f lo ws. However,α-limitset s mus t

negativesolutionthroughx.Wedefinetheα-limitsetασ(x)as{y∈X:σ(tⁿ)→yforsometⁿ→−∞}.Notethatfor

✷✶✶✷

ispositively(weakly)minimalif it is nonempty, closed, positively (weakly negatively) invariant and no proper subset

bedefinedin a slightlydifferentway, as there may be many negative solutions through a given pointx.Letσbeaaangdivweneapkolyinntexg,adtiiffveerlyenitnvnaergiaatnitv.esolutionsmaygivedifferentnegativelimitsets.

Limitsetsinsemiflowsarepositively

whffiechewasltehefirstnvedrsionofthye/YPowinhcearreé–

XBeannddixYsonTheorem.Thiswaspresentedinthethirdpaper[60],publishedin85.

moaiinnclyanalyesticarlcmhecthods. asymptoticP aréal pa atentionto ointsandtheirtypes.Fromthepointofviewofgeometricpicture, hecharac-

poitnte,wasarelsoconesiuderedbyhihme.two-dimensionalPonarécsidere nlythesy m byananalyticfunctionf.However,inmanycasesheconcentrated

thischapterofmathematicswasoffundamentalmeaning.However,mathematiciansusedtoconcentrateonthemethodsof

f r d e ais H – )h ap quat

differentialequations. The main goal of this theory is to investigate properties of solutions f ofdifferentialequationsnotne rl i se s u n.

phiyesnicalphednioemreennatiaofethesystne,mwgitivenbytheinvestigatedequation,especiallythesecondorderdifferentiale quations.

inftedgiffrationioaflpeaqrutiactuiolarstypeofteicquuaations.OfPoincaré’sinteresttherewasageometricpictureoftrajectorie sofasystem

touraesloettofpsipglniifiucaenstrTesearcheirnedpifferentiale quations.Poincaréb egant hew orko ncurvesd efinedb yt hes ol utions

otherwords,atrajectory),i.e.acurveinthe(x;x⁰⁾planeparametrisedbythetimevariablet.Suchacurvewascalled

BthyataaJoJordrdaannacrucrv(aeJJoirndathnecpulravnee)wsuerrmoeuanndsaahpoomienotmpoirfpphiicsicmoangtaeinoefdaincoambpoa uctndseegdmceonmtp[o−n1e;n1t]c(uatufrnoimtctihrcelep)l.anWeebsyaJy.

3. Poincaré and the XIXthcentury

Differentialequationshavebeenaveryimportantareaofmathematicalresearchformanycenturies.IntheXIXthcenturysolving odiffeent kin s of qu t on . It was enri Poincaré (1854 1912 w o was ioneer of li ative theoryo

IPnctehsesaenidAyokfnotqhweénXgI.XthtohhceerenotluwrtyioPsouinbclaisrhéepdufboluisrhpeadpheirssm(themeofirr asbtleinp1a8p8e1r,st[h5e8–l6a1s]tiinnt1h8e86Jo)utrhnaatlgdaeveMtahtehébmegaitniqnuinegsoerent n of par l r type and investigated the global nature of those solutions. He did notw o r k ong v byaff l quatio hout integrating this equation. Such a picture would help inu n d e r s t a n d i n g someParé’sre a o n c e r n e d

propertiesofsolutioncurvesandtheirtopologicalproperties.Poincaréused

toeforiinshecddaiffnedscolnatsiasilifideqdtdaitffioenrenotntytspinegsuolfacrrpiticalpointsttohrauts,apwpheearreineatchhecfiursrvteorsdoel rutpiolanneisedqeunastieonasn.dTthheerfaemiosunsoexcarimtipcalel

edvieinrconntimaoorneqnuaartiroodwocxa/sXe,=i.e.desqtuea,tsioxn0s=wfit(hx)pgoilvyennomariealproiglyhnt-

ohmainadls.sidUen.dMerotrheispraescsisuemlyp,tihoenwhoerkfoerdmounlattehdefithrsetroersduelrtPgiov1iennc8abry

épdaerstcirciubleadrkailnsdoothfediqffuearelintatitaivleeqbueahtaivoinosu,ri.oef.agiflvoewnbinyaannaelyigtibcoufurnh

cotoidonosf.aHneissohloawteeddctrhiatitcafolrptohientefqourafltioownss

x⁰⁽t)=Ax(t)+g(x),whereAislinear,undersomead0ditionalassumptionsongandA,thequalitativebehaviourofthe

isnydsetexmofcaoincrciitdiceaslwpiotihntth(ethbisehtearvmiouwrilolfbt

ehedessycsrtiebmedxin(t)th=enAexx(tt)seinctsioonm)e. neighbourhood of0. He investigated also the Let us mention also some aspects of terminology introduced by Poincaré. He introduced the concept of an orbit (or, in

beqyuPaotiionncsa.réAalsochtahreacteterrmis“tilcim(iitncFyrcelnec”hi:scdaureacttoérPisotiinqcuaer)é..Itcanbeobtainedbyeliminatingthe variabletfromthegiven

4. Bendixson and the period beforeWorld WarII

theauthorobtainedmanyotheradvancedresultsonplanarsystems.Thepaperconsistedof88pagesanduptonow

The Bendixson paper is nowadays famous mainly because of the Poincaré–Bendixson Theorem. However, in the paper Bendixson(1861–1935)publishedinActaMathematicahispaper[11].Therehegaveamoredetaileddescriptionof

✷✶✶✸

The full story of the Poincaré–Bendixson Theorem starts in the verybeginning of the XXth century. In

1901,Ivarbworuitntednedbyli,maimtosentgsoatnhdercsr,itEicmaillepoBinotrself,oHrepllgaenavrodniffKeorcehn,tiRauldeo qlufaLtiiposncsh.itIznathnedsGaömsetavoMluitmtaegt-hLeerffleewr.erepublishedpapers

gives the inspiration to research. Besides the Poincaré–Bendixson Theorem and its several consequences, only some

Theorem4.6.

Theorem4.7.

Theorem4.8.

Theorem 4.1 ([11, Chapter I, TheoremII]).

proofsonthewaytomaintheorems,orconsequencesofmaintheorems.

morepreciselyinthenextsection.

investigated planar systems with much weakerassumptions. He considered the system

investigatedsolutionsofthesystematapoint(x0;y0)givenuniquelyanddefinedforallt.

If the positive semiorbit γ⁺(p)of a point p∈²is bounded and positive limit set ω(p)does not contain any critical

✷✶✶✹

BInetnhdeixvseorny’sbreegsiunlntsinwgiollfbtheempeanpteiornBedenhdeixreso.nnotedthathisresearchisacontinuationof Poincaré’sresults.Bendixson

d d

x

t=X(x;y); d d

y

t=Y(x;y)

andassumedthatX,Y,∂X/∂x,∂X/∂y,∂Y/∂x,∂Y/∂yarecontinuousinacertaindomaincontainedintheplane.H e Letussummarise

brieflysomeofresultsfrom[11],undertheassumptionsgivenabove(inmodernformulation).

R

point, then either γ(p)is a periodic orbit or γ⁺(p)approximates a periodic orbit and this orbit is its limit set.

Bendixson gave also more detailed description of isolated critical points and periodic orbits.

I

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(4.2.2)there exists a point x6=p such that ω(x) ={p}or α(x) ={p}.

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(

p)

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s:

(4.3.2)thereexistsapointx∈/γ(p)suchthatω(x)=γ(p)orα(x)=γ(p).

TwheecparloloitfsthreelileodcavlerpyarsatlrloenligzlayboilnittyheinJoardsmanalClunrevieghTbhoeuorrheomo.dAolfso

a,annoont-

hcerirticcraulcpiaolinfat.ctTwhaespuhseendomtheenrae.wInillmboededrinsctuesrsmesd,Letusmentionalsosomeotherim

portantresultsthatappearinthepaper[11].Theywereeitherresultsusedinthe

If a limit set contains a periodic orbit, then it is equal to this orbit.

If a bounded limit set does not contain a critical point, then it is a periodic orbit.

If a limit set L is not a periodic orbit, then for any x∈L the set ω(x)is a critical point and α(x)is a critical point.In the bounded region given by a periodic orbit there is at least one critical point.

A bounded minimal set is either a periodic orbit or a critical point.

Theorem4.2.

Theorem4.3.

Theorem4.4.

Theorem4.5.

(4.9.1)U isinvariant,

impossibleforplanarsystemgivenbypolynomials.A 1)

dfr d o

Bendixsoncentre.Forexample,(0;0)isaBendixson centre inthesystemgivenbytheequations(inpolarcoordinates)

0 forr=0;

radiusr= 1/n,n= 1;2;3; : : :, is a periodic orbit. The orbits between the circle of radiusr= 1/nand the circle of

Bendixsongotacrucialresultconcerningtheindexofacriticalpoint.Letusdescribeveryroughlythisresult.Assume

✷✶✶✺

K. Ciesielski

Btheenredixwseorne’sarefeawsopniicntgurweass;sraotmheeropfutrheelmygaeroempertersiceanltewdheinreFaisgPuroein1c.aréusedfirstof allanalyticmethods.I n thepaper[11]

Figure 1.

LAetcruitsiccaolmpeoinbtacwkhtiochTfhuelfiolrsemth4e.2c.onFdriotimonth(4is.2r.1e)suisltcBaellneddix asocnenotbreta.inTehdertehearcelatswsiofictaytpioensooffciseonltarteesd.

cNriotwicatlhepyoinatrse.

fproeiqnutepntslyucrheftehrartedthearsePeoxiinstcsaraénceeingthrbeosuarhnododBe Undoifxsponfulcfielnlitnrgest.heBfyollaowPionigncparroépecretnietsre:

(4.9.2)all points in U except p are periodic,

(4.9.3)any periodic orbit contained in U surrounds p.

we mean an isolated critical

For example,(0;0)is a Poincaré centre in the system given by the equations (in polar coordinates)

r⁰(

t)=0; θ⁰( t)=1:

Poincaré centres were considered by Poincaré. However, the situation described below, presented by Bendixson, is

byssγu(mqe)n(coownstehqaute(n4t.l2y.,alhlopldosinatsninoγ(va)nyarpeesruiorroicunodrbeditbγy(qγ) (squ)r).roAuncdriintigcaplpthoeinretseaxtisistfsyiangretghuilsarcopndiinttiovnsiusrcroaullneddeda

r⁰(t)=r²sinπ

r forr6=0;

θ⁰( t)=1:

In this system,(0;0)is an isolated critical point which is surrounded by infinitely many periodic orbits. Any circleofrbaigdgiuesrrci=rcl1e/(annd+t1h)eaωre-

lsimpiirtaslsetwihsitchhesspmiraalllefrrocmircolnee.circletothesecondone.Foreachofthem,theα-limitsetisthe

EAnxaomthpelresveorfyaiPntoeirnecsatrinégcetnotpriecainnd[1a1B]wenadsixasomnocreentdreetaairleedpraenseanlytes disinofFcirgiutirceal2.pointsfulfillingthecondition(4.2.1).thataplanarvectorfieldVisgivenandconsideraJordancurveCwithoutcriticalpoint sonit.TheindexofValongC

neighbourhood ofxand for everyy∈Uthere are a uniquez∈Sand a uniquet∈(−λ; λ)withπ(t;z)=y. Bya By asectionthroughxwe mean a setScontainingxsuch that for someλ>0the setU=π((−λ; λ); S)is a Markov( 1903–1979)i n[48]a ndi n1 9 3 2 b yH asslerW hitney( 1907–1989)[81].( Byt hew ay,A.A.Markov,mentioned

For the proof of the Poincaré–Bendixson Theorem the notion of the transversality was of great importance. In the

very good tool for solving many problems as it is possible to describe very precisely the behaviour of a system ina initsmodernformsaysthatinadynamicalsystemforanynon-criticalpointxthereexistsasectionthroughx.T he

investigated critical point. Bendixson proved that the index is equal to1 + (e−h)/2, whereeis a number of elliptic made by suitably defined separatrices. Those sectors were of two kinds: hyperbolic and elliptic. In a hyperbolic arethesame.Suchanindexiscalledtheindexofp.

The abstractdefinitionof adynamical system (a flow) was formulated independently in 1931 by Andrei Andreievich Whitneyinanothermemorablepaper[82]introducedtheconceptnowknownasparallelizability.Roughlyspeaking,

passingagivenpointwhichisnottangenttothesolutioncurvethroughp.However,inthenon-differentialcasethe 1944)publishedhiscelebratedmonograph[16].Itwasacrucialstepforthedevelopmentofthetheory.Thismonograph commonlyregardedasthefounderofthetheoryofdynamicalsystems.InBirkhoff’sbookwecanseethefoundationof

α-limit pointsandω-limit points, was introduced byBirkhoffin this monograph. The definition of dynamicalsystem

✷✶✶✻

Figure 2.

idsiraecntiuomnb.eIfraofcrroittiactailonpsoitnhtrpouogfhVanisglieso2laπtemda,dtheebnyfotrhesuvffiecctioerntolfyVsmwa hllilJeorpdraonceceudrivnegsasruoruronudnCdiningptheindcoicuenstearlcolnogcktwhiesmeBendixson considered the division of a neighbourhood of an isolated critical point into sectors. This division

wassectorseparatriceswereasymptotesoftrajectories.Inanellipticsectoreachtrajectorywasemergingortendingtothesectors andhis a number of hyperbolic sectors.

5. BetweenWorld WarI andWorld WarII;

dynamicalsystems

1927hasmadeagreatimpactonthequalitativetheoryofdifferentialequations.Then,GeorgeDavidBirkhoff(1884–

wasthebasistoalotofresearchinthenexttwentyyearsand,infact,itstillhasaninfluenceonthetheory.Birkhoffistwoimportantbranchesofthethe ory,i.e.topologicaltheoryandergodictheory.Itshouldbenotedthattheterminology:pTrheesetonpteodloignicCahladpetfienri2tiownasapnpoeta yreetdfosromounlaatfetedrtihte.re,asBirkhoffconsideredonlysystemsgivenbydifferentialequations.

above, was not Andrei Andreievich Markov known by Markov processes and Markov chains, but his son.)

In1933pasarpaallreallilzealbliilniteys.ofthesystemgivesanopportunityoftopologicaltransformationoftrajectoriestogetthosetraj ectories

transversalwe mean a section that is simultaneously a Jordan arc or a Jordan curve.

differentialcase,thereisnoproblemwithit.Thisfollowseasilyasforanynon-

criticalpointpwemayfindacurvessietcutaiotinosnfoturrnflsowosutatnodbperomveudchthmeoreexicsotemnpcleictahteeodr.

eImndwepheicnhdeisntnlyo,wWchailtlneedythine1W9h33itn[8e2y]–

aBnedbuBtoevbuTthoevoirnem19.3T9h[e9]tdheefiorneemdexistence of a section through a given point shows that local parallelizability of the system isfulfilledand givesan e i g h b o u r h o o d o f a n y n o n - c r i t i c a l p o i n t .

way :

Games. Bohr and Fenchel proved in[17]

Wepresent here the theorem most connected to the Poincaré–Bendixsontheory.

(1884–1974)in[25].Poincaréposedthequestionifforaflowonthetorus^2,2givenbyananalyticfunctionf,theo p mn e r

aj re eh tuT n s e hip m,

theseproblemsfromdifferentsides.Theresultsfromthosepaperswereofgreatinfluencetofurtherresearch.Infact,W e

sdr t s sc a i f o n e t ).

Sections were used in the next important result for planarflowsof the Poincaré–Bendixson type. It was

obtainedbyHa dB r n . Bor ou t f ph

h a

Theorem6.1.

ω(p) =ωC(p)∪ωN(p). HereωC(p)is the set of all critical points contained inω(p),ωN(p)the set ofallnon-criticalp t BendixsonTheoremcanbegeneralisedtosuchcase.

systemgivenbythefunctionofclassC^1forwhichtheassertionoftheabovetheoremdidnothold.Hisresultswereoge o

c, v sr o t men cri c ys n

bounded .

✷✶✶✼

K. Ciesielski

Inthepapersmentionedabove,twoauthorsnotonlypresenteddifferentproofsofthetheorem,buttheyevenapproachedhitn y con i e ed noflowbut pe ial f mil es o curves (m re general case tha famili s given by orbi s offlows

veryralgoodohforoatbnadllWpelranyeer,FaemcehmelbeBryofththeewDaya,nishhnawtiaosnaalyfootnbgaelrlbteroatmh,eraowfinhneeraomf oaussilveyrsimciesdtaslNinie1ls90B8oOrlyamndpic

Theorem 5.1.

Let(R²;

R;π)beaflowandp∈R²be

aregularpoint.Thenp∈/ω(p).

Itnypfeacfot,rbfleofworsewtahseoabbtsatirnaecdtbdyynKanmeiscearl.sInys1t9e2m4shweeprreovfoerdmianll[y43i]natrtohdeuocreedm,wa hviaclhunabowlemreasyublteoffortmheulaPtoeidncianrtéh–eBfeonlldoiwxsinogn

Theorem 5.2.

Let(K;R; π)be a flow without critical point on the Klein bottle. Then there exists a periodic point x∈K.

Soonafterit,anextremelyvaluableresultforanothersurfaceinthedifferentiTalcasewasobtainedbyArnaudDenjoyshnolywinogsstihbeleassierit mioanlsisttsruaereepveonintins,apestiroodnigcetrrveercstiooni.sHaendprtohvedwseovleeraolrressult.sDaebojuotytihne1s9y3s2temoslvod ntthestorroubsleT^2.

A

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s

C

2

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um

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h

t

thi

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y

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iv

es

a

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e

t

M

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a

m

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im

al

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e

t

fo

r

t

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y

s

t

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m

.

T

he

n

e

i

th

e

r

is

a

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r

iti

ca

l

point,Mishomeomorphictothecircle(i.e.isaperiodicorbit),orM=T².

M

AswasalsoshownbyDenjoy,theassumptionthatfisofclassC²ⁱsessential.Hepresentedanexampleofthe

infverstiagtaitmiopnsr.taAnmoenggaotehearsg,ohoedcdoenscidieprteidnpohfenhoemepnhaenboasedaonocerugordnigcitiynasundhrostatti eomnsnaumdbewres.reabaseforfurther

6. First generalization afterWorld WarII

Neither Poincaré nor Bendixson investigated limit sets with infinite number of critical points. However, the Poincaré–

Solntzevin1945[74]providedageneralizationofTheorem4.6.H e spliteachcompactlimitsetω(p)intotwoparts,thoeinorsemc:

ontainedinω(p).AnycomponentofωC(p)wascalledasingularcomponent.Solntzevprovedthefollowing

Consideranautonomoussystem

x0=f(x); (1)

p

∈

R

i

s

T

h

e

n

ei

t

h

e

r

Theorem6.2.

Theorem6.3.

suchthathfh⁻¹(Ω_N(p))isahomeomo

b

o

u

n

d

ed

.

N(p).

{x<0;y=0};{x=0;y>0}and{y=x/n;x>0}forn≥1.DenotethedomainsbyDninsuccessivewaysothat x∼yifx=yorx; yare critical points belonging to the same singular component ofωC(p). LetΩ(p) =ω(p)/∼,

manyregulartrajectories(comingfromonesingularcomponenttoanotherone).Ontheotherhand,itmaybeinthe

Anotherexamplemaybedescribedasfollows.Considerthedivisionoftheplanegivenbyhalf-lines:{x>0;y=0};

a regular trajectoryγksuch that in both time directions it goes to(0;0)and its diameter is equal to1/k.For anyn Forexample,considerpointspn,where(inpolarcoordinates(r;θ))pn=(1;π/n),n≥1.Takeascriticalpointsallpn,

Thelimitsetsmaybeofdifferentshape.Forinstance,alimitsetmaybeintheshapeofthecircleandcontaininfinitely Heprovedthatifasemitrajectoryisunboundedandalimitsetassociatedwiththissemitrajectorycontainsnocritical non-singular point precisely once.

setand,moreover,eachseparatingtheplane.Forlimitsets,wheretherearenounboundedcomponentconsistingof

regulartrajectories.Thent

s

bo

u

n

d

e

d

✷✶✶✽

(6.1.1)the positive limit set ω(p)is a periodic trajectory,or

(6.1.2)the set of non-critical trajectories contained in ω(p)is at most countable.

o

f

ω

C

(

p

)

a

n

d

t

h

e

s

et

ω

(q

)

is

c

o

n

t

a

i

n

e

d

in

s

o

m

i

n

g

u

l

a

r

co

m

o

n

e

n

t

o

f

ω

C(

p

)

.

FinotrroadnuoctehearsrpeescuilatloefqSuiovlanlteznecve(rfoerlmatuiolanteidnahelriemiitnsaet.sliDghefitlnyedaiffnere eqnutivwalaeynctehacnlasinsitnheωo(pri)gifnoralappaopienrt),pw.eWneeewdrittoeΩC(p) =ωC(p)/∼,ΩN(p)

=ωN(p)/∼. Of course, we may identifyΩN(p)withωN(p). Now we have

γ

+

(p

)

t

h

ro

ug

h

a

p

oi

n

t

∈

R

2

is

h

e

n

t

he

re

ex

is

t

s

a

c

o

n

tin

u

o

u

s

su

r

je

c

tiv

e

m

a

p

pi

n

g

h

fr

om

a

c

i

r

cle

t

o

Ω

(

p

)

Roughly speaking, the theorem says that we can go along the whole limit set like along “cyclic paths” and meet any

TheworkofSolntzevwascontinuedbyVinogradwhoin[79,80]obtainedsimilarresultconcerningunboundedlimitsets.

points,thenitconsistsofatmostcountablenumberoftrajectories,eachofwhichhavingemptyα-limitsetandω-

limitclyriitnigcailnptohienmts.,VHienoggarvaedacllsaossaifideedscthrieptcioomnpoofnpeanrtasmoefttrhizisatliiomnists oeftcionmtopofinveenttyspseismiinlatretromsthoafttphreebseenhtaevdiobuyroSfotlrnatjzeecvt.ories

shape of a finite-leafed rose or infinite-leafed rose, with only one critical point common for all the leaves.

(a1s;0fr)omanodu(t0s;id0e)..TInhethteraujencittocriirecsleo,fthpeoirnetgsunloatrctroanjteacitnoeridesingtohefroumnitpkc itroclepks+p1i.raltowardsthiscircle,aswellfrominside

D1is the lower half-plane and fork≥3eachDkis bounded byy=x/(k−1),y=x/(k−2). Now, in

eachDktakeatrlaljepcotionrtysswuhrircohunsdpeirdalbsytoγnthaeresectoΓnt=ain{e(d0;i0n)}tr∪ajeSct{oγrnie:snw≥ith1}

boatnhdliΓmiitssietstsωe-qliumailttsoet(.0;0).Nowwecanconstructthe

AScocmoerdoinfgthteoptohsesiabbloevpehtehneoomreemnsawareeaplrseosehnatveedinFigure3.

t

h

r

o

u

g

h

a

p

o

i

n

t

p

∈

R

2

i

a

n

d

th

e

l

im

i

t

s

e

t

ω

(

p)

c

o

n

ta

in

s

pr

ci

s

e

l

y

o

n

e

cr

i

t

ica

l

p

o

i

n

t

a

n

d

i

n

fi

n

i

t

e

ly

m

a

n

y

Theorem6.4.

plane)simplyconnecte

fo

2.

FromthistheoremappliedtoflowsonRit followsthatthelimitsetassociatedwithanunboundedsemitrajectorydoes

sed

ThrevethperoorveimeoftDheenpjhoaysewasspagceenwerasnoetaato1r9u6s3.HbyowAertvheur,rtJh.eScphrowoafrwtza wshnootpcroorvreedct[,6a6]s:waspointedoutbyPeixoto[56].ali in

Theorem6.6.

doesnotcontainanycriticalpoint.Thenω(p)ishomeomorphictothecircle.

Assume that this system gives a flow. Let M be a minimal set for this system. Then either M is a critical point, Mis

✷✶✶✾

K. Ciesielski

Figure 3.

Vinograd gave also another very interesting theorem concerning planar limit sets. He proved

b

e

in

g

th

e

ω

-

u

li

m

i

t

e

t

r

so

m

e

p

o

i

n

t

if

a

n

d

on

l

y

i

f

B

i

s

h

e

b

o

u

nd

a

r

y

o

f

s

o

m

n

o

n

e

m

p

ty

(

a

n

d

d

i

ffe

re

n

t

f

ro

m

th

e

w

h

ol

e

IctosmhpoaucltdfibrestncootuendttahbaletVspinaocgerathdewωo- rlkimeditnaslestooifnsgoemneerpaolinfltoiwssn.oInn-

pcaomrtipcauclta,rt,hheenpirtodvoeedsinno[7t9h]atvheaatnifyincoamflpoawctocnomaplooncaelnlty.not have any boundedcomponent.T resultsofSolntzevandVinogradgavefurtherprecisecharacterizationoflimit

sInetsth.enextyears,followingDenjoyhe Kneser,subsequentresultsformanifoldswerepublished.In[30]asimilar tcchruietoicraelmpfooirndtcsdloosnedma2n-

difiomldesnswiohnicahlograiveanntdabflloewm.aHneifoclldasimweadstshtaattead.limTihteseatutchoonrtacionninsigdeareJodrddaiffnerce unrvtiealwsayssteeqmusalwtiothtohuist

Theorem 6.5.

Assume that(1)is a system on a compact, connected2-dimensional manifold X of class C², where f is of class C².

m

a

n

if

o

ld

X

m

us

t

e

eq

u

a

l

to

t

h

e

t

wo

-

d

i

m

e

n

s

i

on

a

l

to

r

u

s

T

.

Another result of Schwartz was

A

cl

a

s

s

C

2

.

A

s

s

u

m

th

at

t

h

i

s

sy

s

te

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g

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ve

s

a

fl

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w

a

n

d

t

ha

t

th

e

m

a

n

i

fo

l

d

X

i

s

n

o

t

a

m

i

n

i

m

a

ls

et

.

L

e

t

p

e

su

c

h

t

h

a

tω

(p

)

F19o6r9abnyonM-

oarrikelneytab[4l6e].mTahniisfotlhdeoarnedmwmitahyoubtethreegdairffdeerdenatsiaabiglietnyeorafltihzeatflioonwoafsTshuemoerde,ma5 s.2im.ilartheoremwasobtainedin

Let(K;R; π)be a flow on the Klein bottle. Assume that p∈ω(p)or p∈α(p). Then p is either critical or periodic.

Theorem6.7.

pointonecanconstructasectionwhichisanimageofaclosedinterval.

andω(x)

T

ofXwi

a

n

d

n

.

dOinffeeroefntmiaalincarseea.sonswastheproblemwithsections.Theexistenceofsectionswasshowneveninaverygene ralcase,

tHoápjoelko’gsieressaurltesesqhuoawl.edt hat th e Poincaré–

BendixsonTheoremispurelyt op o log ica l and does n ot dependond iff e ren-

topologyinducedfromtheplane.Ontheotherhand,wemaydefineatopologyonaregulartrajectorytakingasthe manifolds,notnecessarilycompact.A2-dimensionalmanifoldXiscalleddichotomic,ifforanyJordancurveJtheset parallelizabilityofflows,guaranteedbytheWhitney–BebutovTheorem.However,thedifficultywasmovedtoanother byOtomarHájek.

hadtobedoneinadifferentwaybecauseofthelackoftheassumptionofdifferentiation.Also,severalothertechniques

transversals through any non-critical point withoutdifficulty.This gave a good local description of solutions close to

7. Thetheorem for dynamical systems: from Hájek toGutiérrez

AseftveerraWloarultdhoWrsarwIeIrseevperroavlinggenmearanlyizraetsiounltssofofrthgeenPeorianlcaflroéw–

sB.eHnodwixesvoenr,Tthheeocrelamssoifcavalrtihoeuosretympewsahsasdtilblekennoowbntaoinnleyd,inaltshoe,

however,notmuchwasprovedontheshapeofsectionseveninthe2-dimensionalcase.Thischangedduetoaresult OneofbasicresultswhichcouldhelpinthegeneralizationofthePoincaré–

BendixsonTheoremforflowswaslocalpoint.Inthecaseofflowsgivenbydifferentialequations,fulfillingsuitablecontinuityassum ptions,wecouldfind

nthoen-

tcorpitoilcoaglicpaolinsthsaapnedotfhseecptriooonfss.mightuseit.Butwithouttheassumptionofdifferentiability,nothingwa sknownaboutItnhefa2c-

td,iWmhenitsnieoynailnc[a82se]smtautsetdbaerJeomrdaarnkoanrcsthoisrtcouprivce.sH.Howeeovenrl,yhsetadtieddn(o atndwrgitaeve(aandnidciedenxoptlapnroavtieo)nt)htahtaatllthsreocutgiohnsaniyn

The problem was solved by Hájek, who proved in 1965 [31]the following

L

is

ei

th

e

ra

Jo

r

d

an

a

r

c

o

r

a

J

o

rd

a

n

c

u

r

ve

.

TbyhiHsáhjeelkpeind[w33it]h).tHháejegkengearvaeliazavteiorynporfetchiesePdoeisnccrairpét–

ioBneonfdliixmsoitnsTethseoinreflmowfosroflnotwhseipnlatnheea2n-ddimdiecnhsoitoonmailcc2a- sdeim(oebntsaioinneadl

X\Jhas two components. Hájek proved, in particular,

L

p

o

in

t

s

a

mo

s

t

c

ou

n

ta

bl

e

f

a

m

i

l

y

{

T

:

n

∈

}

,

A

⊂

N

,

o

f

n

o

n

-

p

c

ri

tic

al

t

ra

j

e

ct

or

ie

s.

M

re

o

v

e

r,

e

a

c

h

c

o

m

p

a

c

t

s

u

b

s

e

t

L

p

∈

X

.

h

e

n

e

ith

e

r

p

is

a

pe

r

i

o

d

ic

p

o

i

nt

,

o

r

ω

(

p

)

is

a

pe

r

i

od

ic

r

b

o

i

t

a

nd

a

li

m

it

c

yc

l

e

,

o

r

f

or

e

v

e

ry

x

∈

ω

(

p

)

b

o

t

h

α

(

x

)

HGeánjeekraalllsyo,tchaerrmieadinsopmoientrsesinulHtsáojefkS’solpnrtozoefvwaenrdeVsiimnoilgarratdomtheonstieonu eseddaibnovtheetodiflffoewresntoinabdleichcoatsoem.iHco2w-

mevaenri,fsoeldvse.ralpartswereused.Oneofthemwasconsideringtheinherenttopology.Foragivenregularorbit,wemayconsidert heEuclideanbaset hei mageso fo peni ntervalst hrought hes olution.Oneo ft hem ainp ointso ft h ep roofw ast os howt hatt hese Theorem7.1.

Theorem7.2.

Theorem7.3.

✷✶✷✵

tdieasbcirliitbyinagssnuomtpotniloynsli.mAitsseittst.urnedo ut a bo u t twentyye a r s late r,i t was relatedto m uc h mo regene ral,d e e p property

Theorem7.4.

s theimplication

(7.4.1)(X;R;π)istopologicallyequivalenttoaC²flowonX,

topologicallyequivalentto(X;R;π).Furthermore,thefollowingconditionsareequivalent:

Letu sp resentm ainraensultshoefG utiérrez’sp aper.Hist h e o remg e n e ralisedn o t o n l y t h e resulto f [54],butf i r s t o f a l l Theoremt hisg a vea tdopologdicale quivalenceo ff l o w s w i t h f i n i t e n u m b e r o f c r i t i c a l p o i n t s o n t hep lanet oCe^∞nflow s.

istopologicallyequivadlenttothaC^∞flowonXundertheassumptionthatintheflowtherearefinitenumberocfacritical propertiesoforbits,in2particsuelar,thepropertiesinvestigatedinthePoincaré–BendixsonTheorem.Notethatepaorlier,i∞n

Ine1v98l6yCuaserldosaGtouptoiélrorgicalfleoxampleaofpaapseuritoanblsemsoeotthgiinvgencobnytiBniunogu.sFfloorw ns=[292],afunnddnam=en3tathlefoprrtohbeletmheworysosft2ll-doipmeenn.-ezplished

However,f orh igherdimenssitont heRp4roblemturnedouttobenon-trivial.Itwassolvedin1974 byChewning[19]who

smiaonnaiflolsdysXtemofs.clAascsorCo∞llaryofthemaintheeqoureivmaleonftthtiosapaCpe1rflwoawsothnaXt.anTywcoo

ncotinntuinouuosuflsowfloownsa(X2;-Rdi;mπen)saionndal(Xco;mRp;πact)a s b

p is pologically 1 2

u b

flows.Inp articular,t hePoincaré–BendixsonTheoremf ort opologicalflowsc anb eregardeda sa c onsequenceo fthePica so

i la Ne t l t

his proof many advanced and complicated modern techniques and results. The other proofs of thePoincaré–

BendixsonTheore ,e w u mor y.

NoteaulcsotnhawtaproupseedrtiesoofPaioninecvarné–mBernedixsonlatryepxeinthse2- dimensionalcaseforflowswereconsiderednotonlyfor

examplief[67c]hofaaveCc¹toflrowewithxoisutts,abclotseiddorbit.foIrnm1994Knup- erbesrgen[4c4]constructedaC^∞counterexample.Laterthis

withoutncroitnicaallflpoowinstsingivhenabsypdecifft,erienitianlaetuqruaaltitoonmoenthen3t- deimeensfeiorntalspherehasaclosedorbit.In1950Seifert[69]

Theorem7.5.

✷✶✷✶

dTehfienpitrioobnleomfdifynaamfloicwalonsyRⁿemmsu.stInbethgeivoenne-

bdyimaendsiffioenreanlticaalseeqtuhaetiaonnsawreorseisnpaotusritailvleyaanftderthinetrpordouocfinigsnthoetadbiffis tcrualctt.pclroveidredanexampleofa win

whichisn eitherisomorphicn ore quivalentt oadifferentiablesysteam.Cihewning

trraejecatoidrietosofe(Xto;Ro;lπog)i,caplrleytoerqvuinivgatlheentn,aitfutrhaelroerieexnitsatstioanhoofmtreaojemcotor priheiss.mToopfoXlogthicaatlteaqkueisvatlreanjecectporreiesserovfes(Xto;R;lπog1)ictaol1978, Neumannprove[54]

at any continuousflowon a2-dimensional compact orientable manifoldXof l ssC points and only perio ic an critical points are contained in their limit sets. Together with the Poincaré–B dixson Gutiérrez used Neum n’s t orem in his proof.

Theorem6.5,whichi (7.4.1)⇒(7.4.3).

Let(X;R; π)be a flow on a compact2-dimensional manifold X of class C^∞. Then there exists a C¹flow on X which is

(7.4.2)(X;R; π)is topologically equivalent to a C^∞flow on X,

(7.4.3)i

(

i.e

.i

s

p

e

r

io

di

c

o

r

b

it

)

,o

r

M

=

X

(

th

e

w

h

o

le

m

a

n

i

fo

ld)

;

in

h

e

l

a

s

t

c

a

s

e

th

e

ma

n

if

o

ld

X

m

u

s

t

b

e

qu

a

lt

o

t

h

e

two-dimensional torusT². i.e

Fromthistheoremitfollowsimmediatelythattopologicalpropertiesofdifferentialplanarautonomoussystemsholdforonré–

BendixnTheoremnthedifferentiacse. vertheless,ishoudbepointedoutthatGuiérrezusedin 3T-

hdeimperombsilfeomrfloofwssmdoosthpitintegisntohteeflaoswy,tfoersrner=eal3lyism,tcohtnhteiobeeeshltemoSfentihtaeraucothnojerc’st ukrneo.wItlesdtgaete,ssttihllatopeveenr.yScopnetaikniunogusabfloouwt

asked su filde ud not ulateonexit

e a sa c onjecture.I n1 9 7 4 S chweitzerg avea nconstrtio s to bt e o regu ample.

sflyoswtesmdsefionnedsuornfacseusb,sebtustoaflsthoeopnlasnueb.seTthseyofgtehneerpalalinseed.TInhe1o9r6e7mS5e.1ibaenrtdaonbd taTinuelldey[68]sho we d a si mil ar t h e o remf or

LetX⊂R²and

(X;R;π)beaflow.Assumethatp∈R²i

saregularpoint.Thenp∈/ω(p).

The reasoning in their proof did not use sections.

positive direction. This leads to the abstract definition of asemiflow,which was first formulated in 1965 by

Hájek[32].Thet ro i l esnboo 3

Thusanenatuwraelllq)uesdtiocnonasriseesabouttheePmoitincard e p e n d i n g notonlyonthe2- dimensionalsemiflows,notonlyforω-limit

givenpoowinthxa,vweemavyeminetnrodducenegantilve“sfeomitraje”,ctbouriesthr questionmaybemanysuchtrajectories,buttheremayoughx(there

dependt.onlysoo,nittosponltorgiicuailngpropertiepsrapnedrontheapopsasritbilityof ctarecoinjustonedirection.movement

systemsn.oAtvtiheewsoafmdeiffetimene,tifromt haetitopsotlogicaolstpioninetrofview elsofsspacest heremaya ppears everalquestionsofinmany

directionn,wapphesaorlntoiwonassgaivenatuuraluteopicafnorfduerfithneerdreosrearchof .egeneralphenomena.It shouldbenotedthatfrommor

amanifold, Theorem8.1.

in [20]. A closed setScontainingxis called asectionthroughxif there are aλ>0and a closed setBsuch that F(λ;B)=S,F([0;2λ];B)isaneighbourhood(notnecessarilyopen)ofxandF(µ;B)∩F(ν;B)=∅for0≤µ<ν≤2λ.

perfectlyfromonesidetotheoppositeoneinthetimeinterval2λ(allsegmentsstartononesideoftheboxandno other trajectory joins these

segments).

forsemiflowsitisimpossibletogivesuchagooddescription,ashereatrajectorycan“glue”withothertrajectories.

The Poincaré–Bendixson Theorem: from Poincaré to the XXIst century

8. ThePoincaré–Bendixson Theorem for semidynamicalsystems

Inflows,wehavethemovementdefinedinbothdirections.However,onemayconsideronlythemovementdefinedin

Teqhueatthoiopeooloygiticfasletmhuefloorwysowfflaoswndsevgneriqeowplfdryomooddifflearteenrtiianlftehqeuaalltitok n∈s[1aR]s.aSfleomwiflioswas,nwatituhratlhgeemneorvaelmizeantitodneofifnaendaountloynionmoonuesthe poit f r

alequ onhem te s t i n g mod emiflowsmaybefoundininfinitedimensional interesAl ii g which o tiesof icularobje nsequences ofdifferentiablestructure,which

Semifl s mo tefinedo y rward ta na tu ra l about negative continuations

arises.Forabenoas an id rnegativl i sets,

pointbutalsoonanegativesemitrajectory.

sInet1s9b7u7tMalcsCoafnonrα[5σ1-]liwmriottseetasn,wimhpeorertaσnitsp

aapneergaét–iBveensdoliuxstioonnptrhorpoeurgtihesx.for

In particular, his results implied that

isnovleustitoignaitsindgefithneedtoopnoltohgeicinatleprvraolpe(−rt∞ies;0o].fsemiaflboowustiosnom2o-

rdpihmiesnmssioonfaslemmaiflnoifwosld.sonecanassumethatanynegative IpnartahleleplirzoaobfioliftythoefPflooiwncsawréa–

sBfeunnddiaxmsoenntTahlefoorretmhefolroflcaolwcsh,atrraancsteverirzsaatlisonanodfstheectinoenisghpblaoyuerhdoao

ndiomfptohretasnytsrtoelme..THhoewleovcearl,Thereweremanyattemptstogeneralisethenotionofsectionsforsemiflows.In1992,thefollowingdefini

tionwasstated

TInhethseecsaescetioonfsflogwivse,tahigsodoedfilnoictiaolndgeisvcersipatioWnhoitfnaeys–

uBiteabbuletonvesiegchtbioonu.rhAolosdo,othfeaenxoisnt- ecnrciteicoaflspeocitniotnisnignesneemrailfloswemsiiflnotwhse.

generalcase,i.e.withouttheassumptionoflocalcompactnessofthephasespace,wasproved[20].Wehave

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if

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it

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t

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ti

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x

.

aAcscuoirtdaibnlgetnoeitghhisbotuhrehooroedm.aTnhdisMnceCigahnbno’surrheosoudltsi,sianpaarpalallnealirzasbelmeif l“boowx”aninywnhoinc-hcraitlilcathleposeingtmxenctasnofbetracjoencttaoirnieesdgino

in1994[21].Inthatpaperitwasalsoshownthatanycompactsectioninasemiflowona2-manifoldXiseitheraJordan

Int hef ollowingt heoremsbyal imits etLw emeane ithera nω-limitset ω(p)oranασ-limitset ασ(p),w h e reσ isa2

✷✶✷✷

Jordan curve, which was important for the reasoning.

noregoantiave2-sdoilmuteionnsiothnraolusgphhearen.onH-

ocwriteivcearl,psooimnetpof.tIhneamllccaintebdetheexoternemdesdthtoerseomisea2ss- udmimeednsthioantawlemahnaivfeoladss.emiflowonR

If a limit set L is connected and does not contain critical points, then L is a single trajectory.

Theorem8.2.

xist s

developedbySaitoin1968[64].TheUra–KimuraTheoremisveryimportantforseveralotherapplications.

wereobtainedafterWorldWarII.ThePoincaré–BendixsonTheoremgavethebeginningtootherinvestigationsand

andaquestionaboutthebehaviourofthesystemintheneighbourhoodofacompactinvariantset,notnecessarilyfor

ofSchwartz.Also,theyprovedtheassertionofthePoincaré–BendixsonTheoremforanycompactminimalsaddlesetin suchflows.

Theoremdependedontheuniquenessofthenegativesemi-solutionsandthepossibilityofuniquecontinuousmovement in bothdirections.

Another generalization of the Poincaré–Bendixson Theorem was given in 1988 by Athanassopoulos and Strantzalos [7].

Becauseofthecomplicatedstructureofsemiflowsandthecharacterofsingularpointsinthefinitedimensionalcase,

AttheendofthissectionitshouldbepointedoutthatthePoincaré–BendixsonTheoremforsemiflowsshowsthatthis

✷✶✷✸

K. Ciesielski

Theorem 8.3.

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c

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it

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s

.

If p∈ω(p)(or p∈ωσ(p))then p is either critical or periodic.

If a compact set A is either positively minimal or weakly minimal, then it is either a critical point or a periodic orbit.

Gademnietraclolmy,ptlhiceapterdoosfistuwaetrieonnsowtahiscihmaprleeiamnpaolosgsiyblteoftohrosfleowfosr.dMiffoerreeonvteiar,lasyllst theemsea(orlrieervepnrotoofsthoofsteheforPflooinwcsa)réa–sBseenmdiiflxoswons

NotethattheGutiérreztheoremaboutthetopologicalequivalenceconcernsonly2- dimensionalflows,notsemiflows.oneshouldnotexpectananalogousresultforsemiflows.

tohfetohreemtimisenvoatrioanbllyeptu.rReloyugtohplyolospgeicaakli,nbgu,ttihnefaecstseitncdeepoefntdhsisotnhleyoornemthiescaonptoi snsuiobuilsitmyoovfeameconnttdineufionuesdmfoorvpeomseitnivtefovrawluaerds

withoutbeingbotheredaboutthebackwarddirection.

9. Some otherresults

ManyfurtherresultswhichmayberegardedassomegeneralizationsoftheclassicalPoincaré–

BendixsonTheoremgmeantehreamliaztaictiaolnms;oanlolgorafpthhe.mHecroeu,ldwenomwenftoiromnaratlahregrebrcioelflleyctoinolny.sA olmlethoofsethoresseugltesnceoraullidzaptrioobnas.blybeatopicforalarge

OhonoedooffthaepaedrivoadnitcatgreasjeocftothryeoProainccarirtéic–

aBlepnodiinxts,oTnhTehoreeomrem4.2waansdaTphreecoirseemd4e.s3c.riTphtiiosnsuofggpelasntsaraspyostsesmibslei ngeanneeraiglihzbaotiuorn-2-dimensional systems. Thiswas obtained (in the general case offlows)in 1960 by Ura and Kimura [77] andlater

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x

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⊂

M

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(

x

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M

.

Further generalization of this theorem, giving a more precise description, was obtained by Bhatia, see

[12,15].TThhiesywparosvaedgetnheartafloirzaatifloonwfoolnloaw2in-

gditmheenrseisounlatslmofaSncifhowldaratzcoamndpatchtemaisnsiemratilosntaobflteheseUtrias–

aKsiminutrhaeTahsesoerretimonsiomfuthlteantheeoourselym.sInio1n9a9l6spAhtehraenaSs2sohpoolduslofsorpraocvte Theorem8.4.

Theorem8.5.

Theorem9.1.

HBeenpdroixvseodnthTehefoorlleomwifnogrflthoewosreomnst.he2-dimen-