To appear Journal of Differential Equations

We consider cyclic nearest neighbor systems of differential delay equations, in which the coupling between neighbors possesses a monotonicity property. Using a discrete (integer-valued) Lyapunov function, we prove that the Poincar'e-Bendixson theorem holds

TheP oicnre-aBndeixso Tneorhem forM notooenCyc li Fecebdcka ysSetm sitwh elDyaoJhnM alle-Patrte iviDion osf ppliAedM ahetmatiscB rownUni vesirtyProvi edcne,IR 20129Ge rgo R.e Slel cSool oh faMtemhaticsUn viesrtyi foM nnieosat inMnapolise,M 5N455

5OTOBCRE 24,199 4T apoear: pJuronlao fD ereinialt quEatoin

s

We consider cyclic nearest neighbor systems of differential delay equations, in which the coupling between neighbors possesses a monotonicity property. Using a discrete (integer-valued) Lyapunov function, we prove that the Poincar'e-Bendixson theorem holds

bstAarctW eocnsderi yccil cnareestnei gbhr oystsmes f do ierenita ldlae yqeautions,i n hiwc hht ceopulin gbetwen neiehbogsrpossess es amo noonititc propertyy. Uisnga di srecet( inetergv-laud)e Lypuano vunctifo,n e prwov thet ahe toiPcnae-rendBxsoi theonemr hodsl or sfcuh sysets. mW easol otabinr esltuso np eciweisem oontonciiyt adnstabilit yof preioicd solutoisn o sfcuh sytses.m

Ke yordWs ndaPh arses:i eredtinl adlea yquetaion P,onicae-rBndixseont hereo, myccil cefdbaeckysset, mnie timensionad invaliratnse .t

We consider cyclic nearest neighbor systems of differential delay equations, in which the coupling between neighbors possesses a monotonicity property. Using a discrete (integer-valued) Lyapunov function, we prove that the Poincar'e-Bendixson theorem holds

1 nItroudciotIn tnish aper wp peorevth ePiocnre-aeBndxiso nthorem efors styem sof ycciclall y(earnset eighnbor )oucplded eiernita ldleayeq uaitnso,in wich thh eocpuingl asiste s amontonicoiy contitdoi. n Atyipacls ch syusetmi sigenvb y hte N+1 equa toinsxi t)(= f ixi(t();x+i1 t(? )); i_0i N;

1(:)1

wehe r 0Na n the dindcesiar e aten mkdo N+,1wh er i e0f ro eac hi a,dnwh reee ahcf i enjo s tyh eomnotnociiy tporpetyr f@ (; ) 6= 0 fori al (l;) 2RI: (12:2 )@We in luce dhe scatar lacs Ne=0 o f(11).,(1.2) i no ruth ore. Wyea tucallyall w oa sightly mloe rgneral ceasslof s ytems thsa (n.11),(12).,na melys ysems permtiting tat w-sodide niteartcoinof t h vaerailb eix wti hboh itstne igborh xs 1i. A di scetre (inegte-rvluad)e Lypauovnf untcino,den teod V, wa sdeelvpeod ni PMS-1]efor a c asl ofsde alys sytms eicnludnig 1(.1), 1.()2 .hTe ufcnton iVi s i,na snse,e ayhbid ofr het milSile unctifnoo f MPSm]-,rst edne din Sml],a d n asacalr vrseio no fV whic wha desn de niMP] . hTe repestnpaper m keasext enive uss eo fte hrsuetl sin M-PSe]1 i,n patrciuar lhe tpoprerteiso Vf,a n itd sohul dbe ocsnidred aeco mpaionn apep.rN verteelesh,swe have sumarmizd ell ahet erevalt rnesluts frmo PMe1]S bloew, an duor perentsaiont ehe irs ssentialeylself -contanedi R.eustls i thned riceiot nfoa oPinarecB-ndiesxn toehorme for id erentiald laye equtiaos bngen awiht arley orwk f Koapln aadnY ore Kk-ao1Y, K]-Yoa],2 and ewrem re oercneltygive n by Wltaher Wa]1 W,3a,]Wa]4 Wa5,],Wa 6], b Hyrez eHr] a,ndb R.y SmthiSmR]1,S R5]. Exmcptefor th workeof mitS, hall htee serulstsc nceonr cslaraeq atiuno, ands egenrlalysl owl oysillctiagn osutlonis. hTe pesrne ptaerp flloos whetap proahcs ineMP- Sm]and F -iM1P,] ad nmaesk ullf use o thfed scietr eLypauovnfunc itnoV . hT OeED acs ef (1.1o, )1.2(,) i whinh cla li=, w0a tsretad eni M-SP]m, hewr eaPoi nareceBdnixsn theooerm aw prosve,dfol olwing pelirinamy rreustslin Ha T-yW-].e hiTs aw sgenraliezd eni El t]o wo-tsidd enterictianso; es ealo sHo-PMS-m f]ora nxeetsnin ofothis esrutl A P.oniarceendBisoxnth oeer wams rovedp i nF-iM1P] ofrt h ecaslra PED

u t=uxx+ fx(;u ux;);1

x2

S 1;

We consider cyclic nearest neighbor systems of differential delay equations, in which the coupling between neighbors possesses a monotonicity property. Using a discrete (integer-valued) Lyapunov function, we prove that the Poincar'e-Bendixson theorem holds

whree S1 denotes ht ecicrl,et ha its,w eavh eeprodiicbounda y rocniditons .Int hse peaeprs,a s hre,ethe m ai noolti n prviog snuc reshlus its adsiretce yLapuovn fncutino .I naprtciluar,of rth ebova PDE ehte ezrocro-ssignn mubr eofMa tno apays ltihs olre .Th erpofos i MP-nSm] na dFi-MP1],hwie bolh rteyilg no nidcrste Lyaepuonvfu nciots, nar eomsweht dia rent; in ete formeh rppare ht aerumgnes trae olngerand omer niovvel,da lhtugo hteyh yied loremi nfrmaoiotn,p ratcilarlu yabuto hcarcatriestc muiltilpiesr ndalo alc niavrian matnfolisd. In he prtseetn aper pewadopt he tsipmlre\ostefr"approah oc fi-MFP1] (es alsoe Fi),] hwchi avidosthe us of escu objhetcs. aMn yo tfeh esrlut sno hacacterrsitcimu tiplielsr, Flquotes lotuino, sadne xonpnteai dlciohtoies, mohweve, rcna be oufd nni P-SeM], 1na aldos in MP-S2e] .Asm entonid eelbw,omany of the rseults fo he tprseent pape fro rhte cslaa requaitno

x()t f (x=t(;)x t(? 1) )_(:3)1wher epD)(= D(? 0 )D (? 1) (D? N )i san N+( 1s)t-dgeere plyoonmai lallof howe soros ater eral,a dn whreef 0 (x ) 6=0 foral lx 2I R:In ede,dupon s etitngx 0(t )=x( ) atnd x+i1( ) t= x it)(?i i x() _ it= 0 e xecp tN= 1. fro 0iN?, on1eobtains asstymeo f th feom (1.r)1, (.1)2 wit,hall ePridiocsol uiton so efuqaitn o1.4( )hvea ebneob atien dn Hiei2]f r o= 1N an, fdr ognerea Nl n ihe tsingluarpe turbrtaoni acs ien HaIv]. For -ou resrlus tot appy to elquatino( 1.4,) itis n ceesarsy hat atllr otso fothe plynomoal p bi real,eso th at het bove traanfsroamtio to n assytm e1.1()can b eadm. eIdnee, dteh rsuletsof H MML]Ss gugse that ti p hfasno n-eralr oost th,n ehe toPicane-renBdxiosn hetrome ca fanl fio r(.41), na tdahtc oplme xan dcaothcidy namic sac noccu.rT h claes sfoe qauiotn wes rteat nit ih sapep rs aictualyls iglhtl moyrege enrlat ha n1(1.), (1.2) .Inp atriulcra,e mway laow lfi to d eenp odnx?i 1t() n ianapp oprirtaef ahsino,a telast of ir6= 0. 2

weredev …… 此处隐藏：29184字，全部文档内容请下载后查看。喜欢就下载吧 ……