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DEGENERATED

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DEGENERATED

    DEGENERATED

    Chin.Ann.o|Math

    21B:2(2000),201210

    ENERATEDHoMoCLINICBIFURCATIoNS DEG

    WITHHIGHERDIMENSIoNS木木木

    JINYINLAI*,ZHUDEMING*

    Abstract

    Thedegeneratedhomoclinicbifurcationfor

    existence,uniqueness,andincoexistenceofthe

    arestudiedunderthenonresonantcondition.

    undertheresonantcondition.

    highdimensionalsystemisconsidered.The

    1-homclinicorbitand1-periodicorbitnearF

    Complicatedbifurcationpatternisdescribed

    KeywordsLocalcoordinates,Poincar6map,Homoclinicorbit,Periodicorbit 2-foldperiodicorbit,Resonantcondition

    1991MRSubjectClassification58F14,58F22

    ChineseLibraryClassificationO175.12DocumentCodeA

    ArticleID0252?9599(2000)02?0201?10

    ?1.IntroductionandHypotheses

    Inrecentyears,withthedevelopmentofnonlinearscienceandthedeepstudyofchaotic phenomena,anincreasinglylargenumberofpapersaredevotedtothebihtrcationproblems ofhomoclinicandheteroclinicorbitsinhighdimensionalspace(see[14).Duetothe

    difficultyencountered,unfortunately,onlyafew(e.g.[1,13,14])areconcernedwiththe periodicorbitsbifurcatedfromsingularloops.Papers[1,13]discussedtheproblemofthe homoclinicloopbifurcationinhighdimensionwithcodimension2,thatis,thesystemhas resonanteigenvaluesandthehomoclinicloopF=z=r(t):t?R,r(士?)=0)satisfies

thenondegeneratedconditioncodim((t)W+(t)W):1.

    Inthispaper,theperiodicandhomoclinicorbitsproducedfromthedegeneratedhomo- clinicbifurcationareconsidered,whichmeansweassumecodim(T,(t)W+ff1)=2. Resultscorrespondingtononresonantandresonantconditionsareobtained.Themethod toestablishasystemoflocalcoordinatesnearthehomoclinicloopsuggestedandusedin [13,14]issimplifiedhere.

    ConsiderthefollowingCsystem

    =f(z)+eg(z,,?)

    wherer4,z?R+",?R,01?11,f(0)=0,g(o,,?)=0

    Weneedthefollowingassumptions.

    ManuscriptreceivedJune7,1999.RevisedOctober25,1999.

    ,DepartmentofMathematics,EastChinaNormalUniversity,Shanghai200062,China. **DepartmentofMathematics,LinyiTeacherCollege,Linyi276005,Shandong,China. }}*ProjectsupportedbytheNationalNaturalScienceFoundationofChina(No.19771037) (1.1)

202CHIN.ANN.oFMATHVo1.21Ser.B

    (H1)ForE=0,System(1.1)hasahomoclinicloopF={z=?'(t):tER}with?'(士?)=

    0.ThestablemanifoldW.andtheunstablemanifoldW"ofz=0aremdimensionaland

    ndimensional,respectively.Moreover,ThelinearizationDI(O)attheequihbriumOhas simplerealeigenvaluesl,2,3and4suchthatanyremainingeigenvalueofDf(O)

    satisfieseitherRe>5>),3>l>0,orRe),<),6<),4<

    ),2<0forsomepositive

    numbers5and6.ForanyP?F,codim(1pW"+1pW.):2.

    (H2)Definee=.

    1,

    im

    .

    (t)/l(t)1.Then,e+?ToW"ande一?ToW.areunit

eigenvectorscorrespondingtoland2,respectively.

    LetW..andW""bethestrongstablemanifoldandthestrongunstablemanifoldofz=0. respectively,+andbeuniteigenvectorscorrespondingto3and4,Wt't'+CWt't'

    andW..CW..betheonedimensionalsolutionmanifoldstangentto+andatz=0.

    respectively,W""CWbethe(n2)dimensionalsolutionmanifoldtangenttothe

    generalizedeigenspacecorrespondingtothoseeigenvalueswithlargerrealpartthan5, andW...CW..bethefm21dimensionalsolutionmanifoldtangenttothegeneralized eigenspacecorrespondingtothoseeigenvalueswithsmallerrealpartthan6.Then,we

    haveToW""=ToW"""0TOW""+TOW..=TOW...0ToW...

    (H3)

    (H4)

    ?

    ((t).n(t)")=e.

    lim((t).nTr(t)W")=e+0+

    tIoo,

    span((t)",(t).,e+,+)=R,t1,

    span((t)",(t).,e,e)=R,t《一1

    WesayFisdegenerateifdim(T~(t)W"n(t).)>1.Indegeneratecases,thepatten ofbifurcationwillbemuchmorecomplicated.Itiseasytoseethatunderthehypothesis (H1),thehypothesis(H2)isgeneric,andsoarethehypotheses(H3)and(H4).Hypothesis (H4)isequivalentto

    (t)W"W""0e0.

    (t)._?...0e+0+

    Thisiscalledthestronginclinationproperty.

    ast}+0(3,

    ast—?一?.

    52.LocalCoordinates

    OurstudywillbebasedontheanalysisofthePoincar6mapdefinedonsomelocal transversalsectionofF.FortheestablishmentofthePoincar6map,weshouldchoose

asuitablecoordinatesystem.ConsiderSystem(1.1)underthehypotheses(H1)(H4).

    SupposethattheneighborhoodUissmallenough.ThenwecanintroduceaCchange

    suchthatSystem(1.1)hasthefollowingformin:

    =

    [Bl(E)+]u,=[B2(E)+]u (2.1)

    wheret(0)=ifori=1,2,3,4,Rea(B1(e))>3andRea(B2(E))<4forsmall

    

    ,?J,?J

    ++

    ,J,J

    ,L,

    24

    ,<

    一一

    ==

    .:

    U

    ++

    ,J,J

    ?

    ,L,

    l3

    ,<

    ?

    【?【==

    .

    .1U

    No.2JINY.L.&ZHU.D.M.DEGENERATEDHOMOCLINICBIFURCATIONS203 enough.Inotherwords,wehavestraightenedthefollowingmanifoldsinU, rn,u={Y=0,=0,=0,u=0,u=0),+={x=0,Y=0,=0,u=0,u=0), rn={=0,=0,=0,u=0,=0),={=0,Y=0,=0,u=0,=0),

    "

    =x=0,Y=0,=0,=0,u=0),={x=0,Y=0,=0,=0,u=0.

    Here,u?R,u?R,and(2.1)isC.

    Takingatimetranslationifnecessary,wemayassumer(-T)=(,0,0,0,0,0),r(T)= (0,,0,0,0,0),whereissmallenoughsuchthat{(,Y,u,v,u,u):ll,ll,ll,ll,lul,

    <35/2}CU.LetA(t)=D,(r(?)).Considerthelinearsystem

    =A(t)z

    anditsadjointsystem

    =A(?)

    Nowwechoosesolutionsof(2.2)asfollowing:

    Zl(?),z2(t)?((?).)n((?)"),

    za(t)=(?)/l()l,z4(t)?(t).n(?)",

    zs(t)=((?),,(?))?(?)"C((?).)n(?)"

    z6(t)=((?),,(t))?(?)...C(?).n((?)")

    (2.2)

    (2.3)

    Zl(T)=(1,0,0,0,0,w16),z2(T):(@21,0,1,0,0,w26),z3(T)=(0,1,0,0,0,0) 4(T)=(0,0,l,0,0,O),zs(-T)=(0,0,0,0,,O),z6(T)=(0,0,0,0,0,) suchthatZ(t)=(Zl(?),2(?),z3(t),zdt),5(?),zdt))isafundamentalsolutionmatrix. Proposition2.1.(H1)-(H4)arevalid,thenthereexistconstantvectorsw16,w26and

    西21suchthatthefollowingsaretrue:

    1(T)=(Wll,w12,w13,w14,wI5,0),2(T)=(w21,w22,w23,w24,w25,0),

    z3(T)=(w31,0,0,0,0,0),z4(T)=(0,w42,0,w44,0,0),

    zs(T)=(w51,w52,w53,w54,w55,w56),6(T)=(w61,w62,w63,w64,w65,w66),

wherew31<0,w44?0,detw55?0,detw66?0andeitherw12W24?0,W22=0or

    w12=0,西21=0,W14W22?0.Moreover,Iorsmallenough,lWliWl1fori?2,

    1w2iw2-41l1fori?4,l西21l1,1w42wgl1,1w5iw551l1,0ri?5,1w6iw661l

    1

    |ori?6.

    ProofiTheexistenceofzs(t)andz6(t)withgivenvaluesatTand-Tisclear.By thedefinitionofz3(t),wehavew31<0immediately.Nowlet1(t)beasolutionof(2.2)

    with21(T)=(1,0,0,0,0,0).Then,z1(t)=1(t)+z6(t)w16isalsoasolutionof(2.2) withzl(T)=(1,0,0,0,0,wI6).Denote1(T)=(Wll,12,13,"to14,5,W

    16)andtake

    w16=wg?M16.ThenwegetZl(-T)asdesiredincasedetw66?0.

    Nowbythedefinition,zl(t)?(?")n((?).),weget(W12)+(W14)?0.

    Firstassumew12?0.Then,similartotheprocedureforgettingthedesiredZl(t), weseethereisavector西26suchthatthereexistsasolutionz2(t)satisfing22(T)= (0,0,1,0,0,w)26)and2(T)=(21,"t~22,"t~23,"t024,W)25,0).Sincew12?

    0,wecande-

    finez2(t)=(t)+Zl(t)~21with西21W22W121andw26=26+7~21W16suchthat

    2(T)=(w21,0,w23,w24,w25,0).

204CHIN.ANN.oFMATHVol_21Ser.B

    Thatz4()hastheexpression(0,w42,0,-H244,0,0)issimplybecauseTr(t)n(t)" isaninvariantsubspaceof(2.2)andbecomestheYplaneastT.

    AsimplecomputationshowsthatdetZ(T)=-w44det11)55,whichturnsoutthat?44?0

    anddetw55?0.

    Nowweshowdet66?0.Infact,ifdetw66=0,then,duetodimW"=rankz6(T)= rankz6(),wehave()nspan{Tr()",e,百一)?0.Noticethathypoth

    esis(H3)meansdim(T~(T1nspan{T~(T),e,百一})3,whichturnsoutthat

    dim(span{T~(),(),e,百一))<n+m.Itcontradictshypothesis(H4). w24?0isadirectconsequenceofdetZ(-T)=--W12W24W31detw66?0.

Ifwl2:0,thenw14?0,andwesimplytakez2(t)=2(t).Thismeans西21=0.Nowit

    followsfromdetZ(T)=wl4w22w3ldetw66?0thatw22?0.

    TheremainderiseasytocheckbyusingtheexpressionsofA(t)att:+?and..,we

    omitthedetail.Theproofisfinished.

    Generically,wehavewl2?0.Sincethediscussionissimilarforthecasew12=0then

    wl4w22?0byProposition2.1),werestrictourselvestothecasew12?0inthispaper.

    Denoter(t)=(rl(t),r2(t),r3(t),r4(t),7';(t),7'(t)),?12=?l?l21.WesaythatFis

    nontwistedas?=1.andtwistedasA=1.

    Inthefollowing,weregardz1(?),z2(t),z3(t),z4(t),zs(t),z6(t)asalocalcoordinate systemalongF.Denote(?):(l(?),2(?),3(?),4(?),5(?),6(?))=(z(?)).Dueto

    15],wehave

    Proposition2.2.(?)isafundamentalsolutionmatrix0(2.3).Moreover,l(?),2(t)

    ?(?W)n(?W)areboundedandtendtozeroexponentiallyast+o0. ?3.Poincar6MapandItsAssociatedSuccessorFunction NowwesetupthePoincardmap.Set

    5=(,,几一2),n6:(n,?,n2),

    s(t)=r(t)+Zl(?)nl+z2(t)~2+z4(t)n4+z5(t)n5+z6(t)n6

    Let

    So=z=s(T):II,Il,ll,lI,luI,IvI<35/2},

    S-=z=s(T):Il,ll,I西I,II,Iul,lvl<35/2)

    becrosssectionsofFatt:Tandt=T,respectively,whereissmallenoughsuchthat So,SlCU.Atfirst,wesetupamapFlfromS1toSowhichisdefinedbytheorbitsof

    (1.1).Secondly,weconsiderthemapF2fromSotoslinducedbytheorbitsof(2.1)in

    U.Thirdly,bycombiningthetwomapswegetthePoincardmapF=FloF2:So卜?.

    Finally,theassociatedsuccessorfunctionisgiven. Letzs(t)beasolutionof(1.1).Substitutingitinto(1.1),weget (?)+(?)(nl,n2,0,n4,n5,;)+z(?)(l,2,0,h4,n5,)

    =

    ,(7'(?))+A(?)z(?)(nl,n2,0,n4,n5,)+eg(r(t),,0)+h.o.t

By(?):,(?'(?))andz(t)=A(?)z(?),itreadsas

    z(?)(l,h2,0,n4,n5,)=E9(7'(?),,0)+h.o.t.

    No.2JIN.Y.L.&ZHU.D.M.DEGENERATEDHOMOCLINICBIFURCATIONS205

    Multiplyingtwosidesoftheequationby(t)andusing(?)z(?)=I,weobtain

    t:E(?)g(r(?),,0)+h.o.t.,i=1,2,4,5,6(3.1)

    System(3.1)yieldsthemapR:S1卜?Sodefinedby(rl,1(),ne(-T),nt(-T),;(),

    n())H(111(),n2(T),n4(T),n(),n()),

    ni(T)=11i(-T)+E()+h.o.t

    where()=(?)g(r(?),,O)dt,i=1,2,4,5,6.

    (3.2)

    Proposition3.1.Fori=1,2,4,5,6,()=(?)g(r(?),,O)dt.

    ProoLItsllf~cestoshow(?)g(r(?),,0)=0forItI>T.Clearly,r(t):(0,r2(?),0,

    0,0,0)andIr2(?)l<5fort>T.Owingto(T)z3(T)=0fori?3andsolving(2.3)

    wegetthey-componentof(?)isequaltozerofort>T.Meanwhile,(2.1)'impliesthat

    g(r(?),,0)=(0,g2(r(?),,0),0,0,0,0)fort>T.Thuswehave(?)g(r(?),,0)=0

    fort>T.Similarly,wehave(?)(r(),,0)=0fort<T.Thustheproofiscomplete. FunctionsM1(),(),(),()and()arecalledMelnikovfunctions. Nowweconsiderthemap:SoS1,qo(xo,Yo,72o,0o,Uo,)ql(Xl,Yl,Ul,V)l,

    ,)whichisinducedbytheorbitof(2.1)inU,whereui=(,,?),vi=

    (,,)fori=0,1.

    Firstassume12.Inordertoguaranteethedifferentiabilityofthemapatthe

    origin,weset8=e-Al(,whereistheflyingtimefromqo(Xo,yo,Uo,Vo,uo,)to

    ql(Xl,Yl,721,E1,U1,).

    Ifweapplythemethodof[2orsuccessivesubstitutionswithinitialsolutionvalue z=eA1()(一一)z1

    ,

    Y=e-A2()(t-T)Yo

    ,

=e3()(T一下)1

    ,

    =e-A4()(t--T)0

    ,

    :e

    B1(E)(t-T--r)1

    ,

    =e-B2()(t--T)0

    to(2.1),andneglectthehigherordertermsforsufficientlysmall,thenweget

    z0:el(E)z1:8x1

    ,

    Y1=e-A2()=8A2()/l()

    ,

    0=e3(E)下面1=s3()/l()1,1=e-A4()0=s4()/l()0 ,

    0:eBl(E)1=sB1()/l()1,1=e-B2()0=sB2()/l()0 Herewehaveusedthefactthat

    z)0()e^(T,Y)0()e-A2(T),面)0()e3(T,

    ^0()e(,^0()eB(T,^0()eB2(.

    Nowweseekthenewcoordinatesofqoandq1.Let

    Then,

    q0(xo,yo,Uo,vo,Uo,vo

    q1(z1,yl,Ul,Vl,u1,

    =r(T)+Z(T)(nol,no2,0,'104,1105,1106), =r(T)+Z(-T)(n11,n12,0,n14,1115,1116) usingr(T)=(0,,0,0,0,0)andr(T)=(,0,0,0,0,0),weget (n01,n02,0,n04,n5,11o6)=Z1()(z0,Yo,0,0,,), (n11,n12,0,n14,n:5,1116)=Z1(T)(Xl),yl,1,E1,u1,) (3.3)

    (3.4)

(3.5)

206CHIN.

    ANN.OFMATHVo1.21Ser.B Let

    nW)W21叫叫

    ,ns=Wls一叫zs叫叫,n=WW25W2

    4

    1

    ,

    bxW51?-U21W53,b2W52W42w~41w54, 66W56W16W51)(W26(v21w16)w53,C1=W61W21W64,

    3W63W23W24~W64

    ,

    C4W64Wl4WW62,C5W65W25W

    24

    1

    W64

    whereIla~12ill1fori=1,3,5,ll1fori=1,2,6,ffc.lI1for= assmallenough.DuetothehypothesisW12?0andProposition2.1,wehave

    n01=o一面21.b1w51u0,Tt02=.W53uo ,

    Y0?,o4=W441(V0W54W551Uo), no5=uo,no6=W16zo(26WI67-U21).66uo+o n=(一叫ez"),nz=(W+云一cw6-61"), XX?5,n14=a3wl21yx+1W23W

    24

    1

    Vxc3a3叫叫62)"1,

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