Figure1:Left:Schematicofthebifurcationdiagramfortheunperturbedsystem(2 - PDF document

Figure1:Left:Schematicofthebifurcationdiagramfortheunperturbedsystem(2.1)with"=0,withillustrativesolutionproles.Thedasheddarkbluesnakingbranchconsistsoftwobranchesofevenparitysolutions,whilethedottedorangebranchconsistsoftwobranchesofoddparitysolutions.Solidgreencross-connectingbranchesconsistoffoursetsofasymmetricsolutions.Right:Schematicofthebifurcationdiagramforaperturbedsystemasin(2.1)with"6=0.Snakingbranchesforevensymmetricpatternswithcentralmaximumandminimumareshownindarkandlightbluedashed,respectively.Asymmetricbranchesareinsolidblackandgreen.Dashedverticallinesindicatethesaddlenodelocationsfromtheunperturbedsystem.ThoughweusetheSwift{Hohenbergequationtodemonstrateourresultsnumerically,wewishtoemphasizethatthehypothesesweimposearenotspecictotheSwift{Hohenbergsetting.Inparticular,asshownin[3],wedonotrequirethesystemtobeconservativewhenposedasaspatialdynamicalsystem,meaningthattheunderlyingPDEsystemneednotbevariational.Furthermore,whilewehavefocusedonsymmetry-breakingterms,thisapproachisapplicabletoanyperturbativetermspreservingspatialreversibility.Thepaperisorganizedasfollows:inSection2,wereviewthenumericalresultsofHoughtonandKnobloch[15]onsymmetrybreakinginthe1DSwift{Hohenbergmodel,whichinitiatedthepresentwork.InSection3,welinktheframeworkintroducedin[3]withaformalgluingapproachoffrontsandbacks,rsttoindicatethebroadapplicabilityoftheapproachtakenin[3]and,second,tounderstandsomeofthemajorfeaturesobservedinsystemsadmittinglocalizedpatterns.Wenotethatthissectionisintendedtoprovideintuitionandmotivatetheparticularhypothesesemployedinthefollowingsection,ratherthantopresentpreciseresultsforaparticularsystem.InSection4,wedetailpredictionsontheevolutionofbifurcationdiagramsupontheintroductionofperturbativesymmetrybreakingterms.InSection5weprovidenumericalcontinuationstudiesonplanarstripeandspotpatterns,studiedpreviouslyin[2];thesenumericalresultsconrmouranalyticalpredictionsfromSection4,includingtheexistenceofisolasproducedbytherearrangementofodd-symmetricandasymmetricsolutionbranches.Wealsoexplainandillustratehowthefullbifurcationdiagramforawidevarietyofperturbationscanbeobtainedfromthebifurcationdiagramofanunperturbedsystem,withoutactuallycarryingoutthenumericalcontinuation.Finally,inSection6weidentifyareasforfuturework.2 3.1BasicassumptionsFirst,wewillprovideanintuitivetreatmentoftherealizationoflocalizedrollpatternsfromsimplerbuildingblocks.Inthisspirit,ratherthanspecifyingaformforthePDEorODEgoverningoursystemofinterest,wemakethebasicassumptionthatthesystemadmitsfronts,i.e.,solutionsevolvingfromaconstantstatetoaspatiallyoscillatoryone.Moreprecisely,ifwewritestationarysolutionsasu(x)2Rnwithx2R,weassumethatthereexiststeadystatesuf(x)suchthatuf(x)!u0asx!�1anduf(x)!v(x)asx!+1,wherev(x)isperiodicinxwithnonzerominimalperiod.Infact,wecanmoregenerallyconsidersystemsadmittingsolutionsuf(x;y),wherey2 bRd�1,whichsatisfyuf(x;y)!w(y)asx!�1,wherew(y)isanyfunctionindependentofx,aswellasuf(x;y)!v(x;y)asx!+1,wherev(x;y)isperiodicinx.Inessencewerequireonlythattheevolutioninspaceoccuralongonedimension,perhapsafteranappropriatecoordinatetransformation.Forsimplicityinthissectionwewritesolutionsasu(x),butinsubsequentsectionswewillusethemoregeneralformulationtopredictsomeinterestingeectsofsymmetrybreaking,includingtheformationofisolasinplanarsystemsunderappropriateconditions,andtoverifythesepredictionsnumerically. Figure2:Illustrationofafrontandaback,relatedbyx7!�x.Wewillassumethroughoutthatoursystemisreversibleinthefrontevolutionvariable.Withtheabovenotation,thismeanswehavex7!�xsymmetry,sothatgivenanysolutionu(x),u(�x)isalsoasolution.Inthecasethatuf(x)isafront,ub(x):=uf(�x)istermedaback;seeFigure2.Theperiodicorbitv(x)approachedbyfrontsisassumedtobeinvariantunderx7!�x.OurprimaryinterestwillbeinsystemsthathaveanadditionalZ2symmetry,andinSection4wewilldescribetheresultsofbreakingthissymmetryinaperturbativemanner.Consequently,inthefollowingwewilllooktounderstandthecharacteristicsofsystemswithandwithouttheZ2symmetry.Forconcreteness,inthissectionwewillassumethesymmetryis:u7!�u,andinthiscasefurtherassumethattheconstantsolutionapproachedasx!1isu0=0.Wewillnallyassumethatthelimitingoscillatorysolutionv(x)iscompatiblewiththesymmetry,whichimpliesthat�v(x)=v(x+).Then,givenafrontsolutionuf(x),wewillalsohavethefrontsolutionuf2(x):=�uf(x),aswellasthebacksolutionsub1(x):=uf(�x)andub2(x):=�uf(�x).3.2ConstructionoflocalizedsolutionsviagluingWenowwishto\glue"togetherfrontandbacksolutionstoformalocalizedstationarysolutionuloc(x)whichisinvariantunderu(x)7!u(�x).Clearlythisisonlypossibleifwehaveamaximumorminimumatthecenterofthelocalizedoscillatorystructure.Deningthephase'atthecenterofthelocalizedsolutiontobethedistancetraveledpastamaximum,andrescalingxifnecessarysothatthespatiallyoscillatorylimitingsolutionv(x)mentionedabovehasperiod2,thisisequivalenttorequiringthatthephaseatthecenterofthestructuresatises'=0or'=;seeFigure3.Inthecasethatwehavetheadditionalsymmetry:u7!�u,werecallthattheexistenceofafrontsolutionuf(x)impliestheexistenceofthefrontsolutionuf2(x):=�uf(x)andthebacksolutionsub1(x):=uf(�x)and4 Figure9:Illustrationthatthefunctionzwillbe-periodicwhenevertheperiodicorbitv(x)respectsbothx7!�xandu7!�usymmetries.Left:Afrontuf(x)withcharacteristiclengthl,whichweassumeexistsatsome0.Right:Thefrontuf2(x):=�uf(x)willalsoexistforthis0,andwillhavecharacteristiclengthl+.Thus,asinthecasewherewehadonlyx7!�xsymmetry,weagainseethatthezero-levelsetofthefunctionZ(L;'):=z(L+')�z(L�')describesallbifurcationbranchesoflocalizedoscillatorystructures.TheR-symmetricsolutionbranchesarethosewith'=0and'=,whiletheRbranchescorrespondto'= 2and'=3 2.BoththesesolutiontypesexistforallvaluesofL.Finally,asymmetricsolutionsexistonlyforthosevaluesofLand'=2f0; 2;;3 2gsuchthatZ(L;')=0.SeeFigure1forthebifurcationdiagramwhenzhastheshapeoutlinedinFigure7(i).Similartothecasewithoutsymmetry,theseresultshavebeenderivedrigorouslyin[3]:ifaZ2symmetryispresent,thefunctionzwillberatherthan2-periodic,andtwoadditionalsnakingbrancheswithRsymmetrywillexistfor=(L;'0)=z(L+'0)+O(e�KL)forsomeK�0with'02f 2;3 2g.4MainresultsandpredictionsOurgoalnowistostartwithasystemthatrespectstheZ2symmetryforall,andtodescribewhathappensunderforcedsymmetrybreaking.Toillustrateourapproach,westartwiththecasewherez(L)possessesonemaximumandoneminimumforeachperiod,wherethe-periodicityisenforcedbythepresenceofaZ2symmetry.Wewillbeinterestedinperturbativetermsbreakingthesymmetrywhenasecondparameter"isswitchedon,i.e.,when"6=0.InFigure10(a)weprovidetwoequivalentrenderingsofthesolutionbranchesoflocalizedstructuresinasystempossessingsymmetryandwithz(L)havingasinglemaximumperperiod.Weillustratethebranchesofevenandoddsymmetricstructures(R-andR-symmetric,respectively)aswellastheasymmetricsolutionbranches.Theleftpanelshowsthephase'alongthex-axis,andthehalf-pulselengthLalongthey-axis,whilethecenterpanelshowsthesolutionsinthe(;L)planeviathefunction=z(L+')forsolutions(L;')ofZ(L;'):=z(L+')�z(L�')=0:Thisisanalogoustoourusualbifurcationdiagram,withlengthLbeingequivalenttotheL2norm.Theformu-lationintheleftpanelwillprovideanaturalwaytounderstandtheeectsofsymmetrybreakingperturbations,whilethecenterpanelprovidesthelinktofamiliarbifurcationdiagrams.Weseethat,beforeperturbation,theR-symmetricsolutionsat'=0and'=coincideinthe(;L)plane,asdotheR-symmetricsolutionsat'= 2and'=3 2(latternotshown).Wealsonotethat,duetothe-periodicityofz,allinformationisactuallycontainedinasinglequadrantoftheleftpanel,butweshowthelargerdiagramhereforeasiercomparisonwiththediagramaftersymmetrybreaking.9 Inparticular,wecansolvefor(~L0;~'0)(")as:~F(L;';")=~F(L0;'0;0)+D~F(L0;'0;0)0B@L�L0'�'0"1CA+O("2)= 00!so 2z00(L0+'0)(L�L0)+(z01(L0+'0)+z01(L0�'0))"�2z00(L0+'0)('�'0)+(�z01(L0+'0)+z01(L0�'0))"!+O("2)= 00!yieldingL�L0="�z01(L0+'0)�z01(L0�'0) 2z00(L0+'0))+O("2)'�'0="�z01(L0+'0)+z01(L0�'0) 2z00(L0+'0))+O("2)or(~L0;~'0)(")=(L0;'0)+" 2z00(L0+'0)(�z01(L0+'0)�z01(L0�'0);�z01(L0+'0)+z01(L0�'0))+O("2):Incase(i)where'02f0;g,the2-periodicityofz1(L)yields(~L0;~'0)(")=(L0;'0)+" 2z00(L0+'0)(�2z01(L0+'0);0)+O("2):Infact,for'02f0;g,the2-periodicityof~z(L;")inLimpliesthat~F(L;'0;")=2 ~z0(L+'0)0!:Sotheunique(~L0;~'0)(")near(L0;'0)satisfying~F((~L0;~'0)(");")=0mustbeoftheform(~L0(");'0)where~L0(")satisesz0(~L0(")+'0)+"z01(~L0(")+'0)=0.Thisthenimplies~Z(~L0(");'0;")=z(~L0(")+'0)+"z1(~L0(")+'0)�z(~L0(")�'0)+"z1(~L0(")�'0)=0as~zis2-periodic.Thuswehaveshown(i).Incase(ii)where'02f 2;3 2gwehave~Z((~L0;~'0)(");")=zL0+'0�"z01(L0+'0) z00(L0+'0)+"z1L0+'0�"z01(L0+'0) z00(L0+'0)�zL0�'0�"z01(L0�'0) z00(L0+'0)�"z1L0�'0�"z01(L0�'0) z00(L0+'0)+O("2):(4.5)WeexpandzL0+'0�"z01(L0+'0) z00(L0+'0)=z(L0+'0)+2z0(L0+'0)�"z01(L0+'0) z00(L0+'0)+O("2)andsimilarlyforzL0�'0�"z01(L0�'0) z00(L0+'0).Wealsorecallthatz0(L0+'0)=z0(L0�'0)=0,andz(L0+'0)�z(L0�'0)=0.Thuswerewrite(4.5)as~Z((~L0;~'0)(");")="z1(L0+'0)�"z1(L0�'0)+O("2)sothat~Z((~L0;~'0)(");")6=0aslongasz1(L0+'0)6=z1(L0�'0).Thiscompletestheproofof(ii). ThekeypointoftheaboveisthatsaddleequilibriacorrespondingtopitchforkbifurcationsfromtheR-symmetricbranchesgenericallydonotremaininthezero-levelsetofZoncethesymmetryisbroken,sothattheR-symmetricbranchesarethemselvesbrokeninamannerconsistentwiththeHamiltonianvectoreldformulationdescribedabove.12 Figure12:Bifurcationstructureofasystemcharacterizedby-periodiczpossessingtwodistinctmaximaperperiod.Again,weillustratethesolutionbranchesinthe(';L)plane,bothwithandwithoutthevectoreldinterpretation,aswellasinthe(=z(L+');L)plane,wheretheactualbifurcationbrancheswillbeexponentiallycloseinLtotheonesshown.Asbefore,R-symmetricsolutionbranchesareshownindashedblue,andR-symmetricindottedorange.Particularasymmetricsolutionbranchesareshowninsolidpurpleandgreen.Forclarity,notallasymmetricsolutionbranchesareshownintheright-mostrendering;branchesnotshownontherightarerenderedinthindottedgrayinthecenterillustration.ThelightdashedhorizontallinesshowthecorrespondencebetweenthehyperbolicequilibriaatA,a,B,b,etc.andthemaximaandminimaontheright.infactqualitativelyequivalentasbothresultinaseriesofalternatingcross-connectingandself-connectingasymmetricbranches,eachwithtwosaddlenodes.Intermsofthefamiliarbifurcationdiagraminthe(;L)plane,self-connectingbrancheswillappearas`S'shapedcurvesandcross-connectingas`Z'shapedcurvesforperturbationssuchthatthesignof~Zisthesame[(+;+)or(�;�)]forsaddleequilibriawithL2[0;)near'==2.Theoppositeistrueforperturbationssuchthatthesignof~Zis(+;�)or(�;+).Aswewillseebelow,whenzhastwoormoremaxima,dierentsymmetrybreakingperturbationsmayresultindistinctbifurcationdiagrams,whicharenotreducibleviare ectionsortranslations.Wenotethattheseresultsareapplicabletolocalizedrollsolutionsoftheone-dimensionalSwift{Hohenbergmodelut=�(1+@2x)2u�u+u3�u5;x2R(4.6)withtheadditionofperturbativeterms,regardlessofwhetherthesetermspreservethevariationalstructure.In-deed,weobservethatthesendingsareentirelyconsistentwiththenumericalresultsofHoughtonandKnobloch,includingthebreakingupoftheoddparitybranches,broadeningofthesnakingregion,andappearanceofSandZasymmetricbranches.4.2SystemssuchthatzhasatleasttwomaximaperperiodWenowturntothesomewhatmorecomplicatedsituationwherez(L)possessestwomaximaandminimaperperiod;ofcoursetheperiodicityimpliesthatmaximaandminimamustoccurinpairs.14 (a)Bifurcationdiagramforplanarstripesandspots,withnumbersindicatingthelocationsoftwoR-symmetricsolutions,andtheintermediatecross-connectingasymmetricsolutionshownatright;see[2]. (b)Bifurcationdiagramforplanarstripesandspots,withnumbersindicatingthelocationsofthetwoR-symmetricandtheinter-mediateself-connectingasymmetricsolutionshownatright;see[2].Figure15:Bifurcationdiagramforplanarstripesandspotsin(5.1),alongwithlocationsofsolutionprolesshownatright.R-symmetricbranchshownindashedblue,R-symmetricbranchindottedorange,andrepresentativeasymmetricbranchesinsolidgray.Thecolorbaristhesameforallsolutionproles,andrecallingthatsolutionsareperiodicinthex-direction,weshow6periodsforeachsolution.18 5.2ComputationofsplittingdistanceandcomparisonwithcontinuationresultsIntheirnumericalstudyofsymmetry-breakingintheone-dimensionalcubic-quinticSwift{Hohenbergmodel,HoughtonandKnoblochnotedthatthesplittingdistanceisunequalontheleftandrightsideofthesnakingdiagram;thatis,thesymmetrybreakingtermcausesthesetofsaddlenodestoshiftmoretoboththeinsideandoutsideononesidethanontheotherside.Tounderstandthisobservation,andtoexplainasimilarphenomenonwhichoccursinthetwo-dimensionalcase{seeFigures16and19,inwhichthedisplacementoftheoutersaddlenodesontheleftismuchlessthanontheright{wecanlookatthederivativeoftheperturbedSwift{Hohenbergequation(2.1)or(5.2)withrespectto".Startingwiththeone-dimensionalcase,wedeneF(u;;"):=�(1+@2x)2u�u+bu3�u5+"g(u)whereg(u)isourperturbativeterm,e.g.,u2oru2x.Wecanthenparameterizeasolutionbranchfortheunperturbedsystemas(u(s);(s)),wheresis,forinstance,arclengthalongthebranch,sothatF(u(s);(s);0)=0foralls.Wedenotethetangentvectortothissolutionbranchby(v;):=d ds(u;)(s):For"nonzero,thepersistingR-symmetricbranchwillbegivenbyF(u(s;");(s;");")=0;anddierentiatingthiswithrespectto"weobtainFuu"+F"+F"=0:DeningL=�(1+@2x)2�+3bu2�5u4foraparticularsolution(u;),thisyieldsthesystem(Lu0�u0+g(u)=0hu0;vi+0=0(5.4)whosesolutionis(u0;0)=d d"(u;):Thustheosetalongthesolutionbranchwillbegivenby0"+O("2).Wecannd0anywherealongthesolutionbranchbysolvingthelinearsystem(5.4).Alternatively,wenotethat,atasaddlenode,wehave=0sothatLv=0;sinceLisself-adjointinL2,applyinghv;
itotherstequationin(5.4)yieldshv;0ui+hv;g(u)i=0or0=hv;g(u)i hv;ui:Thusweneedonlytocalculatethesolution(u;)anditsassociatedeigenfunctionvtocomputetheosetatasaddlenode.Whilethemethodofdirectlysolvingthelinearsystemissomewhatmorerobustnumerically,thelattermethodprovideshelpfulinsight,particularlyintheone-dimensionalcase.Weemphasizethat,whichevermethodweuse,thiscalculationallowsustodeterminethesignof~ZasdenedinSection4,whichinturndetermineswhichclassofbifurcationdiagramtheperturbedsystemwillexhibit;thatis,wecandescribethefullbifurcationdiagramwithouttheneedforanycomputationsontheperturbedsystem.InFigure20weshowfoursuccessivesaddlenodesfortheone-dimensionalcubic-quinticSwift{Hohenbergequa-tionpriortoperturbation.Weseethatthesolutionu(x)atsuccessiveleft-handsaddlenodesisrelatedby21 leadingorderinanydimension,andhaveshownthatthismethodagreeswellwiththeresultsofnumericalcontinuationintheplanarcase.Thismethodcanbeemployedfurthertodeterminewhichperturbationsleadtowhichbifurcationscenarios,usingmeasurementsfromonlytheunperturbedbifurcationstructure.Finally,weobservethatthissortofanalysiscouldbeusedtointerpretorpredictresultsofvaryingforvariouspatterntypes,someofwhichwerereportedin[2].Severalareasremainforfutureexploration.Althoughitseemsclearatthisstage,ithasnotyetbeenshownanalyticallythatasymmetricsolutionsareconstructedbygluingtogethersymmetricsolutions,evenintheone-dimensionalcase.Provingthisrigorouslywillaidinaddressingthestabilityofplanarpatterns;whilesomecomputationshavebeendonenumerically,stabilityintheplanarcasehasyettobestudiedanalytically.Beyondthis,localizedhexagonpatches(see,forexample,[18])andotherfullylocalizedstructuresintwoorhigherdimensionsremainchallengingphenomenawhereeventhebifurcationstructuresthemselvesremainpoorlyunderstood.Furthermore,ashighlightedrecentlyin[24],therearestrongconnectionsbetweenthedescriptionoflocalizedstructuresviaSwift{Hohenberg-typemodelsandthetransitionfroma uidtocrystallinestate;understandingtheserelationshipspromisestobeafruitfulareaforongoingwork.AcknowledgementsMakrideswassupportedbytheNSFundertheIGERTgrant\ReverseEcology:Com-putationalIntegrationofGenomes,Organisms,andEnvironments"DGE-0966060.SandstedewaspartiallysupportedbytheNSFundergrantDMS-0907904.References[1]Y.AstrovandY.Logvin.Formationofclustersoflocalizedstatesinagasdischargesystemviaaself-completionscenario.Phys.Rev.Lett.79(1997)2983{2986.[2]D.Avitabile,D.J.B.Lloyd,J.Burke,E.KnoblochandB.Sandstede.TosnakeornottosnakeintheplanarSwift-Hohenbergequation.SIAMJ.Appl.Dyn.Syst.9(2010)704{733.[3]M.Beck,J.Knobloch,D.J.B.Lloyd,B.SandstedeandT.Wagenknecht.Snakes,ladders,andisolasoflocalizedpatterns.SIAMJ.Math.Anal.41(2009)936{972.[4]S.Blanch ower.Magnetohydrodynamicconvectons.Phys.A261(1999)74{81.[5]U.Bortolozzo,M.G.ClercandS.Residori.Solitarylocalizedstructuresinaliquidcrystallight-valveexperiment.NewJ.Phys.11(2009)093037.[6]J.BurkeandE.Knobloch.LocalizedstatesinthegeneralizedSwift-Hohenbergequation.Phys.Rev.E73(2006)056211.[7]J.BurkeandE.Knobloch.Homoclinicsnaking:Structureandstability.Chaos17(2007)037102.[8]J.BurkeandE.Knobloch.Snakesandladders:localizedstatesintheSwift-Hohenbergequation.Phys.Lett.A360(2007)681{688.[9]S.J.ChapmanandG.Kozyre.Exponentialasymptoticsoflocalisedpatternsandsnakingbifurcationdiagrams.Phys.D238(2009)319{354.[10]P.Coullet,C.RieraandC.Tresser.Stablestaticlocalizedstructuresinonedimension.Phys.Rev.Lett.84(2000)3069{3072.[11]M.CrossandP.Hohenberg.Patternformationoutsideofequilibrium.Rev.Mod.Phys.65(1993)851{1112.24