Bifurcations and a chaos strip in states of long J

arXiv:cond-mat/0003375v1[cond-mat.supr-con]23Mar2000BifurcationsandachaosstripinstatesoflongJosephsonjunctionsK.N.Yugay,N.V.Blinov,andI.V.ShirokovOmskStateUniversity,55aMiraAve.,Omsk644077,Russia(February1,2008)AbstractStationaryandnonstationary,inparticular,chaoticstatesinlongJosephsonjunctionsareinvestigated.Bifurcationlinesontheparametricbiascurrent–externalmagneticfieldplanearecalculated.Thechaosstripalongthebi-furcationlineisobserved.Itisshownthattransitionsbetweenstationarystatesarethetransitionsfrommetastabletostablestatesandthatthether-modynamicalGibbspotentialofthesestablestatesmaybelargerthanforsomemetastablestates.Thedefinitionofadynamicalcriticalmagneticfieldcharacterizingthestabilityofthestationarystatesisgiven.74.50+r,05.45.+bTypesetusingREVTEX1INTRODUCTIONDynamicalchaosinlongJosephsonjunctionsisofgreatinterestbecauseitcanbeasourceofdynamicalnoiseindevicesbasedonthem,inparticular,inSQUIDs,limitingthesensitivityofthesedevices.Furthermore,dynamicalchaosinlongJosephsonjunctions(LJJ)isaveryinterestingphysicalphenomenontakingplaceinnonlinearsystemsintheabsentofanexternalstochasticforce.1–9DynamicalchaosinaLJJiseasilyexcitedandthereforeitmayalsobeinvestigatedexperimentallyrathereasily.10,11Inourpreviousworks12,13wehaveshownthatamongasetofsolutionsoftheFerrell-PrangeequationdescribingstationarystatesoftheLJJinanexternalmagneticfield14arebothstableandunstableones.Atthesametime,thesestationarystatesareasymptoticsolutionsofthenonstationarysine-Gordonequationandwehavealsoshownthataselectionofthestablesolutionscanbegovernedbyarapiddampingintimeoftheinitialperturba-tionenteringintothenonstationarysine-Gordonequationthroughtheboundaryconditions.Changingtheintensityofthisperturbationatfixedshape,wecanobtainvariousstation-arystatesfortheLJJwithoutabiascurrentorthreeclustersofstates(stationary,andtimedependentregularandchaotic)inthepresenceofabiascurrent.Itturnedoutthatasymptoticstatesareverysensitivetoanexternalperturbation,itsvalueandshapedefinethestate(stationary,regularorchaotic)towhichthesystemwilltendatt→∞(wehavecalledthisinfluenceontheselectionofasymptoticstatesofthesmallrapidlydampinginitialperturbationintimeaneffectofmemory).Thefactofcoexistenceofallthesethreecharac-teristicasymptoticstatesselectedonlybytheformoftheinitialperturbationseemstobeastonishing.ItisevidentlyenoughthattheFerrell-Prangeequationwillnothavesolutionsatalargebiascurrentβ.Thereforethequestionarises:atwhichvaluesofβdostationarystatesofaLJJdisappearorwhatwillbeaboundaryintheparametricβ−H0plane(H0isanexternalmagneticfield)thatseparatesthisplaneontheregionswherestationarystatesdoanddonotexist?SincethenumberofsolutionsoftheFerrell-Prangeequationchangesatvariationoftheparameters(H0,β),anotherquestionarises:whatistheformofbifurcation2linesintheplaneβ−H0thatseparatetheparametricplaneontheregionswithadifferentnumberofstationarysolutionsoftheFerrell-Prangeequation?TheexistenceofseveralstablesolutionsoftheFerrell-PrangeequationisequivalenttothefactthatthermodynamicalGibbspotentialGassociatedwiththedistributionofthemagneticfieldalongthejunctionhasminima,andeachminimumcorrespondstoacertainsolutionoftheFerrell-Prangeequation.DoesaglobalminimumofGcorrespondtothemoststablestate(e.g.,intheLyapunovsense)?InthecaseofthejunctionofthefinitelengthbothMeissnerandone-fluxonstatesarethermodynamicallyadvantageoussimultaneously,soitisinterestingtoinvestigatedynamicalpropertiesofthesestates.Answeringthisquestion,weintroduceadynamiclalcriticalfieldthatdescribesthestabilitycharacteristicofthejunctions.InSec.1bifurcationlinesontheparametricβ−H0planearecalculated.InSec.2thedefinitionofthedynamicalcriticalmagneticfieldisgivenandthedependenceofthisfieldonβandthelengthofthejunctionLiscalculated.InSec.3transitionsbetweenstatesaredescribed.ItisshowninSec.4thatachaosstriparisesalongthebifurcationlineontheparametricβ−H0plane.ThelastSec.5containsthediscussionofourcalculationandbriefconclusions.I.BIFURCATIONLINESStationarystatesofaLJJareinvestigatedusingthenumericalintegrationoftheFerrell-Prangeequation:ϕxx(x)=sinϕ(x)−β,(1)whereϕ(x)isthestationaryJosephsonphasevariable,βisthedcbiascurrentdensitynormalizedtothecriticalcurrentjc,xisthedistancealongthejunctionnormalizedtotheJosephsonpenetrationlengthλJ=qCΦ0/8π2jcd,Φ0isthefluxquantum,d=2λL+b,λListheLondonpenetrationlength,bisthethicknessofthedielectricbarrier.TheboundaryconditionsforEq.(1)havetheform3ϕx(x)|x=0=ϕx(x)|x=L=H0,(2)whereListhetotallengthofthejunctionnormalizedtoλJandH0istheexternalmagneticfieldperpendiculartothejunctionandnormalizedto˜H=Φ0/2πλJd.NumericalintegrationofEqs.(1)–(2)allowsustofindtheregionswithacertainnumberofsolutionsontheparametricβ−H0plane(Fig.1).Itiseasytoshowthatthesetofpointscorrespondingtotheevennumberofsolutionsformstwo-dimensionaldomainsonthisplane,whereasthesetcorrespondingtotheoddonesmayformjustone-dimensionalcurves.Mostly,thelinescorrespondingtotheoddnumberofthesolutionsoftheFerrell-Prangeboundaryproblemcoincidewiththebifurcationlines.Usingtheshootingmethodforsolvingoftheboundaryproblemonecanprovethatthe2π–peri