加工厂.docx
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加工厂.docx
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加工厂
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昨天去买烟,买了包20的给了老板50找了我40,我装进兜里!
没走多远老板喊我:
你的烟没拿,我留下了感动的泪水,拿出10块钱给老板,你多找了我10块钱!
老板也留下了感动的泪水,小伙子把烟拿来,我给你换一包。
抽着老板新换给我的烟,那纯正的味道在一次感动了我,老板,把刚才那张50的拿来我给你换一张吧。
老板接过那50也在一次感动:
小伙子,把刚才那找你的钱给我!
我也给你换了,接过老板从新找我的钱,我也再次感动,从口袋拿出一部手机:
老板手机还你吧。
老板热泪盈眶,颤抖着掏出一个钱包:
小伙子,钱包还你。
此时我再也把持不住了,扑在地上抱着老板的大腿器道:
老哥,你去隔壁宾馆304房把你那上小学的女儿领回家吧!
老板听后一惊,吧了口气,回头朝小店大喊道:
红杏,别躲了,出来跟你老公回Theorem1.1[15].Let
and
bedefinedasabove.Then
restrictedto
issemiconjugateto
.Furthermore,if
uniformlyin
as
then
restrictedto
isconjugateto
.
的家吧!
Asimpleexampleisthatofapointparticlemovinginatwo-wellpotentialwithfriction,asinFigure1(a).Duetothefriction,allinitialconditions,exceptthoseat\(x=dx/dt=0\)oronitsstablemanifoldeventuallycometorestateither\(x=x_0\)or\(x=-x_0\,\)whicharethetwoattractorsofthesystem.Apointinitiallyplacedontheunstableequilibriumpoint,\(x=0,\)willstaythereforever;andthisstatehasaone-dimensionalstablemanifold.Figure1(b)showsthebasinsofattractionofthetwostableequilibriumpoint,\(x=\pmx_0,\)wherethecrosshatchedregionisthebasinfortheattractorat\(x=x_0\)andtheblankregionisthebasinfortheattractorat\(x=-x_0\.\)Theboundaryseparatingthesetwobasinsisthestablemanifoldoftheunstableequilibrium\(x=0.\)
Figure2:
Acasewherethebasinboundaryisafractalcurve.
Fractalbasinboundaries
Intheaboveexample,thebasinboundarywasasmoothcurve.However,otherpossibilitiesexist.Anexampleofthisoccursforthemap\[x_{n+1}=(3x_n)\mod1\,\]\[y_{n+1}=1.5+\cos2\pix_n\.\]Foralmostanyinitialcondition(exceptforthosepreciselyontheboundarybetweenthebasinsofattraction),\(\lim_{n\rightarrow\infty}y_n\)iseither\(y=+\infty\)or\(y=-\infty\,\)whichwemayregardasthetwoattractorsofthesystem.Figure2showsthebasinstructureforthismap,withthebasinforthe\(y=-\infty\)attractorblackandthebasinofthe\(y=+\infty\)attractorblank.Incontrasttothepreviousexample,thebasinboundaryisnolongerasmoothcurve.Infact,itisafractalcurvewithabox-countingdimension1.62....Weemphasizethat,althoughfractal,thisbasinboundaryisstillasimplecurve(itcanbewrittenasacontinuousparametricfunctionalrelationship\(x=x(s),y=y(s)\)for\(1>s>0\)suchthat\((x(s_1),y(s_1))\neq(x(s_2),y(s_2))\)if\(s_1\neqs_2\.\))
Figure3:
Basinsofattractionforaforceddampedpendulum(picturemadebyH.E.Nusse).
Anotherexampleofasystemwithafractalbasinboundaryistheforceddampedpendulumequation,\[d^2\theta/dt^2+0.1d\theta/dt+\sin\theta=2.1\cost\.\]Fortheseparameters,therearetwoattractorswhicharebothperiodicorbits(Grebogi,OttandYorke,1987).Figure3showsthebasinsofattractionofthesetwoattractorswithinitial\(\theta\)valuesplottedhorizontallyandinitialvaluesof\(d\theta/dt\)plottedvertically.Thefigurewasmadebyinitializingmanyinitialconditionsonafinerectangulargrid.Eachinitialconditionwasthenintegratedforwardtoseewhichattractoritsorbitapproached.Iftheorbitapproachedaparticularoneofthetwoattractors,ablackdotwasplottedonthegrid.Ifitapproachedtheotherattractor,nodotwasplotted.Thedotsaredenseenoughthattheyfillinasolidblackregionexceptnearthebasinboundary.Thespeckledappearanceofmuchofthisfigureisaconsequenceoftheintricate,finescaledstructureofthebasinboundary.Inthiscasethebasinboundaryisagainafractalset(itsbox-countingdimensionisabout1.8),butitstopologyismorecomplicatedthanthatofthebasinboundaryofFigure2inthattheFigure3basinboundaryisnotasimplecurve.Inbothoftheaboveexamplesinwhichfractalbasinboundariesoccur,thefractalityisaresultofchaoticmotion(seetransientchaos)oforbitsontheboundary,andthisisgenerallythecaseforfractalbasinboundaries(McDonaldetal.,1985).
BasinBoundaryMetamorphoses
Wehaveseensofarthattherecanbebasinboundariesofqualitativelydifferenttypes.Asinthecaseofattractors,bifurcationscanoccurinwhichbasinboundariesundergoqualitativechangesasasystemparameterpassesthroughacriticalbifurcationvalue.Forexample,forasystemparameter\(p
TheUncertaintyExponent
Fractalbasinboundaries,likethoseillustratedabove,areextremelycommonandhavepotentiallyimportantpracticalconsequences.Inparticular,theymaymakeitmoredifficulttoidentifytheattractorcorrespondingtoagiveninitialcondition,ifthatinitialconditionhassomeuncertainty.ThisaspectisalreadyimpliedbythespeckledappearanceofFigure3.Aquantitativemeasureofthisisprovidedbytheuncertaintyexponent(McDonaldetal.,1985).Fordefiniteness,supposewerandomlychooseaninitialconditionwithuniformprobabilitydensityintheareaofinitialconditionspacecorrespondingtotheplotinFigure3.Then,withprobabilityone,thatinitialconditionwilllieinoneofthebasinsofthetwoattractors[thebasinboundaryhaszeroLebesguemeasure(i.e.,'zeroarea')andsothereiszeroprobabilitythatarandominitialconditionisontheboundary].Nowassumethatwearealsotoldthattheinitialconditionhassomegivenuncertainty,\(\epsilon\,\)and,forthesakeofillustration,assumethatthisuncertaintycanberepresentedbysayingthattherealinitialconditionlieswithinacircleofradius\(\epsilon\)centeredatthecoordinates\((x_0,y_0)\)thatwererandomlychosen.Weaskwhatistheprobabilitythatthe\((x_0,y_0)\)couldlieinabasinthatisdifferentfromthatofthetrueinitialcondition,i.e.,whatistheprobability,\(\rho(\epsilon)\,\)thattheuncertainty\(\epsilon\)couldcauseustomakeamistakeinadeterminationoftheattractorthattheorbitgoesto.Geometrically,thisisthesameasaskingwhatfractionoftheareaofFigure3iswithinadistance\(\epsilon\)ofthebasinboundary.Thisfractionscalesas\[\rho(\epsilon)\sim\epsilon^\alpha\,\]where\(\alpha\)istheuncertaintyexponent(McDonaldetal.,1985)andisgivenby\(\alpha=D-D_0\)where\(D\)isthedimensionoftheinitialconditionspace(\(D=2\)forFigure3)and\(D_0\)isthebox-countingdimensionofthebasinboundary.FortheexampleofFigure3,since\(D_0\cong1.8\,\)wehave\(\alpha\cong0.2\.\)Forsmall\(\alpha\)itbecomesverydifficulttoimprovepredictivecapacity(i.e.,topredicttheattractorfromtheinitialcondition)byreducingtheuncertainty.Forexample,if\(\alpha=0.2\,\)toreduce\(\rho(\epsilon)\)byafactorof10,theuncertainty\(\epsilon\)wouldhavetobereducedbyafactorof\(10^5\.\)Thus,fractalbasinboundaries(analogoustothebutterflyeffectofchaoticattractors)poseabarriertoprediction,andthisbarrierisrelatedtothepresenceofchaos.
RiddledBasinsofAttraction
Wenowdiscussatypeofbasintopologythatmayoccurincertainspecialsystems;namely,systemsthat,throughasymmetryorsomeotherconstraint,haveasmoothinvariantmanifold.Thatis,thereexistsasmoothsurfaceorhypersurfaceinthephasespace,suchthatanyinitialconditioninthesurfacegeneratesanorbitthatremainsinthesurface.Thesesystemscanhaveaparticularlybizarretypeofbasinstructurecalledariddledbasinofattraction(Alexanderetal.,1992;Ottetal.,1994).Inordertodiscusswhatthismeans,wefirsthavetoclearlystatewhatwemeanbyan"attractor".Forthepurposesofthisdiscussion,weusethedefinitionofMilnor(1985):
asetinstatespaceisanattractorifitisthelimitsetoforbitsoriginatingfromasetofinitialconditionsofpositiveLebesguemeasure.Thatis,ifwerandomlychooseaninitialconditionwithuniformprobabilitydensityinasuitablesphereofinitialconditionspace,thereisanon-zeroprobabilitythattheorbitfromthechoseninitialconditiongoestotheattractor.Thisdefinitiondiffersfromanothercommondefinitionofanattractorwhichrequiresthatthereexistssomeneighborhoodofanattractorsuchthatallinitialconditionsinthisneighborhoodgenerateorbitsthatlimitontheattractor.Asweshallsee,an"attractor"withariddledbasinconformswiththefirstdefinition,butnottheseconddefinition.Thefailuretosatisfytheseconddefinitionisbecausetherearepointsarbitrarilyclosetoanattractorwithariddledbasin,suchthatthesepointsgenerateorbitsthatgotoanotherattractor(hencetheneighborhoodmentionedabovedoesnotexist.)
Figure4:
Schematicillustrationofasituationwithariddledbasinofattraction.
Wearenowreadytosaywhatwemeanbyariddledbasin.Supposeoursystemhastwoattractorswhichwedenote\(A\)and\(C\)withbasins\(\hatA\)and\(\hatC\.\)Wesaythatthebasin\(\hatA\)isriddled,if,foreverypoint\(p\)in\(\hatA\,\)an\(\epsilon\)-radiusball,\(B_\epsilon(p)\)centeredat\(p\)containsapositiveLebesguemeasureofpointsin\(\hatC\)forany\(\epsilon>0\.\)Thiscircumstancehasthefollowingsurprisingimplication.Sayweinitializeastateat\(p\)andfindthattheresultingorbitgoesto\(\hatA\.\)Nowsaythatweattempttorepeatthisexperiment.Ifthereisanyerrorinourresettingoftheinitialcondition,wecannotbesurethattheorbitwillgoto\(A\)(ratherthan\(C\)),andthisisthe
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