系統(tǒng)動力學3PPT課件

系統(tǒng)動力學(SystemDynamics),主講：張學民,(3),一階系統(tǒng),一階系統(tǒng),所謂系統(tǒng)的階數(shù)就是系統(tǒng)狀態(tài)變量的維數(shù)，在系統(tǒng)動力學中也就是水平(變量)的個數(shù)。在社會經(jīng)濟系統(tǒng)中高階(可能是數(shù)十至數(shù)百階)系統(tǒng)到處可見。任一高階系統(tǒng)都可以視為由一階系統(tǒng)關(guān)聯(lián)而成。一個復雜系統(tǒng)的行為往往是由某些主回路和某些主要的變量決定的，換言之，復雜系統(tǒng)中往往存在一些起主導作用的主回路和主要變量。一階反饋系統(tǒng)就是只含一個水平(變量)的反饋系統(tǒng)。正反饋系統(tǒng)行為的共同特征就是指數(shù)式地趨于無窮。一階正反饋系統(tǒng)的行為是單調(diào)地指數(shù)式地趨于無窮。負反饋系統(tǒng)行為的共同特征就是尋找目標。一階負反饋系統(tǒng)的行為是單調(diào)地趨于目標。,一階反饋系統(tǒng)的復雜性,一階反饋系統(tǒng)也會呈現(xiàn)復雜的系統(tǒng)行為！考察某商品的銷售問題。假定：銷售量正比于還沒有購買該商品的人數(shù)每人只需一件這樣的商品總?cè)藬?shù)是恒定的商品更新的周期遠大于商品普及所需的時間,一階反饋系統(tǒng)的復雜性,(1)FRACTION=0.06Units:dmnl/Month(2)TOTALBUYERS=100Units:person(3)buyrate=FRACTION*potentialbuyersUnits:person/Month(4)Buyers=INTEG(buyrate,1)Units:person(5)potentialbuyers=TOTALBUYERS-BuyersUnits:person,FINALTIME:60TIMESTEP:0.5Units:Month,一階負反饋系統(tǒng)的行為是單調(diào)地趨于目標,一階反饋系統(tǒng)的復雜性,FINALTIME:60TIMESTEP:0.5Units:Month,FRACTION=0.06dmnl/MonthTOTALBUYERS=100person,一階負反饋系統(tǒng)的行為是單調(diào)地趨于目標,一階反饋系統(tǒng)的復雜性,FINALTIME:60TIMESTEP:0.5Units:Month,FRACTION=0.1dmnl/MonthTOTALBUYERS=100person,一階負反饋系統(tǒng)的行為是單調(diào)地趨于目標,一階反饋系統(tǒng)的復雜性,FINALTIME:60TIMESTEP:0.5Units:Month,TOTALBUYERS=100person,FRACTION=0.1,FRACTION=0.06,FRACTION=0.01,一階負反饋系統(tǒng)的行為是單調(diào)地趨于目標,一階反饋系統(tǒng)的復雜性,FRACTION=0.1,FRACTION=0.06,FRACTION=0.01,一階反饋系統(tǒng)的復雜性新假設(shè),對于一種新商品人們并不了解，一開始就購買的人不會太多大部分人只是在聽到已經(jīng)購買的人的介紹或者看到別人在搶購后才逐漸對該商品有所了解，購買者也才會多起來已經(jīng)購買該商品的戶數(shù)越多，越會促成更多的銷售更符合實際的新假定是：銷售量正比于還沒有購買該商品的戶數(shù)和已經(jīng)購買該商品的戶數(shù)的乘積,一階反饋系統(tǒng)的復雜性新假設(shè),(1)FRACTION=0.002Units:dmnl/(Month*person)(2)TOTALBUYERS=100Units:person(3)buyrate=FRACTION*Buyers*possiblebuyersUnits:person/Month(4)Buyers=INTEG(buyrate,1)Units:person(5)possiblebuyers=TOTALBUYERS-BuyersUnits:person,FINALTIME:60TIMESTEP:0.5Units:Month,銷售量正比于還沒有購買該商品的戶數(shù)和已經(jīng)購買該商品的戶數(shù)的乘積,一階反饋系統(tǒng)的復雜性新假設(shè),FINALTIME:60TIMESTEP:0.5Units:Month,FRACTION=0.002dmnl/(Month*person),TOTALBUYERS=100person,一階反饋系統(tǒng)的復雜性新假設(shè),FINALTIME:60TIMESTEP:0.5Units:Month,FRACTION=0.001FRACTION=0.002FRACTION=0.004,TOTALBUYERS=100person,一階反饋系統(tǒng)的復雜性新假設(shè),左半段曲線的斜率為正，表明兩個反饋環(huán)中正反饋環(huán)起主導作用左半段曲線的斜率隨著水平變量Buyers的增加而遞減至零，意味著正反饋環(huán)的力量逐漸削弱，水平變量Buyers的行為呈亞指數(shù)增長的特性，購買率buyrate則隨著Buyers的增長而增至其最大值右半段曲線的斜率為負，且其絕對值隨著水平變量Buyers的增加由零逐漸遞增的，表明負反饋環(huán)不僅起了主導作用，而且其力量在不斷加強，水平變量Buyers的行為呈超漸近增長的特性，購買率buyrate則隨著Buyers的增長由最大值逐漸衰減至零,一階反饋系統(tǒng)的復雜性S型增長,S型增長是一種常見而重要的系統(tǒng)行為，它是系統(tǒng)中正負反饋環(huán)交互作用的結(jié)果。在S型增長的過程中先是正反饋環(huán)占主導地位，因而系統(tǒng)行為呈指數(shù)增長的特性。按指數(shù)規(guī)律無限增長的正反饋過程是不會無止境地持續(xù)下去的：系統(tǒng)崩潰水平變量達到某一較高水平時，正反饋環(huán)的主導地位由負反饋環(huán)取而代之,一階反饋系統(tǒng)的復雜性新假設(shè),一階反饋系統(tǒng)的復雜性新假設(shè),FRACTION=0.001,FRACTION=0.002,FRACTION=0.004,buyrate,S型增長系統(tǒng)的特點,兩個特點：純速率狀態(tài)關(guān)系曲線可分為兩段，前一段的斜率為正，稱正反饋軌線，后一段的斜率為負，稱負反饋軌線。當系統(tǒng)運行在正反饋軌線上時，主導反饋環(huán)為正反饋環(huán)，而當系統(tǒng)運行在負反饋軌線上時，主導反饋環(huán)為負反饋環(huán)。主導反饋環(huán)的轉(zhuǎn)移發(fā)生在前后兩段軌線的交接處。純速率狀態(tài)關(guān)系與橫坐標有兩個交點，左邊一點是系統(tǒng)的不穩(wěn)定平衡點，右邊一點是系統(tǒng)的穩(wěn)定平衡點，即S型增長趨近的目標。,一階線性反饋系統(tǒng)行為模式,非線性與主導反饋環(huán)的轉(zhuǎn)移,復雜系統(tǒng)內(nèi)部存在相互作用的或正或負的多重反饋環(huán)。所謂主導反饋環(huán)就是在多重反饋環(huán)中起主導作用的反饋環(huán)。當系統(tǒng)行為表現(xiàn)出指數(shù)增長(或指數(shù)崩潰)特性時，可以推斷系統(tǒng)中必定存在正反饋環(huán)，并且正起著主導作用。當系統(tǒng)行為表現(xiàn)出尋找目標特性時，則可以推斷系統(tǒng)中必定存在負反饋環(huán)，并且正起著主導作用。系統(tǒng)行為是由多重反饋環(huán)相互作用共同產(chǎn)生的，其行為模式主要由主導反饋環(huán)決定。主導反饋環(huán)并非是固定不變的，它(們)往往隨著系統(tǒng)狀態(tài)的變化而在各反饋環(huán)中轉(zhuǎn)移，由此產(chǎn)生了多種多樣的復雜的系統(tǒng)行為。實際系統(tǒng)幾乎都具有非線性的特征。非線性關(guān)系是導致主導反饋環(huán)極性轉(zhuǎn)移的根本原因。不僅要研究正反饋環(huán)或負反饋環(huán)的作用，而且要研究主導反饋環(huán)轉(zhuǎn)移的作用。S型增長是主導反饋環(huán)由正反饋環(huán)向負反饋環(huán)轉(zhuǎn)移的實例。,二階系統(tǒng),二階系統(tǒng),所謂二階系統(tǒng)就是包含有兩個水平(變量)的系統(tǒng)。二階系統(tǒng)也可分為二階負反饋系統(tǒng)和二階正反饋系統(tǒng)。二階線性負反饋系統(tǒng)的行為特征也是尋找目標，但逼近終值的方式更為復雜。一般可分為兩種，一種為欠阻尼形式，另一種為過阻尼形式。二階線性正反饋系統(tǒng)的行為特征仍為趨于無窮，但呈兩種不同的模式：一種模式與一階正反饋的場合相似仍是單調(diào)地趨于無窮，另一種模式是振蕩。如果二階系統(tǒng)是非線性的，那么系統(tǒng)行為將更為復雜。,二階系統(tǒng)典型行為,增幅振蕩,一、二階系統(tǒng)行為模式是高階系統(tǒng)行為模式的基礎(chǔ),指數(shù)增長線性增長漸近增長S型增長指數(shù)崩潰線性衰減漸近衰減S型衰減增幅振蕩等幅振蕩減幅振蕩,高階系統(tǒng)的行為高階系統(tǒng)具有互相連接的多重反饋環(huán)，因而其行為比二階系統(tǒng)更為復雜。在結(jié)構(gòu)上，高階系統(tǒng)可以看作是由若干個典型的一階結(jié)構(gòu)或二階結(jié)構(gòu)相互關(guān)聯(lián)而成。由于有了關(guān)聯(lián)以及系統(tǒng)中可能存在的非線性，高階系統(tǒng)的行為已經(jīng)不僅僅是指數(shù)增長、線性增長、漸近增長、S型增長、指數(shù)崩潰、線性衰減、漸近衰減、S型衰減、增幅振蕩、等幅振蕩及減幅振蕩等幾種形式，而是它們的復雜組合。,Second-OrderSystemsonHowTwoLoversInteract,RomeoandJulietbyWilliamShakespeare,GonewiththeWindbyMargaretMitchell,TheRedandtheBlackbyStendhal,Second-OrderSystemsonHowTwoLoversInteract,Differentpeoplehavedifferentpersonalitiesthatdictate(指示)howtheywillrespondtotheirpartnerslove.Somepeoplelovetheirpartnersmorethemoretheirpartnerslovethem.Theygetdiscouragedandgiveupiftheirpartnersdonotlovethem.Otherpeople,ontheotherhand,getannoyediftheirloverslovethemtoomuch.Theyareattractedtopeoplewhoarenotattractedtothem.Thispaperwillmatchtogetherloverswithdifferentpersonalitiesandobservehowtheirrelationshipsunfold.Eachpairwillberepresentedbyadifferentsetofparametersinthemodel.Bylookingatthedynamicbehaviorgeneratedbymatchingdifferentpersonalities,thatis,bysimulatingthemodelwithdifferentcombinationsofparameters,thispaperwillstudythebehaviorofsimplesecond-ordersystems.,RomeoandJuliet,RomeoandJulietaremadlyinlovewitheachother.Witheachsecretmeeting,RomeosloveforJulietgrows.Becausehelovesher,hedoeseverythinghecantoimpressher.Julietisflatteredbyhisattentionand,inreturn,herloveforRomeoalsogrows.BecauseRomeosensesthatJulietloveshim,heallowshispassiontosoar(驟升,升騰).,RomeoandJulietbyWilliamShakespeare,TheModelforRomeoandJuliet,Inthemodeleachstockrepresentstheaccumulationofoneloversloveforanother.TherateatwhichRomeoslovechangesdependsontheextentofJulietsloveandRomeospersonality.Hispersonalityisrepresentedbyaparameterthatmeasureshowhereactstoherlove.Likewise,thechangeinJulietsloveequalstheamountofRomeoslovetimestheextenttowhichshereactstohislove.,MeasurementforLove,ThelovethatRomeoandJulietfeelforeachothercanbequantifiedinseveralways.Lovecanbemeasuredintermsofkissesperday,ortenderthoughtsperhour,forexample.Theunitsofthestocksofloveinthemodelsarebaseduponascaleof“l(fā)oveunits.”Ifthetwoprotagonistsareindifferent(不感興趣)toeachother,theyfeelzeroloveunits.Positiveloveunitsrepresenthowmuchtheylikeeachother,whilenegativeloveunitsrepresenthowmuchtheydislikeeachother.Rightnow,assumethatinitiallyRomeoandJulietfeelkindlydisposedtowardseachother.Theyeachfeeloneunitoflovefortheother.(Scenario1),ParameterforPersonality,Theparameter“JULIETSREACTION”representshowJulietreactstoRomeoslove,thatis,towhatextentRomeoslovechangesJulietsloveovertime.AssumethateveryunitofRomeosloveincreasesJulietslovebyoneuniteachday.ThereforeJulietsreactiontoRomeosloveisoneloveunitperloveunitperday,oroneperday.JULIETSREACTION=1Units:1/dayRomeohasthesamepersonalityasJuliet,soRomeosreactiontoJulietsloveisalsooneperday.ROMEOSREACTION=1Units:1/day,Equations,(01)changeinJulietslove=RomeosLoveforJuliet*JULIETSREACTIONUnits:loveunit/day(02)changeinRomeoslove=JulietsLoveforRomeo*ROMEOSREACTIONUnits:loveunit/day(03)JulietsLoveforRomeo=INTEG(changeinJulietslove,1)Units:loveunit(04)JULIETSREACTION=1Units:1/day(05)RomeosLoveforJuliet=INTEG(changeinRomeoslove,1)Units:loveunit(06)ROMEOSREACTION=1Units:1/dayFINALTIME=12dayINITIALTIME=0dayTIMESTEP=1day,Theirlovegrowtogetherexponentially!,RomeosloveandJulietslovegrowtogetherexponentially.Bytheendof12days,theyeachfeel4,096loveunits!Inotherwords,theylikeeachother4,096timesmorethantheydidinitially.,Positivefeedbackloop,Apositivefeedbackloopjoinsthetwostocks.AnincreaseinJulietsloveforRomeoincreasesthechangeinRomeoslove,whichincreasesRomeosloveforJuliet.TheincreaseinRomeosloveleadstoanincreaseinthechangeinJulietslove,furtherincreasingJulietsloveforRomeo.Withtime,thestocksreinforceeachotherandincreaseexponentially.,InitiallyRomeolovedJuliettwice,WhatwouldhappenifinitiallyRomeolovedJuliettwiceasmuchasshelovedhim?(Scenario2)JulietsLoveforRomeo=INTEG(changeinJulietslove,1)RomeosLoveforJuliet=INTEG(changeinRomeoslove,2)WillJulietsloveevergrowtoequalRomeoslove?,Scenario2vs.Scenario1,JulietslovequicklycatchesupwithRomeoslove.ThechangeinJulietsloveisproportionaltoRomeoslove.IfRomeoinitiallylovesJuliettwiceasmuch,thenJulietsloveincreasestwiceasmuchasitwouldhaveincreasedifinitiallyheonlyfeltoneunitoflove.Theadditionalincreaseprecipitates(促進，加速)theexponentialgrowthoftheirlove,soafter12daystheyloveeachotherevenmorethaninthefirstscenario.,Scenario1,Scenario2,Scenario3,ConsiderthescenarioinwhichJulietisentirelyindifferenttoRomeo.Romeo,however,catchesaglimpseofJulietatapartyandfindsherreallyattractive.Inotherwords,JulietinitiallyfeelszeroloveunitsforRomeo,butRomeofeelsoneloveunitforJuliet.（Scenario）JulietsLoveforRomeo=INTEG(changeinJulietslove,0)RomeosLoveforJuliet=INTEG(changeinRomeoslove,1),InitiallyRomeolikesanindifferentJuliet,Again,bothstocksgrowtogetherexponentially.BecauseofJulietsinitialindifference,however,onthetwelfthday,theirloveis“only”2,048timesasstrongasRomeosinitialreactiontoJuliet.IfJuliethadinitiallyrespondedinthesamewayasRomeo,asinthefirstscenario,theirlovewouldhavebeentwiceasstrong.Tosummarize,ifjustoneofthetwoloversfeelssomepositiveemotionfortheother,theloveofbothloverswillfeedoffofeachother,andbothstockswillgrowexponentially.,4,000,3,000,2,000,1,000,0,0,3,6,9,12,Time(day),Scenario4,NowimagineascenarioinwhichRomeoandJulietaresoinfluencedbytheanimosity(仇恨)betweentheirfamiliesthatwhentheyfirstmeet,eachfeelssomeinstinctivedislikefortheother.(Scenario4)JulietsLoveforRomeo=INTEG(changeinJulietslove,-1)RomeosLoveforJuliet=INTEG(changeinRomeoslove,-1)Whatwillhappentothem?,InitiallyRomeoandJulietdislikeeachother,ItshowshowRomeoandJulietsdislikeforeachotherexperiencesexponentialdeclineandturnsintoadamant(堅硬的)hatred.JulietdislikesRomeomorebecausehedislikesher.Hedislikeshermorebecauseshedislikeshim.Themodelnowproducesexponentialdeclinetowardsaninfinitelylargenegativevalue.,Scenario5,Nowimagineascenarioinwhich,althoughRomeowasinitiallyenchanted(迷惑)byJuliet,shefeltaninitialdislikeforhim.(Scenario5)JulietsLoveforRomeo=INTEG(changeinJulietslove,-1)RomeosLoveforJuliet=INTEG(changeinRomeoslove,1)Whatwillhappentotheirlove?,InitiallyJulietandRomeohaveoppositereactionstoeachother,Onafirstglance,thebehaviorisverysurprising.JulietsdislikeforRomeodecreaseshisloveforher.Aslongasherloveforhimisnegative,hisloveforherdecreases.Whilehisloveforherispositive,however,herloveforhimincreases,becominglessnegative.AsJulietslovebecomeslessnegative,Romeoslovestilldecreases,butbyless.Eventually,theystabilizeattotalindifferencetowardseachother.,Scenario6,IfRomeoandJulietareinitiallyentirelyindifferenttoeachother,thenthesystemwillremainatequilibrium.JulietsLoveforRomeo=INTEG(changeinJulietslove,-1)RomeosLoveforJuliet=INTEG(changeinRomeoslove,1),InitiallyJulietandRomeoareindifferenttoeachother,Becauseneitherpersonlovestheother,theratesofchangearealsozero,andthesystemneverleavestheequilibriumstate.RomeoandJulietneverclick,sonothinghappens.,InitiallyRomeofeels1/100ofaloveunitforanindifferentJuliet,Theequilibriumthattheyreach,however,isveryunstable.If,duetoanyexternalreason,eitherRomeoorJulietstopsfeelingindifferenttothe