系統(tǒng)動力學(xué)

(SystemDynamics)主講：張學(xué)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(xué)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(fù)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(fù)雜系統(tǒng)中往往存在一些起主導(dǎo)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(biāo)。一階負反饋系統(tǒng)的行為是單調(diào)地趨于目標(biāo)。一階反饋系統(tǒng)的復(fù)雜性一階反饋系統(tǒng)也會呈現(xiàn)復(fù)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(fù)雜性(1) FRACTION=0.06 Units:dmnl/Month(2) TOTALBUYERS=100 Units:person(3) buyrate=FRACTION*potentialbuyers Units:person/Month(4) Buyers=INTEG(buyrate,1) Units:person(5) potentialbuyers=TOTALBUYERS-Buyers Units:person

FINALTIME:60TIMESTEP:0.5Units:MonthTOTALBUYERSpotentialbuyersFRACTIONbuyrateBuyers-++一階負反饋系統(tǒng)的行為是單調(diào)地趨于目標(biāo)一階反饋系統(tǒng)的復(fù)雜性FINALTIME:60TIMESTEP:0.5Units:MonthBuyers100755025003060Time(Month)FRACTION=0.06dmnl/MonthTOTALBUYERS=100personTOTALBUYERSpotentialbuyersFRACTIONbuyrateBuyers-++一階負反饋系統(tǒng)的行為是單調(diào)地趨于目標(biāo)一階反饋系統(tǒng)的復(fù)雜性FINALTIME:60TIMESTEP:0.5Units:MonthFRACTION=0.1dmnl/MonthTOTALBUYERS=100personBuyers100755025003060Time(Month)TOTALBUYERSpotentialbuyersFRACTIONbuyrateBuyers-++一階負反饋系統(tǒng)的行為是單調(diào)地趨于目標(biāo)一階反饋系統(tǒng)的復(fù)雜性FINALTIME:60TIMESTEP:0.5Units:MonthTOTALBUYERS=100personBuyers1007550250015304560Time(Month)FRACTION=0.1FRACTION=0.06FRACTION=0.01TOTALBUYERSpotentialbuyersFRACTIONbuyrateBuyers-++一階負反饋系統(tǒng)的行為是單調(diào)地趨于目標(biāo)一階反饋系統(tǒng)的復(fù)雜性Buyers100755025015304560Time(Month)FRACTION=0.1FRACTION=0.06FRACTION=0.01potentialbuyers10075502503060Time(Month)buyrate107.552.503060Time(Month)Buyersvs.buyrate107.552.50100Buyers(person)buyrate一階反饋系統(tǒng)的復(fù)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(fù)雜性––新假設(shè)(1) FRACTION=0.002 Units:dmnl/(Month*person)(2) TOTALBUYERS=100 Units:person(3) buyrate=FRACTION*Buyers*possiblebuyers Units:person/Month(4) Buyers=INTEG(buyrate,1) Units:person(5) possiblebuyers=TOTALBUYERS-Buyers Units:personFINALTIME:60TIMESTEP:0.5Units:MonthTOTALBUYERSpossiblebuyersFRACTIONbuyrateBuyers-+++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(fù)雜性––新假設(shè)FINALTIME:60TIMESTEP:0.5Units:MonthTOTALBUYERSpossiblebuyersFRACTIONbuyrateBuyers-+++Buyers1007550250015304560Time(Month)FRACTION=0.002dmnl/(Month*person)TOTALBUYERS=100person一階反饋系統(tǒng)的復(fù)雜性––新假設(shè)FINALTIME:60TIMESTEP:0.5Units:MonthTOTALBUYERSpossiblebuyersFRACTIONbuyrateBuyers-+++FRACTION=0.001FRACTION=0.002FRACTION=0.004Buyers1007550250015304560Time(Month)TOTALBUYERS=100person一階反饋系統(tǒng)的復(fù)雜性––新假設(shè)64.531.5011937557391Buyers(person)buyrate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(dǎo)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(dǎo)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(fù)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(dǎo)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(dǎo)地位由負反饋環(huán)取而代之一階反饋系統(tǒng)的復(fù)雜性––新假設(shè)6person/Month100person3person/Month50person0person/Month0person091827364554Time(Month)buyrateBuyerspossiblebuyers一階反饋系統(tǒng)的復(fù)雜性––新假設(shè)FRACTION=0.001FRACTION=0.002FRACTION=0.004Buyers1007550250015304560Time(Month)possiblebuyers1007550250015304560Time(Month)buyrate107.552.50015304560Time(Month)Buyersvs.buyrate107.552.5011937557391Buyers(person)buyrateS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(dāng)系統(tǒng)運行在正反饋軌線上時，主導(dǎo)反饋環(huán)為正反饋環(huán)，而當(dāng)系統(tǒng)運行在負反饋軌線上時，主導(dǎo)反饋環(huán)為負反饋環(huán)。主導(dǎo)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(biāo)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(biāo)。64.531.5011937557391Buyers(person)buyrateBuyers1007550250015304560Time(Month)一階線性反饋系統(tǒng)行為模式一階線性正反饋系統(tǒng)可能有的三種行為模式指數(shù)增長行為指數(shù)崩潰行為恒值行為漸近衰減行為漸近增長行為恒值行為一階線性負反饋系統(tǒng)可能有的三種行為模式非線性與主導(dǎo)反饋環(huán)的轉(zhuǎn)移復(fù)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(dǎo)反饋環(huán)就是在多重反饋環(huán)中起主導(dǎo)作用的反饋環(huán)。當(dāng)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(dǎo)作用。當(dāng)系統(tǒng)行為表現(xiàn)出尋找目標(biāo)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(dǎo)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(dǎo)反饋環(huán)決定。主導(dǎo)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(fù)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(dǎo)致主導(dǎo)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(dǎo)反饋環(huán)轉(zhuǎn)移的作用。S型增長是主導(dǎo)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(biāo)，但逼近終值的方式更為復(fù)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(fù)雜。二階系統(tǒng)典型行為二階正反饋系統(tǒng)的典型行為二階負反饋系統(tǒng)的典型行為欠阻尼過阻尼增幅振蕩一、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(fù)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(fù)雜組合。Second-OrderSystemsonHowTwoLoversInteractRomeoandJuliet

byWilliamShakespeareGonewiththeWindbyMargaretMitchellTheRedandtheBlackbyStendhalSecond-OrderSystemsonHowTwoLoversInteractDifferentpeoplehavedifferentpersonalitiesthatdictate(指示)

howtheywillrespondtotheirpartners’love.Somepeoplelovetheirpartnersmorethemoretheirpartnerslovethem.Theygetdiscouragedandgiveupiftheirpartnersdonotlovethem.Otherpeople,ontheotherhand,getannoyediftheirloverslovethemtoomuch.Theyareattractedtopeoplewhoarenotattractedtothem.Thispaperwillmatchtogetherloverswithdifferentpersonalitiesandobservehowtheirrelationshipsunfold.Eachpairwillberepresentedbyadifferentsetofparametersinthemodel.Bylookingatthedynamicbehaviorgeneratedbymatchingdifferentpersonalities,thatis,bysimulatingthemodelwithdifferentcombinationsofparameters,thispaperwillstudythebehaviorofsimplesecond-ordersystems.RomeoandJulietRomeoandJulietaremadlyinlovewitheachother.Witheachsecretmeeting,Romeo’sloveforJulietgrows.Becausehelovesher,hedoeseverythinghecantoimpressher.Julietisflatteredbyhisattentionand,inreturn,herloveforRomeoalsogrows.BecauseRomeosensesthatJulietloveshim,heallowshispassiontosoar(驟升,升騰).RomeoandJuliet

byWilliamShakespeareTheModelforRomeoandJulietInthemodeleachstockrepresentstheaccumulationofonelover’sloveforanother.TherateatwhichRomeo’slovechangesdependsontheextentofJuliet’sloveandRomeo’spersonality.Hispersonalityisrepresentedbyaparameterthatmeasureshowhereactstoherlove.Likewise,thechangeinJuliet’sloveequalstheamountofRomeo’slovetimestheextenttowhichshereactstohislove.ROMEO'SREACTIONJULIET'SREACTIONchangeinJuliet'slovechangeinRomeo'sloveJuliet'sLoveforRomeoRomeo'sLoveforJulietMeasurementforLoveThelovethatRomeoandJulietfeelforeachothercanbequantifiedinseveralways.Lovecanbemeasuredintermsofkissesperday,ortenderthoughtsperhour,forexample.Theunitsofthestocksofloveinthemodelsarebaseduponascaleof“l(fā)oveunits.”Ifthetwoprotagonistsareindifferent(不感興趣)toeachother,theyfeelzeroloveunits.Positiveloveunitsrepresenthowmuchtheylikeeachother,whilenegativeloveunitsrepresenthowmuchtheydislikeeachother.Rightnow,assumethatinitiallyRomeoandJulietfeelkindlydisposedtowardseachother.Theyeachfeeloneunitoflovefortheother.(Scenario1)ParameterforPersonalityTheparameter“JULIET’SREACTION”representshowJulietreactstoRomeo’slove,thatis,towhatextentRomeo’slovechangesJuliet’sloveovertime.AssumethateveryunitofRomeo’sloveincreasesJuliet’slovebyoneuniteachday.ThereforeJuliet’sreactiontoRomeo’sloveisoneloveunitperloveunitperday,oroneperday.JULIET'SREACTION=1Units:1/dayRomeohasthesamepersonalityasJuliet,soRomeo’sreactiontoJuliet’sloveisalsooneperday.ROMEO'SREACTION=1Units:1/dayEquations(01) changeinJuliet'slove=Romeo'sLoveforJuliet*JULIET'SREACTION Units:loveunit/day(02) changeinRomeo'slove=Juliet'sLoveforRomeo*ROMEO'SREACTIONUnits:loveunit/day(03) Juliet'sLoveforRomeo=INTEG(changeinJuliet'slove,1)Units:loveunit(04) JULIET'SREACTION=1Units:1/day(05) Romeo'sLoveforJuliet=INTEG(changeinRomeo'slove,1)Units:loveunit(06) ROMEO'SREACTION=1Units:1/dayFINALTIME =12dayINITIALTIME =0dayTIMESTEP =1dayTheirlovegrowtogetherexponentially!Romeo’sloveandJuliet’slovegrowtogetherexponentially.Bytheendof12days,theyeachfeel4,096loveunits!Inotherwords,theylikeeachother4,096timesmorethantheydidinitially.RomeoandJulietinitiallylikeeachother6,0004,5003,0001,5000036912Time(day)Juliet'sLoveforRomeo-CURRENTloveunitRomeo'sLoveforJuliet-CURRENTloveunitPositivefeedbackloopApositivefeedbackloopjoinsthetwostocks.AnincreaseinJuliet’sloveforRomeoincreasesthechangeinRomeo’slove,whichincreasesRomeo’sloveforJuliet.TheincreaseinRomeo’sloveleadstoanincreaseinthechangeinJuliet’slove,furtherincreasingJuliet’sloveforRomeo.Withtime,thestocksreinforceeachotherandincreaseexponentially.ROMEO'SREACTIONJULIET'SREACTIONchangeinJuliet'slovechangeinRomeo'sloveJuliet'sLoveforRomeoRomeo'sLoveforJulietInitiallyRomeolovedJuliettwiceWhatwouldhappenifinitiallyRomeolovedJuliettwiceasmuchasshelovedhim?(Scenario2)Juliet'sLoveforRomeo=INTEG(changeinJuliet'slove,1)Romeo'sLoveforJuliet=INTEG(changeinRomeo'slove,2)WillJuliet’sloveevergrowtoequalRomeo’slove?Scenario2vs.Scenario1Juliet’slovequicklycatchesupwithRomeo’slove.ThechangeinJuliet’sloveisproportionaltoRomeo’slove.IfRomeoinitiallylovesJuliettwiceasmuch,thenJuliet’sloveincreasestwiceasmuchasitwouldhaveincreasedifinitiallyheonlyfeltoneunitoflove.Theadditionalincreaseprecipitates(促進，加速)theexponentialgrowthoftheirlove,soafter12daystheyloveeachotherevenmorethaninthefirstscenario.8,0006,0004,0002,0000036912Time(day)Scenario1Scenario2Scenario3ConsiderthescenarioinwhichJulietisentirelyindifferenttoRomeo.Romeo,however,catchesaglimpseofJulietatapartyandfindsherreallyattractive.Inotherwords,JulietinitiallyfeelszeroloveunitsforRomeo,butRomeofeelsoneloveunitforJuliet.（Scenario３）Juliet'sLoveforRomeo=INTEG(changeinJuliet'slove,0)Romeo'sLoveforJuliet=INTEG(changeinRomeo'slove,1)InitiallyRomeolikesanindifferentJulietAgain,bothstocksgrowtogetherexponentially.BecauseofJuliet’sinitialindifference,however,onthetwelfthday,theirloveis“only”2,048timesasstrongasRomeo’sinitialreactiontoJuliet.IfJuliethadinitiallyrespondedinthesamewayasRomeo,asinthefirstscenario,theirlovewouldhavebeentwiceasstrong.Tosummarize,ifjustoneofthetwoloversfeelssomepositiveemotionfortheother,theloveofbothloverswillfeedoffofeachother,andbothstockswillgrowexponentially.4,0003,0002,0001,0000036912Time(day)Scenario4NowimagineascenarioinwhichRomeoandJulietaresoinfluencedbytheanimosity(仇恨)

betweentheirfamiliesthatwhentheyfirstmeet,eachfeelssomeinstinctivedislikefortheother.(Scenario4)Juliet'sLoveforRomeo=INTEG(changeinJuliet'slove,-1)Romeo'sLoveforJuliet=INTEG(changeinRomeo'slove,-1)Whatwillhappentothem?InitiallyRomeoandJulietdislikeeachotherItshowshowRomeoandJuliet’sdislikeforeachotherexperiencesexponentialdeclineandturnsintoadamant(堅硬的)

hatred.JulietdislikesRomeomorebecausehedislikesher.Hedislikeshermorebecauseshedislikeshim.Themodelnowproducesexponentialdeclinetowardsaninfinitelylargenegativevalue.0-1,500-3,000-4,500-6,000024681012Time(day)Scenario5Nowimagineascenarioinwhich,althoughRomeowasinitiallyenchanted(迷惑)

byJuliet,shefeltaninitialdislikeforhim.(Scenario5)Juliet'sLoveforRomeo=INTEG(changeinJuliet'slove,-1)Romeo'sLoveforJuliet=INTEG(changeinRomeo'slove,1)Whatwillhappentotheirlove?InitiallyJulietandRomeohaveoppositereactionstoeachotherOnafirstglance,thebehaviorisverysurprising.Juliet’sdislikeforRomeodecreaseshisloveforher.Aslongasherloveforhimisnegative,hisloveforherdecreases.Whilehisloveforherispositive,however,herloveforhimincreases,becominglessnegative.AsJuliet’slovebecomeslessnegative,Romeo’slovestilldecreases,butbyless.Eventually,theystabilizeattotalindifferencetowardseachother.1.50-.5-1024681012Time(day)Juliet'sLoveforRomeoloveunitRomeo'sLoveforJulietloveunitScenario6IfRomeoandJulietareinitiallyentirelyindifferenttoeachother,thenthesystemwillremainatequilibrium.Juliet'sLoveforRomeo=INTEG(changeinJuliet'slove,-1)Romeo'sLoveforJuliet=INTEG(changeinRomeo'slove,1)InitiallyJulietandRomeoareindifferenttoeachotherBecauseneitherpersonlovestheother,theratesofchangearealsozero,andthesystemneverleavestheequilibriumstate.RomeoandJulietneverclick,sonothinghappens..2.15.1.050024681012Time(day)InitiallyRomeofeels1/100ofaloveunitforanindifferentJulietTheequilibr