系統動力學(SystemDynamics),主講：張學民,(3),一階系統,一階系統,所謂系統的階數就是系統狀態變量的維數，在系統動力學中也就是水平(變量)的個數。在社會經濟系統中高階(可能是數十至數百階)系統到處可見。任一高階系統都可以視為由一階系統關聯而成。一個復雜系統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。一階反饋系統就是只含一個水平(變量)的反饋系統。正反饋系統行為的共同特征就是指數式地趨于無窮。一階正反饋系統的行為是單調地指數式地趨于無窮。負反饋系統行為的共同特征就是尋找目標。一階負反饋系統的行為是單調地趨于目標。,一階反饋系統的復雜性,一階反饋系統也會呈現復雜的系統行為！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每人只需一件這樣的商品總人數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,一階負反饋系統的行為是單調地趨于目標,一階反饋系統的復雜性,FINALTIME:60TIMESTEP:0.5Units:Month,FRACTION=0.06dmnl/MonthTOTALBUYERS=100person,一階負反饋系統的行為是單調地趨于目標,一階反饋系統的復雜性,FINALTIME:60TIMESTEP:0.5Units:Month,FRACTION=0.1dmnl/MonthTOTALBUYERS=100person,一階負反饋系統的行為是單調地趨于目標,一階反饋系統的復雜性,FINALTIME:60TIMESTEP:0.5Units:Month,TOTALBUYERS=100person,FRACTION=0.1,FRACTION=0.06,FRACTION=0.01,一階負反饋系統的行為是單調地趨于目標,一階反饋系統的復雜性,FRACTION=0.1,FRACTION=0.06,FRACTION=0.01,一階反饋系統的復雜性新假設,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購買該商品的戶數越多，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和已經購買該商品的戶數的乘積,一階反饋系統的復雜性新假設,(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和已經購買該商品的戶數的乘積,一階反饋系統的復雜性新假設,FINALTIME:60TIMESTEP:0.5Units:Month,FRACTION=0.002dmnl/(Month*person),TOTALBUYERS=100person,一階反饋系統的復雜性新假設,FINALTIME:60TIMESTEP:0.5Units:Month,FRACTION=0.001FRACTION=0.002FRACTION=0.004,TOTALBUYERS=100person,一階反饋系統的復雜性新假設,左半段曲線的斜率為正，表明兩個反饋環中正反饋環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，意味著正反饋環的力量逐漸削弱，水平變量Buyers的行為呈亞指數增長的特性，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的復雜性S型增長,S型增長是一種常見而重要的系統行為，它是系統中正負反饋環交互作用的結果。在S型增長的過程中先是正反饋環占主導地位，因而系統行為呈指數增長的特性。按指數規律無限增長的正反饋過程是不會無止境地持續下去的：系統崩潰水平變量達到某一較高水平時，正反饋環的主導地位由負反饋環取而代之,一階反饋系統的復雜性新假設,一階反饋系統的復雜性新假設,FRACTION=0.001,FRACTION=0.002,FRACTION=0.004,buyrate,S型增長系統的特點,兩個特點：純速率狀態關系曲線可分為兩段，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。當系統運行在正反饋軌線上時，主導反饋環為正反饋環，而當系統運行在負反饋軌線上時，主導反饋環為負反饋環。主導反饋環的轉移發生在前后兩段軌線的交接處。純速率狀態關系與橫坐標有兩個交點，左邊一點是系統的不穩定平衡點，右邊一點是系統的穩定平衡點，即S型增長趨近的目標。,一階線性反饋系統行為模式,非線性與主導反饋環的轉移,復雜系統內部存在相互作用的或正或負的多重反饋環。所謂主導反饋環就是在多重反饋環中起主導作用的反饋環。當系統行為表現出指數增長(或指數崩潰)特性時，可以推斷系統中必定存在正反饋環，并且正起著主導作用。當系統行為表現出尋找目標特性時，則可以推斷系統中必定存在負反饋環，并且正起著主導作用。系統行為是由多重反饋環相互作用共同產生的，其行為模式主要由主導反饋環決定。主導反饋環并非是固定不變的，它(們)往往隨著系統狀態的變化而在各反饋環中轉移，由此產生了多種多樣的復雜的系統行為。實際系統幾乎都具有非線性的特征。非線性關系是導致主導反饋環極性轉移的根本原因。不僅要研究正反饋環或負反饋環的作用，而且要研究主導反饋環轉移的作用。S型增長是主導反饋環由正反饋環向負反饋環轉移的實例。,二階系統,二階系統,所謂二階系統就是包含有兩個水平(變量)的系統。二階系統也可分為二階負反饋系統和二階正反饋系統。二階線性負反饋系統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地趨于無窮，另一種模式是振蕩。如果二階系統是非線性的，那么系統行為將更為復雜。,二階系統典型行為,增幅振蕩,一、二階系統行為模式是高階系統行為模式的基礎,指數增長線性增長漸近增長S型增長指數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,高階系統的行為高階系統具有互相連接的多重反饋環，因而其行為比二階系統更為復雜。在結構上，高階系統可以看作是由若干個典型的一階結構或二階結構相互關聯而成。由于有了關聯以及系統中可能存在的非線性，高階系統的行為已經不僅僅是指數增長、線性增長、漸近增長、S型增長、指數崩潰、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“loveunits.”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