系统动力学3PPT课件

系统动力学(SystemDynamics),主讲：张学民,(3),一阶系统,一阶系统,所谓系统的阶数就是系统状态变量的维数，在系统动力学中也就是水平(变量)的个数。在社会经济系统中高阶(可能是数十至数百阶)系统到处可见。任一高阶系统都可以视为由一阶系统关联而成。一个复杂系统的行为往往是由某些主回路和某些主要的变量决定的，换言之，复杂系统中往往存在一些起主导作用的主回路和主要变量。一阶反馈系统就是只含一个水平(变量)的反馈系统。正反馈系统行为的共同特征就是指数式地趋于无穷。一阶正反馈系统的行为是单调地指数式地趋于无穷。负反馈系统行为的共同特征就是寻找目标。一阶负反馈系统的行为是单调地趋于目标。,一阶反馈系统的复杂性,一阶反馈系统也会呈现复杂的系统行为！考察某商品的销售问题。假定：销售量正比于还没有购买该商品的人数每人只需一件这样的商品总人数是恒定的商品更新的周期远大于商品普及所需的时间,一阶反馈系统的复杂性,(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,一阶负反馈系统的行为是单调地趋于目标,一阶反馈系统的复杂性,FINALTIME:60TIMESTEP:0.5Units:Month,FRACTION=0.06dmnl/MonthTOTALBUYERS=100person,一阶负反馈系统的行为是单调地趋于目标,一阶反馈系统的复杂性,FINALTIME:60TIMESTEP:0.5Units:Month,FRACTION=0.1dmnl/MonthTOTALBUYERS=100person,一阶负反馈系统的行为是单调地趋于目标,一阶反馈系统的复杂性,FINALTIME:60TIMESTEP:0.5Units:Month,TOTALBUYERS=100person,FRACTION=0.1,FRACTION=0.06,FRACTION=0.01,一阶负反馈系统的行为是单调地趋于目标,一阶反馈系统的复杂性,FRACTION=0.1,FRACTION=0.06,FRACTION=0.01,一阶反馈系统的复杂性新假设,对于一种新商品人们并不了解，一开始就购买的人不会太多大部分人只是在听到已经购买的人的介绍或者看到别人在抢购后才逐渐对该商品有所了解，购买者也才会多起来已经购买该商品的户数越多，越会促成更多的销售更符合实际的新假定是：销售量正比于还没有购买该商品的户数和已经购买该商品的户数的乘积,一阶反馈系统的复杂性新假设,(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,销售量正比于还没有购买该商品的户数和已经购买该商品的户数的乘积,一阶反馈系统的复杂性新假设,FINALTIME:60TIMESTEP:0.5Units:Month,FRACTION=0.002dmnl/(Month*person),TOTALBUYERS=100person,一阶反馈系统的复杂性新假设,FINALTIME:60TIMESTEP:0.5Units:Month,FRACTION=0.001FRACTION=0.002FRACTION=0.004,TOTALBUYERS=100person,一阶反馈系统的复杂性新假设,左半段曲线的斜率为正，表明两个反馈环中正反馈环起主导作用左半段曲线的斜率随着水平变量Buyers的增加而递减至零，意味着正反馈环的力量逐渐削弱，水平变量Buyers的行为呈亚指数增长的特性，购买率buyrate则随着Buyers的增长而增至其最大值右半段曲线的斜率为负，且其绝对值随着水平变量Buyers的增加由零逐渐递增的，表明负反馈环不仅起了主导作用，而且其力量在不断加强，水平变量Buyers的行为呈超渐近增长的特性，购买率buyrate则随着Buyers的增长由最大值逐渐衰减至零,一阶反馈系统的复杂性S型增长,S型增长是一种常见而重要的系统行为，它是系统中正负反馈环交互作用的结果。在S型增长的过程中先是正反馈环占主导地位，因而系统行为呈指数增长的特性。按指数规律无限增长的正反馈过程是不会无止境地持续下去的：系统崩溃水平变量达到某一较高水平时，正反馈环的主导地位由负反馈环取而代之,一阶反馈系统的复杂性新假设,一阶反馈系统的复杂性新假设,FRACTION=0.001,FRACTION=0.002,FRACTION=0.004,buyrate,S型增长系统的特点,两个特点：纯速率状态关系曲线可分为两段，前一段的斜率为正，称正反馈轨线，后一段的斜率为负，称负反馈轨线。当系统运行在正反馈轨线上时，主导反馈环为正反馈环，而当系统运行在负反馈轨线上时，主导反馈环为负反馈环。主导反馈环的转移发生在前后两段轨线的交接处。纯速率状态关系与横坐标有两个交点，左边一点是系统的不稳定平衡点，右边一点是系统的稳定平衡点，即S型增长趋近的目标。,一阶线性反馈系统行为模式,非线性与主导反馈环的转移,复杂系统内部存在相互作用的或正或负的多重反馈环。所谓主导反馈环就是在多重反馈环中起主导作用的反馈环。当系统行为表现出指数增长(或指数崩溃)特性时，可以推断系统中必定存在正反馈环，并且正起着主导作用。当系统行为表现出寻找目标特性时，则可以推断系统中必定存在负反馈环，并且正起着主导作用。系统行为是由多重反馈环相互作用共同产生的，其行为模式主要由主导反馈环决定。主导反馈环并非是固定不变的，它(们)往往随着系统状态的变化而在各反馈环中转移，由此产生了多种多样的复杂的系统行为。实际系统几乎都具有非线性的特征。非线性关系是导致主导反馈环极性转移的根本原因。不仅要研究正反馈环或负反馈环的作用，而且要研究主导反馈环转移的作用。S型增长是主导反馈环由正反馈环向负反馈环转移的实例。,二阶系统,二阶系统,所谓二阶系统就是包含有两个水平(变量)的系统。二阶系统也可分为二阶负反馈系统和二阶正反馈系统。二阶线性负反馈系统的行为特征也是寻找目标，但逼近终值的方式更为复杂。一般可分为两种，一种为欠阻尼形式，另一种为过阻尼形式。二阶线性正反馈系统的行为特征仍为趋于无穷，但呈两种不同的模式：一种模式与一阶正反馈的场合相似仍是单调地趋于无穷，另一种模式是振荡。如果二阶系统是非线性的，那么系统行为将更为复杂。,二阶系统典型行为,增幅振荡,一、二阶系统行为模式是高阶系统行为模式的基础,指数增长线性增长渐近增长S型增长指数崩溃线性衰减渐近衰减S型衰减增幅振荡等幅振荡减幅振荡,高阶系统的行为高阶系统具有互相连接的多重反馈环，因而其行为比二阶系统更为复杂。在结构上，高阶系统可以看作是由若干个典型的一阶结构或二阶结构相互关联而成。由于有了关联以及系统中可能存在的非线性，高阶系统的行为已经不仅仅是指数增长、线性增长、渐近增长、S型增长、指数崩溃、线性衰减、渐近衰减、S型衰减、增幅振荡、等幅振荡及减幅振荡等几种形式，而是它们的复杂组合。,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(骤升,升腾).,RomeoandJulietbyWilliamShakespeare,TheModelforRomeoandJuliet,Inthemodeleachstockrepresentstheaccumulationofoneloversloveforanother.TherateatwhichRomeoslovechangesdependsontheextentofJulietsloveandRomeospersonality.Hispersonalityisrepresentedbyaparameterthatmeasureshowhereactstoherlove.Likewise,thechangeinJulietsloveequalstheamountofRomeoslovetimestheextenttowhichshereactstohislove.,MeasurementforLove,ThelovethatRomeoandJulietfeelforeachothercanbequantifiedinseveralways.Lovecanbemeasuredintermsofkissesperday,ortenderthoughtsperhour,forexample.Theunitsofthestocksofloveinthemodelsarebaseduponascaleof“loveunits.”Ifthetwoprotagonistsareindifferent(不感兴趣)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(促进，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(坚硬的)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