系統(tǒng)的結(jié)構(gòu)和系統(tǒng)的行為

系統(tǒng)結(jié)構(gòu)與系統(tǒng)行為

提綱系統(tǒng)動(dòng)力學(xué)基本原理一階系統(tǒng)二階系統(tǒng)（振蕩）（美國(guó)）約翰.D.斯特曼.商務(wù)動(dòng)態(tài)分析方法[M].北京，清華大學(xué)出版社，2008王其藩.系統(tǒng)動(dòng)力學(xué)[M]，上海財(cái)經(jīng)大學(xué)出版社，2009李旭.社會(huì)系統(tǒng)動(dòng)力學(xué)-政策研究的原理、方法和應(yīng)用[M]，復(fù)旦大學(xué)出版社，2008[美]Ford環(huán)境模擬－環(huán)境系統(tǒng)的系統(tǒng)動(dòng)力學(xué)模型導(dǎo)論[M]，復(fù)旦大學(xué)出版社，2009

系統(tǒng)動(dòng)力學(xué)的基本原理

系統(tǒng)的行為由它的結(jié)構(gòu)決定！如何提煉和整理產(chǎn)生復(fù)雜行為的那些簡(jiǎn)單結(jié)構(gòu)，成為研究復(fù)雜系統(tǒng)的關(guān)鍵問(wèn)題！系統(tǒng)的結(jié)構(gòu)（1）系統(tǒng)的物理和制度結(jié)構(gòu)－客觀結(jié)構(gòu)；（2）系統(tǒng)主體決策結(jié)構(gòu)－主觀結(jié)構(gòu)系統(tǒng)的基本行為：指數(shù)增長(zhǎng)；尋的行為；振蕩系統(tǒng)的基本結(jié)構(gòu)？如何設(shè)計(jì)？系統(tǒng)的結(jié)構(gòu)系統(tǒng)結(jié)構(gòu)：指組成系統(tǒng)的各個(gè)要素之間相互依賴關(guān)系的秩序?；芈肥墙M成系統(tǒng)結(jié)構(gòu)的基本單元，決策總是存在于回路之中并通過(guò)流率變量具體實(shí)施，流率變量調(diào)節(jié)流位變量的取值，為決策提供信息。認(rèn)識(shí)系統(tǒng)結(jié)構(gòu)的三個(gè)層次：閉合的邊界，反饋回路和流率方程。系統(tǒng)的結(jié)構(gòu)閉合的邊界：閉合的邊界內(nèi)包含了影響系統(tǒng)行為的主要因素和連接關(guān)系－系統(tǒng)流圖反饋回路：反饋環(huán)及反饋環(huán)的鏈接

閉合的邊界和反饋回路從定性的角度描述了系統(tǒng)結(jié)構(gòu)的整體框架。流率變量方程：決策是運(yùn)用信息對(duì)系統(tǒng)實(shí)施控制的過(guò)程，流率變量方程描述了過(guò)程規(guī)律。

流率變量方程從定量角度描述了系統(tǒng)局部的子系統(tǒng)。目標(biāo)、觀察值、偏差和控制是流率變量方程的內(nèi)部次級(jí)子結(jié)構(gòu)系統(tǒng)行為1.ExponentialGrowth2.ExponentialDecay

3.GoalSeekingBehavior4.S-ShapedGrowth5.SystemOscillation系統(tǒng)結(jié)構(gòu)和系統(tǒng)行為complexsystemBehaviorinteractionbetweenstockandflow

setsofstocks/flowswithinlargerfeedbackloops.

系統(tǒng)行為：ExponentialGrowthExactAnalyticalSolutionThefirstorderpositivefeedbackloopsystemisalinearsystem.Asaresult,inadditiontoitssimulatedsolution,itpossessesanexactanalyticalsolution.Thissolutionis:Stockt=Stock0*eCoef*t

InitializationPhase（初始值）Createalistofallequationsinrequiredorderofevaluation.Calculateinitialvaluesforallstocks,flowsandauxiliaries(inorderofevaluation).IterationPhase（迭代）Euler'smethod

EstimatethechangeinstocksovertheintervalDT.Calculatenewvaluesforstocksbasedonthisestimate.Usenewvaluesofstockstocalculatenewvaluesforflowsandauxiliaries.AddDTtosimulationtime.StopiteratingwhenTime>=StopTime

TimeConstant（時(shí)間常數(shù)）Thetimeconstantofthissystem,Tc,measuredinunitsoftime,isdefinedtobe1/Coef.（描述什么？）

TimeConstantGiventhedefinitionofthesystemtimeconstant,thefollowingquestioncanbeasked:Howmuchexponentialgrowthoccursduringonetimeconstant'sworthoftime?TimeConstantStockt=Stock0*eCoef*tStockt=Stock0*eCoef*TcStockt=Stock0*eCoef*(1/Coef)Stockt=Stock0*e1Stockt=Stock0*2.71828（什么意思？）TimeConstantInotherwords,theamountofexponentialgrowththatoccursduringonetimeconstantis2.71828timestheinitialamountinthesystem'sstock.Pushingthislogictothenextstep,theamountofexponentialgrowththatoccursduringtwotimeconstantsis7.3890(i.e.,2.71828*2.71828)timestheinitialamountinthesystem'sstock.TimeConstantDoublingTime（倍增時(shí)間）Thesecondmeasureofhowquicklyexponentialgrowthisproceedinginafirstorder,linear,positivefeedbacksystemisitsdoublingtime.Asystem'sdoublingtimeissimplytheamountoftimeittakesforthecontentsofitsstocktodouble.DoublingTimeThedoublingtimeofafirstorder,linear,positivefeebackloopsystem,Td,isequaltoseventypercentofitstimeconstant.Thatis:Td=.7*Tc.DoublingTime2*Stock0=Stock0*eCoef*TdSincethesystem'stimeconstant,Tc,is1/Coef,thenthesystem'sCoef=1/Tc.Thus:2*Stock0=Stock0*e1/Tc*TdDividingbothsidesbyStock0yields:2=eTd/TcTakingthenaturallogarithmofbothsidesyields:ln(2)=ln(eTd/Tc)Or:.69=Td/TcOr:.7=Td/TcOr:Td=.7*TcDoublingTimeDoublingTimeTheconceptofdoublingtimeleadsdirectlytoaheuristicknownastheRuleof70.DoublingTimetheRuleof70saysthatthenominalsizeofU.S.economywilldoublein(70/.07055)*100=9.86years.ExponentialGrowth小結(jié)要點(diǎn)：基本結(jié)構(gòu)；Positiveloopsgenerateexponentialgrowth(firstorderpositivefeedbackloop)增長(zhǎng)速度：CoefTheTimeConstant:TCDoubleTime:Td關(guān)系：TC＝1∕Coef；Td＝0.7TCExponentialDecayMeasuresofExponentialDecay:HalfLife（半衰期）Onemeasureofhowlongitwilltakeforafirstorder,linear,negativefeedbackloopsystemtoreachitsgoalofzeroisitshalflife.Thehalflifeofsuchasystemisthetimeittakestodrainhalfofthecontentsfromthesystem'sstock.Analogoustothecaseofthedoublingtimeforafirstorderpositivefeedbackloopsystemabove,thesystem'stimeconstantmustbedeterminedbeforeitshalflifecanbefound.ExponentialDecayAswaspreviouslyshown,theanalyticalsolutionstoEquations1ofExponentialGrowthandEquation2ofGoalSeekingBehaviordifferonlybysign.

ExponentialDecayStockt=Stock0*e-(1/TimeCoef)*TcStockt=Stock0*e-(1/TimeCoef)*TimeCoefStockt=Stock0*e-1Stockt=Stock0*(1/e)Stockt=Stock0*.37ExponentialDecaythesystemloses63%ofthecontentsofitsstockafteronetimeconstanthaselapsed.thesystemloses86%ofthecontentsofitsstockaftertwotimeconstantshaveelasped(Stockt=Stock0*.37*.37=Stock0*.14)and95%ofthecontentsofitsstockafterthreetimeconstantshaveelapsed(Stockt=Stock0*.37*.37*.37=Stock0*.05).ExponentialDecayAsinthecaseofafirstorderpositivefeedbackloopsystem,thenegativeloopsystem'shalflifeis70%ofitstimeconstant.Thatis:Th=.7*Tc.ExponentialDecayExponentialDecayTheanalysisofthefirstordernegativefeedbackloopsystemwithanimplicitgoalofzerocanbeextendedtothemoregeneralnegativefeedackloopmodeloriginallypresentedinFigure2ofGoalSeekingBehavior,givingthefollowinginsight:Afirstorder,negative,feedbackloopsystemreachesapproximately95%ofitsgoalafterthreetimeconstantshaveelapsed.ExponentialDecay小結(jié)要點(diǎn)：結(jié)構(gòu)；TIMECONSTANT;HALFLIFE.Th=.7*Tcapproximately95%ofitsgoalafterthreetimeconstantsGoalSeekingBehaviorthetwotypesoffeedbackloopsthatexistintheworld:positiveloopsandnegativeloops.Positiveloops“generate”exponentialgrowth(orrapiddecline)negativeloopsproducegoal-seekingbehavior.GenericFirstOrderNegativeFeedbackLoopSystemGoalSeekingBehaviorFigure1presentsageneric,firstorder,linear,negativefeedbackloopsystem.Themodelisinitializedinequilibriumbysettingtheinitialvalueofthestockequaltoitsgoal,andthen"shocked"outofequilibrium(attime=1)byastepfunction

thatchangesthesystem'sgoalfromtentofifteen.GoalSeekingBehaviorAsystemcontainingnegativefeedbackloopswillbeinequilibriumonlywhenallofitsstocksareequaltoallofitsgoalssimultaneously.GoalSeekingBehaviorTheimplicationofthisideaisveryimportant.Intherealworlditisraretofindactualsystems(particularlyactualsocialsystems)thatexistinstatesofequilibriumbecauseactualsystemscontainmanynegativefeedbackloopsthatdonotreachtheirgoalssimultaneously.Thus,systemdynamicistsoftenstarttheirmodelsinstatesofequilibrium,notbecausetheyfeelthatthisreflectsreality,butratherbecauseitisausefulreferencepointforanalyzingthe"pure"behaviorsoftheirmodels

GoalSeekingBehaviorGoalSeekingBehaviorThetimeittakesthesystemtoreachitsnewgoalisdeterminedbythesizeofTimeCoef.ThebiggerTimeCoefis,thelongerittakesforthesystemtoadjusttoitsnewgoal.GoalSeekingBehavior

ThreeSimulationsoftheGenericFirstOrderNegative

FeedbackLoopSystemgiveastepreductionattime=1

GoalSeekingBehavior討論：GoalSeekingBehavior產(chǎn)生的本質(zhì)原因是什么？SystemOscillationSystemOscillationOscillationreferstoabehaviorinwhichthevaluesofthestocksvaryaroundsomeaveragevalueinarepeatingpattern.SystemOscillationSustainedoscillation:the“peak“isthesameeachcycle.Expandingoscillation:the“peak“becomesgreatereachperiod.Dampedoscillation:the“peak“becomessmallereachperiod.SystemOscillationThemathematicsfunctionofoscillationStock=A?sin(??time+?)WhereA,?,and?areconstantAcademicperformancemodel二級(jí)反饋環(huán)與系統(tǒng)振蕩AcademicperformancemodelWhythesystemdemonstratesthebehaviorofsustainedoscillation?Thestockstheflowsgap26hoursaweekDebriefoftheacademicperformancemodelAnoscillatingsystemrequirestwostocks.Oscillatingsystemhavesomemomentum-gap.Gap---increaseheramountofstudying--….---currentgrades:desiredlevel流位變量的積累作用：延遲Whysecond-ordersystemcanoscillrteThesystemmustbeanegativefeedbackloop:negativefeedbackloopsalwaysattempttoclosethegapbetweensomedesiredstateofthesystemandtheactualstateofthestate,knowasgoal-seekingbehaviorThesystemmustbeatleastsecond-order:thesecondstockcancontinuetochangethesystemevenifoneofthestockisatitsgoalvalue.SystemOscillationSystemoscillationisthecharacteristicsymptomofnegativefeedbackstructuresinwhichtheinformationusedtotakegoal-seekingactionisdelayed.Insuchcases,acontrolactionisnotbasedonthecurrentstateofasystembutonsomepreviousstateorvalue.Asyoucanimagine,usingdatedinformationtocontroltheapproachtoatargetislikelytocausethesystemtomissorovershootitsgoal.Specificstructuralcharacteristicsofasystem,suchasdelaydurationanddelayorder,willdeterminewhethertheresultingoscillationwilldiminishovertime,reachsomesustainedamplitudeandfrequency,orexperienceanincreasingamplitude.SystemOscillationSystemOscillationSystemOscillationGiventhisstructure,Figure3showswhathappensifthegoal,DesiredSystemLevel,isincreasedby5units.Becauseofthedelay,thesystemovershootsitstargetatabouttime=6,reachesamaximumof22unitsattime=10,thenundershootsatabouttime=12,andsoon.Eventually,thesystemdoesfinditstarget50timeunitsafterthegoalchanged!SystemOscillationTypesofOscillationsInformationdelaysareprevalentintherealworldsoit'snosurprisethatoscillationsarethemostcommondynamicbehaviors.Therearefourbasictypes:SustainedOscillation;DampedOscillation;ExplodedOscillation;ChaosOscillation.SustainedOscillation·CharacterizedbyperiodicityofoneConstantamplitudeNostablefixedequilibriumisreachedDampedOscillation·AmplitudeofoscillationdissipatesovertimeStableequilibriumisreachedSystemOscillationExplodedOscillation·NotcommonintherealworldAmplitudeincreasesovertimeGrowtheitherleadstoasustainedpatternordeteriorationofthesystemChaosOscillation·RandomtimepaththatneverrepeatsIrregularamplitudeandinfiniteperiodChaososcillationsmaybefoundinmanynaturalsystemssuchasweatherpatternsSystemOscillation

Systemoscillationiscreatedwhendecisioninformationinnegativefeedbackstructuresisdelayed.SystemOscillation的原因S-ShapedGrowth

S-shapedgrowthisthecharacteristicbehaviorofasysteminwhichapositiveandnegativefeedbackstructurefightfordominancebutresultinlong-runequilibrium.thepositivefeedbackexponentialgrowthloop(showninblue)initiallydominatesthesystem,causingthesystemvariabletoincreaseatanincreasingrate.However,asthesystemapproachesitslimitor"carryingcapacity",thegoal-seekingnegativefeedbackloopbecomesthedominantloop.Inthisnewmode,thesystemapproachesstasisasymptotically.This"shiftingdominance"eventfromonelooptoanotherischaracteristicofnonlinearbehaviorandisfoundinavarietyofphysical,social,andeconomicsystems.S-ShapedGrowthS-ShapedGrowthS-ShapedGrowthS-ShapedGrowthThepositiveBirthRateloop,showninblue,isresponsiblefortheexponentialgrowthexperiencedduringtheearlystagesofherdgrowth.Asthepopulationoftheherdincreases,thebirthrate-givensomeFecundity(fertility)constant-increases.Increasingthebirthratefurtherpromotesthegrowthoftheherd.Inbalance,thenegativeDeathRateloop,showninred,isresponsiblefordrawingdowntheherdpopulation.S-ShapedGrowthS-ShapedGrowthO