系统的结构和系统的行为

系统结构与系统行为

提纲系统动力学基本原理一阶系统二阶系统（振荡）（美国）约翰.D.斯特曼.商务动态分析方法[M].北京，清华大学出版社，2008王其藩.系统动力学[M]，上海财经大学出版社，2009李旭.社会系统动力学-政策研究的原理、方法和应用[M]，复旦大学出版社，2008[美]Ford环境模拟－环境系统的系统动力学模型导论[M]，复旦大学出版社，2009

系统动力学的基本原理

系统的行为由它的结构决定！如何提炼和整理产生复杂行为的那些简单结构，成为研究复杂系统的关键问题！系统的结构（1）系统的物理和制度结构－客观结构；（2）系统主体决策结构－主观结构系统的基本行为：指数增长；寻的行为；振荡系统的基本结构？如何设计？系统的结构系统结构：指组成系统的各个要素之间相互依赖关系的秩序。回路是组成系统结构的基本单元，决策总是存在于回路之中并通过流率变量具体实施，流率变量调节流位变量的取值，为决策提供信息。认识系统结构的三个层次：闭合的边界，反馈回路和流率方程。系统的结构闭合的边界：闭合的边界内包含了影响系统行为的主要因素和连接关系－系统流图反馈回路：反馈环及反馈环的链接

闭合的边界和反馈回路从定性的角度描述了系统结构的整体框架。流率变量方程：决策是运用信息对系统实施控制的过程，流率变量方程描述了过程规律。

流率变量方程从定量角度描述了系统局部的子系统。目标、观察值、偏差和控制是流率变量方程的内部次级子结构系统行为1.ExponentialGrowth2.ExponentialDecay

3.GoalSeekingBehavior4.S-ShapedGrowth5.SystemOscillation系统结构和系统行为complexsystemBehaviorinteractionbetweenstockandflow

setsofstocks/flowswithinlargerfeedbackloops.

系统行为：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（时间常数）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（倍增时间）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小结要点：基本结构；Positiveloopsgenerateexponentialgrowth(firstorderpositivefeedbackloop)增长速度：CoefTheTimeConstant:TCDoubleTime:Td关系：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小结要点：结构；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讨论：GoalSeekingBehavior产生的本质原因是什么？SystemOscillationSystemOscillationOscillationreferstoabehaviorinwhichthevaluesofthestocksvaryaroundsomeaveragevalueinarepeatingpattern.SystemOscillationSustainedoscillation:the“peak“isthesameeachcycle.Expandingoscillation:the“peak“becomesgreatereachperiod.Dampedoscillation:the“peak“becomessmallereachperiod.SystemOscillationThemathematicsfunctionofoscillationStock=A•sin(?•time+?)WhereA,?,and?areconstantAcademicperformancemodel二级反馈环与系统振荡AcademicperformancemodelWhythesystemdemonstratesthebehaviorofsustainedoscillation?Thestockstheflowsgap26hoursaweekDebriefoftheacademicperformancemodelAnoscillatingsystemrequirestwostocks.Oscillatingsystemhavesomemomentum-gap.Gap---increaseheramountofstudying--….---currentgrades:desiredlevel流位变量的积累作用：延迟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