《系统动力学》PPT课件.ppt

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1、系 统 动 力 学(System Dynamics),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

2、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)TOTAL BUYERS=100Units:person(3)buy rate=FRACTION*potential buyersUnits:person/Month(4)Buyers=INTEG(buy rate,1)Units:person(5)potential buyers=TOTAL BUY

3、ERS-BuyersUnits:person,FINAL TIME:60TIME STEP: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,FINAL TIME:60TIME STEP:0.5Units:Month,FRACTION=0.06 dmnl/MonthTOTAL BUYERS=100 person,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,FINAL TIME:60TIME STEP:0.5Units:Month,FRACTION=0.1 dmnl/MonthTOTAL BUYERS=100 person,一阶负反馈系统的

4、行为是单调地趋于目标,一阶反馈系统的复杂性,FINAL TIME:60TIME STEP:0.5Units:Month,TOTAL BUYERS=100 person,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

5、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)TOTAL BUYERS=100Units:person(3)buy rate=FRACTION*Buyers*possible buyersUnits:person/Month(4)Buyers=INTEG(buy rate,1)Units:person(5)possible buyers=TOTAL BUYERS-BuyersUnits:person,FINAL TIME

6、:60TIME STEP: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,FINAL TIME:60TIME STEP:0.5Units:Month,FRACTION=0.002 dmnl/(Month*person),TOTAL BUYERS=100 person,一阶反馈系统的复杂性 新假设,FINAL TIME:60TIME STEP:0.5Units:Month,FRACTION=0.001FRACTION=0.002FRACTION=0.004,TOTAL BUYERS=100 person,一阶反馈系统的复

7、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buy rate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buy rate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

8、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,buy rate,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

9、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

10、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

11、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

12、等几种形式，而是它们的复杂组合。,Second-Order Systems on How Two Lovers Interact,Romeo and Juliet by William Shakespeare,Gone with the Windby Margaret Mitchell,The Red and the Blackby Stendhal,Second-Order Systems on How Two Lovers Interact,Different people have different personalities that dictate(指示)how they will

13、 respond to their partnerslove.Some people love their partners more the more their partners love them.They get discouraged and give up if their partners do not love them.Other people,on the other hand,get annoyed if their lovers love them too much.They are attracted to people who are not attracted t

14、o them.This paper will match together lovers with different personalities and observe how their relationships unfold.Each pair will be represented by a different set of parameters in the model.By looking at the dynamic behavior generated by matching different personalities,that is,by simulating the

15、model with different combinations of parameters,this paper will study the behavior of simple second-order systems.,Romeo and Juliet,Romeo and Juliet are madly in love with each other.With each secret meeting,Romeos love for Juliet grows.Because he loves her,he does everything he can to impress her.J

16、uliet is flattered by his attention and,in return,her love for Romeo also grows.Because Romeo senses that Juliet loves him,he allows his passion to soar(骤升,升腾).,Romeo and Juliet by William Shakespeare,The Model for Romeo and Juliet,In the model each stock represents the accumulation of one lovers lo

17、ve for another.The rate at which Romeos love changes depends on the extent of Juliets love and Romeos personality.His personality is represented by a parameter that measures how he reacts to her love.Likewise,the change in Juliets love equals the amount of Romeos love times the extent to which she r

18、eacts to his love.,Measurement for Love,The love that Romeo and Juliet feel for each other can be quantified in several ways.Love can be measured in terms of kisses per day,or tender thoughts per hour,for example.The units of the stocks of love in the models are based upon a scale of“love units.”If

19、the two protagonists are indifferent(不感兴趣)to each other,they feel zero love units.Positive love units represent how much they like each other,while negative love units represent how much they dislike each other.Right now,assume that initially Romeo and Juliet feel kindly disposed towards each other.

20、They each feel one unit of love for the other.(Scenario 1),Parameter for Personality,The parameter“JULIETS REACTION”represents how Juliet reacts to Romeos love,that is,to what extent Romeos love changes Juliets love over time.Assume that every unit of Romeos love increases Juliets love by one unit e

21、ach day.Therefore Juliets reaction to Romeos love is one love unit per love unit per day,or one per day.JULIETS REACTION=1 Units:1/dayRomeo has the same personality as Juliet,so Romeos reaction to Juliets love is also one per day.ROMEOS REACTION=1 Units:1/day,Equations,(01)change in Juliets love=Rom

22、eos Love for Juliet*JULIETS REACTION Units:love unit/day(02)change in Romeos love=Juliets Love for Romeo*ROMEOS REACTION Units:love unit/day(03)Juliets Love for Romeo=INTEG(change in Juliets love,1)Units:love unit(04)JULIETS REACTION=1 Units:1/day(05)Romeos Love for Juliet=INTEG(change in Romeos lov

23、e,1)Units:love unit(06)ROMEOS REACTION=1 Units:1/dayFINAL TIME=12 dayINITIAL TIME=0 dayTIME STEP=1 day,Their love grow together exponentially!,Romeos love and Juliets love grow together exponentially.By the end of 12 days,they each feel 4,096 love units!In other words,they like each other 4,096 time

24、s more than they did initially.,Positive feedback loop,A positive feedback loop joins the two stocks.An increase in Juliets love for Romeo increases the change in Romeos love,which increases Romeos love for Juliet.The increase in Romeos love leads to an increase in the change in Juliets love,further

25、 increasing Juliets love for Romeo.With time,the stocks reinforce each other and increase exponentially.,Initially Romeo loved Juliet twice,What would happen if initially Romeo loved Juliet twice as much as she loved him?(Scenario 2)Juliets Love for Romeo=INTEG(change in Juliets love,1)Romeos Love f

26、or Juliet=INTEG(change in Romeos love,2)Will Juliets love ever grow to equal Romeos love?,Scenario 2 vs.Scenario 1,Juliets love quickly catches up with Romeos love.The change in Juliets love is proportional to Romeos love.If Romeo initially loves Juliet twice as much,then Juliets love increases twic

27、e as much as it would have increased if initially he only felt one unit of love.The additional increase precipitates(促进，加速)the exponential growth of their love,so after 12 days they love each other even more than in the first scenario.,Scenario 1,Scenario 2,Scenario 3,Consider the scenario in which

28、Juliet is entirely indifferent to Romeo.Romeo,however,catches a glimpse of Juliet at a party and finds her really attractive.In other words,Juliet initially feels zero love units for Romeo,but Romeo feels one love unit for Juliet.（Scenario）Juliets Love for Romeo=INTEG(change in Juliets love,0)Romeos

29、 Love for Juliet=INTEG(change in Romeos love,1),Initially Romeo likes an indifferent Juliet,Again,both stocks grow together exponentially.Because of Juliets initial indifference,however,on the twelfth day,their love is“only”2,048 times as strong as Romeos initial reaction to Juliet.If Juliet had ini

30、tially responded in the same way as Romeo,as in the first scenario,their love would have been twice as strong.To summarize,if just one of the two lovers feels some positive emotion for the other,the love of both lovers will feed off of each other,and both stocks will grow exponentially.,4,000,3,000,

31、2,000,1,000,0,0,3,6,9,12,Time(day),Scenario 4,Now imagine a scenario in which Romeo and Juliet are so influenced by the animosity(仇恨)between their families that when they first meet,each feels some instinctive dislike for the other.(Scenario 4)Juliets Love for Romeo=INTEG(change in Juliets love,-1)R

32、omeos Love for Juliet=INTEG(change in Romeos love,-1)What will happen to them?,Initially Romeo and Juliet dislike each other,It shows how Romeo and Juliets dislike for each other experiences exponential decline and turns into adamant(坚硬的)hatred.Juliet dislikes Romeo more because he dislikes her.He d

33、islikes her more because she dislikes him.The model now produces exponential decline towards an infinitely large negative value.,Scenario 5,Now imagine a scenario in which,although Romeo was initially enchanted(迷惑)by Juliet,she felt an initial dislike for him.(Scenario 5)Juliets Love for Romeo=INTEG

34、(change in Juliets love,-1)Romeos Love for Juliet=INTEG(change in Romeos love,1)What will happen to their love?,Initially Juliet and Romeo have opposite reactions to each other,On a first glance,the behavior is very surprising.Juliets dislike for Romeo decreases his love for her.As long as her love

35、for him is negative,his love for her decreases.While his love for her is positive,however,her love for him increases,becoming less negative.As Juliets love becomes less negative,Romeos love still decreases,but by less.Eventually,they stabilize at total indifference towards each other.,Scenario 6,If

36、Romeo and Juliet are initially entirely indifferent to each other,then the system will remain at equilibrium.Juliets Love for Romeo=INTEG(change in Juliets love,-1)Romeos Love for Juliet=INTEG(change in Romeos love,1),Initially Juliet and Romeo are indifferent to each other,Because neither person lo

37、ves the other,the rates of change are also zero,and the system never leaves the equilibrium state.Romeo and Juliet never click,so nothing happens.,Initially Romeo feels 1/100 of a love unit for an indifferent Juliet,The equilibrium that they reach,however,is very unstable.If,due to any external reas

38、on,either Romeo or Juliet stops feeling indifferent to the other,their love will once again grow or decline exponentially.What will happen if Juliet is indifferent to Romeo but Romeo feels a very slight inclinationa hundredth of a love unitfor Juliet?Any perturbation,that is,any change to either Rom

39、eo or Juliets love,will cause both stocks to grow or decline exponentially.,The Red and the Black,The two protagonists(主角)in The Red and the Black,Julien and Claire,have a love-hate relationship.Claire is attracted to Julien when he courts her friends and shows no interest in her.When Julien begins

40、to love her,he becomes pathetic(可怜的)in her eyes,and she develops a disgust(厌恶)for him.Julien is likewise attracted to Claire only when she gives him a cold shoulder and treats him with contempt.Then she becomes the prize that his honor mandates that he catch.What will happen when these two character

41、s meet?,The Red and the Blackby Stendhal,Gone with the Wind,Scarlet always wants what she cannot have.When Rhett is indifferent to her,she is attracted to him.When Rhett is attracted to her,she scorns(嘲笑)him.Rhett,however,becomes frustrated when Scarlet treats him with contempt.He is attracted to her whenshe is attracted to him,but he loses interest when she snubs(冷落)him.What will happen when these two characters meet?,Gone with the Windby Margaret Mitchell,