Unexpected Behaviors in Higher- Order Positive Feedback Loops

More documents

Recommendations

Info

18 D-4455-22.001: a 2: b 3: c 4: da(0)=11c(0)=0341340.0023b(0)=-141d(0)=0223412-2.000.00 3.00 6.00 9.00 12.00TimeFigure 11: Sustained oscillations within afourth-order positive feedback loop2.001: a 2: b 3: c 4: d1a(0)=1c(0)=0.0013410.002b(0)=-134d(0)=012 234-2.000.00 3.00 6.00 9.00 12.00TimeFigure 12: Sustained oscillations destabilizing to exponential growth(fourth-order positive feedback loop)
D-4455-2 193. SUMMARYAlthough positive feedback loops are usually associated with exponential growth,that is not the only behavior possible within them. Convergence and oscillation can alsobe found. We looked at four different positive feedback loops.The first order loop only behaves exponentially. All initial conditions will reinforcethe system state, and push it out to infinity (be it up or down). The only exception is thedegenerate behavior at the origin. There it stays fixed at one point, but note that thisspecial behavior, like in the other systems, is unstable. Exponential growth is still thestable behavior.The second order loop showed that exponential growth was not the onlypossibility for positive loops. While in perfect balance, the two levels settled to unstableequilibrium. Notice that the fixed behavior of the origin is still present (and unstable).The higher— order positive feedback loops contain many repeated behaviors.The third order loop brought us an interesting twist: oscillations! As in thesecond order loop, the levels had to be in a special balance, but this time their decent toequilibrium was done with a rhythmic dance.The fourth order loop, the final loop examined, went a step further than the thirdand produced stable oscillations. This result clearly demonstrated the surprising characterof positive feedback. A feedback loop that has, at each individual step, only positiveinfluences can produce behavior that is undeniably negative in appearance.We have found great behavioral diversity even within the simplest of positiveloops. The truly stable behavior in each case is exponential growth. It is stable in thatsmall perturbations do not change the behavior greatly, and yet it is unstable in that itsbehavior looks most unsettled. On the other hand, the other behaviors we have seen areunstable because small perturbations do drastically change their behavior, even thoughthey all give the appearance of stability. We can almost always find unusual unstablebehavior disguised as what truly would be stable in other models.As it happens, the behaviors we have seen here demonstrate all the possibilitiesthat models of this simple nature can exhibit. Remember that most models of the realworld are not nearly so constrained or simplified as what we have seen in this paper. Mostpositive feedback loops in real world models will show (for the most part) only
Page 1 and 2: D-4455-1 1Unexpected Behaviors in H
Page 3 and 4: D-4455-2 3Table of Contents1. ABSTR
Page 5 and 6: D-4455-2 5
Page 7 and 8: D-4455-2 71. INTRODUCTIONWe natural
Page 9 and 10: D-4455-2 9+a+x++b. . .+c+Figure 2:
Page 11 and 12: D-4455-2 115.00a(0)=2a(0)=1a(0)=00.
Page 13 and 14: D-4455-2 13many initial values on t
Page 15 and 16: D-4455-2 15The balance for converge
Page 17: D-4455-2 17Fourth-order feedback ha
Page 21: D-4455-2 215.001-10: a v . b0.00-5.
Page 25 and 26: D-4455-1 25Vensim Examples:Unexpect
Page 27 and 28: D-4455-2 27Units: **undefined**(4)
Page 29 and 30: D-4455-2 292Graph of Unstable Conve
Page 31 and 32: D-4455-2 31(06) dc=BUnits: **undefi
Page 33 and 34: D-4455-2 333.4 Fourth-Order LoopAIN
Page 35 and 36: D-4455-2 35Units: noneThe frequency

Unexpected Behaviors in Higher- Order Positive Feedback Loops

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?