Unexpected Behaviors in Higher- Order Positive Feedback Loops

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6 D-4455-2ABSTRACTPositive feedback loops typically show the behavior of exponential growth.However, simple unbounded growth is not the only behavior a single loop is able toexhibit. Other possible behaviors include: exponential decay, damped oscillation,sustained oscillation, and expanding oscillation. Each of these additional behavior modesis unstable, and each will turn to exponential growth if perturbed out of the stable mode.This paper demonstrates and explores these unexpected behaviors in simple positivefeedback loops. The loops used for experimentation are all elementary, with directpositive influences and no minor loops. The loops vary from first to fourth order.
D-4455-2 71. INTRODUCTIONWe naturally expect positive feedback to grow exponentially. It is the behavior weexperience daily. Our savings accounts gather interest, as do our debts from loans, andtypically at an alarming rate. The world population is exploding about us, and the AIDSepidemic quickly spreads within it. The level of technology doubles every few years.Unchecked nuclear explosions power the sun, which gives us the energy we live by. In allof these examples, the rule is simple: the more you’ve got, the faster you get it. Thisbehavior is a part of any system whose rate of advancement is directly dependent on thelevel of its current state— what is commonly called positive feedback. Thus we expectloop structures that have positive influences between its connections to quickly growwithout bound— exponentially.However, simple unbounded growth is not the only behavior a single loop is ableto exhibit. Other possible behaviors include: exponential decay, damped oscillation,sustained oscillation, and expanding oscillation. These are behaviors that we expect to seein negative feedback loops; they are signs of stability. As we shall see, in positivefeedback loops each of these behavior modes is unstable, and each will turn to exponentialgrowth if perturbed out of the uncharacteristic mode.To help gain a better understanding of this special kind of behavior, we willexperiment with simple loops only. Each level in a loop will have a direct positiveinfluence on the rate of the next level. There will be no other inputs to the rates. Inaddition, our investigation will start with a second-order loop, and we will increasecomplexity one level at a time. To explore the behavior of each loop, we will change theinitial conditions of the level variables, but we will not alter the model structure. Althoughchanging the structure leads to a wealth of interesting investigations, there will be nochanges to the rate equations. I recommend that the reader, while proceeding through thepaper, build each model independently and experiment with it freely.
Page 1 and 2: D-4455-1 1Unexpected Behaviors in H
Page 3 and 4: D-4455-2 3Table of Contents1. ABSTR
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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 and 18: D-4455-2 17Fourth-order feedback ha
Page 19 and 20: D-4455-2 193. SUMMARYAlthough posit
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

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