STABILITY 5/11 



Ex. 1 : A bus service starts with its buses equally spaced along the route. If a 

 bus is delayed, extra passengers collect at the stopping points, so it has to 

 take up, and set down, more passengers than usual. The bus that follows 

 it, being closer than usual, has fewer passengers to handle and is delayed 

 less than usual. Are irregularities of spacing self-correcting or self- 

 aggravating ? 



Ex. 8 : What would happen if an increase of carbon dioxide in the blood made 

 the respiratory centre less active? 



Ex. 9: Is the system x' = ^y, y' = \x stable around (0,0)? 



5/10. Positive feedback. The system described in the last exercise 

 deserves closer attention. 



From (10,10) it goes to (5,5) 

 „ (10,12),, „ ,,(6,5); 



so an increase in y (from 10 to 12) leads to an increase in .v (from 

 5 to 6). (Compare S.4/13.) Similarly, 



from (10,10) it goes to (5,5) 

 „ (12,10),, „ ,,(5,6) 



so an increase in x (from 10 to 12) leads to an increase in v (from 

 5 to 6). Each variable is thus having a positive effect on the other 

 and if the system were discussed in plain words these facts might be 

 used to "prove" that it is unstable, for a vicious circle seems to be 

 acting. 



The system's behaviour, by converging back to (0,0), declares 

 indisputably that the system is stable around this state of equilibrium. 

 It shows clearly that arguments based on some short cut, e.g. by 

 showing that the feedback is positive, may not be rehable. (It 

 shows also that feedback can be positive and yet leave the system 

 stable; yet another example of how unsuitable is the concept of 

 feedback outside its particular range of applicability.) 



5/11. Undesirable stability. Stability is commonly thought of as 

 desirable, for its presence enables the system to combine something 

 of flexibility and activity in performance with something of per- 

 manence. Behaviour that is goal-seeking is an example of behaviour 

 that is stable around a state of equilibrium. Nevertheless, stability 

 is not always good, for a system may persist in returning to some 

 state that, for other reasons, is considered undesirable. Once petrol 

 is lit it stays in the lit state, returning to it after disturbance has 

 changed it to "half-lit" — a highly undesirable stabihty to a fireman. 

 Another example is given by the suggestion that as the more 

 intelligent members of the community are not reproducing their 



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