STABILITY 5/14 



system, an input-value that makes the other's state to be one of 

 equiHbrium. Then neither can change, and the whole cannot 

 change; and thus the whole must be at a state of equilibrium. 



Thus each implies the other. Formally: the whole is at a state of 

 equilibrium if and only if each part is at a state of equilibrium in the 

 conditions provided by the other part. (If there are several parts the 

 last word is merely changed to "parts".) 



5/13. Power of veto. The same thesis can be stated more vividly, 

 making it more useful conceptually. Suppose A and B are coupled 

 and suppose we are interested only in the occurrence of a state of 

 equilibrium (not of cycles). When the whole is started from some 

 initial state, and goes along some trajectory, A and B will pass through 

 various states. Suppose it happens that at some moment 5's state 

 provides conditions that make ^'s present state one of equilibrium. 

 A will not change during the next step. If B is not itself at a state of 

 equilibrium in the conditions provided by ^, it will move to a new 

 state, y^'s conditions will thereby be changed, its states of equilib- 

 rium will probably be changed, and the state it is at will probably 

 no longer be one of equilibrium. So A will start moving again. 



Picturesquely, we can say that A proposed a state of equilibrium 

 (for A was willing to stop), but B refused to accept the proposal, or 

 vetoed the state. We can thus regard each part as having, as it 

 were, a power of veto over the states of equilibrium of the whole. 

 No state {of the whole) can be a state of equilibrium unless it is accept- 

 able to every one of the component parts, each acting in the conditions 

 given by the others. 



Ex.: Three one-variable systems, with Greek-letter parameters, are: 

 x' = - X + a, y' = Ifiy + 3, z' = - yz + S. 

 Can they be coupled so as to have a state of equilibrium at (0,0,0) ? (Hint : 

 What value would jS have to have?) 



5/14. The homeostat. This principle provides a simple way of 

 looking at the homeostat and of understanding its working. It 

 can be regarded as a part A coupled to a part B (Fig. 5/14/1). 



Part A consists essentially of the four needles (with ancillary coils, 

 potentiometers, etc.) acting on one another to form a four-variable 

 system to which 5's values are input. y4's state is specified by the 

 positions of the four needles. Depending on the conditions and 

 input, A may have states of equilibrium with the needles either 

 central or at the extreme deviation. 



Part B consists essentially of a relay, which can be energised or 



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