5/7 AN INTRODUCTION TO CYBERNETICS 



to, say, 5°; and T eventually will bring this back to 0°. With the 

 cone (having transformation U, say) D can be the same displace- 

 ment, but the limit, whatever it is, of U"D{a) is certainly not a tilt 

 of 0°; the equiHbrium is unstable. With the bilhard ball, at position 

 a, the dynamic laws will not bring it back to a after displacement, so 

 it is not stable by the definition given here. It has the peculiarity, 

 however, that the limit is D{a); i.e. it retains the displacement, 

 neither annulling it nor exaggerating it. This is the case of neutral 

 equilibrium. 



(It will be noticed that this study of what happens after the system 

 has been displaced from a is worth making only if a is a state of 

 equiHbrium.) 



Ex. 1: Is the state of equilibrium c stable to T under the displacement Z) if T 

 and D are given by: 



a b c d e 



T c d c a e 

 D h a d e d 



Ex. 2: (Continued.) What if the state of equilibrium is e? 

 Ex. 3: The region composed of the set of states b, c and d is stable under U: 



, a b c d e f 



U d c b b c a 

 E b e f f f d 



What is the effect of displacement E, followed by repeated action of t/? 

 (Hint: Consider all three possibilities.) 



5/7. When the dynamic system can vary continuously, small 

 disturbances are, in practice, usually acting on it incessantly. 

 Electronic systems are disturbed by thermal agitation, mechanical 

 systems by vibration, and biological systems by a host of minor 

 disturbances. For this reason the only states of equilibrium that 

 can, in practice, persist are those that are stable in the sense of the 

 previous section. States of unstable equilibrium are of small 

 practical importance in the continuous system (though they may be 

 of importance in the system that can change only by a discrete 

 jump). 



The concept of unstable equiHbrium is, however, of some theoreti- 

 cal importance. For if we are working with the theory of some 

 mechanism, the algebraic manipulations (S.5/3) will give us all the 

 states of equiHbrium — stable, neutral, and unstable — and a good deal 

 of elimination may be necessary if this set is to be reduced to the set 

 of those states that have a real chance of persistence. 



Ex. : Make up a transformation with two states of equilibrium, a and b, and two 

 disturbances, D and E, so that a is stable to D but not to E, and b is stable 

 to E but not to D. 



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