4/9 AN INTRODUCTION TO CYBERNETICS 



4/9. Algebraic coupling. The process of the previous sections, 

 by treating the changes that each state and parameter undergo 

 individually, shows the relations that are involved in "coupling" 

 with perfect clarity and generality. Various modifications can 

 be developed without any loss of this clarity. 



Thus suppose the machines are specified, as is common, in terms 

 of vectors with numerical components; then the rule for couphng 

 remains unaltered : each machine must have one or more parameters, 

 and the coupling is done by specifying what function these parameters 

 are to be of the other machine's variables. Thus the machines 

 Afand A'^ 



j^, J u — ^ -\- pb C c' = rsc -{■ ud^ 



' \b' = — qa N: < 



e = uce 



qa N: < d' = 2tue 



might be joined by the transformations U and V: 



r = a -{- b 





"■■<"--% 



s = a — b 

 t = — a 



u = b'~ 



1/ is a shorthand way of writing a whole set of transitions from a 

 value of (c,d,e) to a value of (p,q), e.g. 



jj. I (0,0,0) (0,0,1) (1,3,5) (2,2,4) 

 "^ • ^ (0,0) (0,0) (2,75) (4,32) 



Similarly for V, a transformation from (a,b) to {r,s,t,u), which 

 includes, e.g. (5,7) -> (12,-2,-5,49) (and compare P of S.6/9). 



The result of the coupling is the five-variable system with repre- 

 sentation: 



a' = fl2 + 2bc 



b' = — ade'^ 



c' = (a2 _ b^)c + /,2^2 



d'= - lab^e 

 e' = bh-e 



(Illustrations of the same process with differential equations have 

 been given in Design for a Brain, S.21/6.) 



Ex. 1. : Which are the parameters in M? Which in A'^? 

 Ex. 2.: Join M and N hy W and X, and find what state (1, 0, 0, 1, 0), a 

 value of (a, b, c, d, e), will change to : 



{r= 



H^: •< '^ ~ 'j. 



52 



