DESIGN FOR A BRAIN 



19/10 



Example 1: The state-determined system of S. 19/8, treated in 

 this way, yields the differential equations, in canonical form: 



da\ 

 lit 

 dx 2 _ | 

 dt - 2 J 



The second system may not be treated in this way as it is not 

 state-determined and the group property does not hold. 

 Corollary: 



d 



fi(x v 



x n ) 



di Fi(Xli 



««; 



(t = i f 



n) 



*=o 



= 2 



(2) Given the differential equations, they may be written 



dx t =fi{x v . . . , x n ).dt (i = 1, . . . , n) 



and this shows that a given set of values of x v . . . , x n , i.e. 

 a given state of the system, specifies completely what change, 

 dx iy will occur in each variable, x { , during the next time-interval, 

 dt. By integration this defines the line of behaviour from that 

 state. The system is therefore state-determined. 

 Example 2 : By integrating 



dx x 

 ~dt 

 dx 2 

 ~dt 



the group equations of the example of S. 19/8 are regained. 



19/10. Definition. The system is linear when the functions 

 fv • • • 9 fn are a N nnear functions of the arguments x v . . . , x n . 



19/11. Example 3: The equations of the Homeostat may be 

 obtained thus : — If x t is the angle of deviation of the i-th magnet 

 from its central position, the forces acting on x t are the momentum, 

 proportional to x i3 the friction, also proportional to x { , and the 

 four currents in the coil, proportional to x v x 2 , x z and x A . If 

 linearity is assumed, and if all four units are construct ionally 

 identical, we have 



dt 



(mx t ) 



kx { + l(p - qftoiM + . . . + a u x A ) 



{i = 1, 2, 3, 4) 



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