DESIGN FOR A BRAIN 



21/8 



as a null-function, remained at its initial value of a ' are merely 

 two ways of describing the same facts. 



21/8. The fact that the field is changed by a change of parameter 

 implies that the stabilities of the lines of behaviour may be 

 changed. For instance, consider the system 



dxjdt = — x t -f- «#2> dxjdt = x± — x 2 + 1. 



When a = 0, 1, and 2 respectively, the system has the three 

 fields shown in Figure 21/8/1. 



Figure 21 /8/1 : Three fields of x x and x 2 when a has the values (left to 

 right) 0, 1, and 2. 



When a = there is a stable state of equilibrium at x 1 = 0, 

 x 2 = 1 ; when a = 1 there is no state of equilibrium ; when a = 2 

 there is an unstable state of equilibrium at x 1 = — 2, x 2 = — 1. 

 The system has as many fields as there are values to a. 



Joining systems 



21/9. (Again the basic concepts have been described in /. to C, 

 S. 4/7; here we will describe the theory in continuous systems.) 



The simple physical act of joining two machines has, of course, 

 a counterpart in the equations, shown more simply in the canonical 

 than in the equations of S. 19/7. 



One could, of course, simply write down equations in all the 

 variables and then simply let some parameter a have one value 

 when the parts are joined and another when they are separated. 

 This method, however, gives no insight into the real events in 

 4 joining ' two systems. A better method is to make the para- 

 meters of one system into defined functions of the variables of the 

 other. When this is done, the second dominates the first. If 

 parameters in each are made functions of variables in the other, 



266 



