PARAMETERS 21/6 



null-functions into the system or remove them from it as we 

 please. 



It also follows that the statements : ' parameter a was held con- 

 stant at a \ and c the system was re-defined to include a, which, 

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

 two ways of describing the same facts. 



21/5. The fact that the field is changed by a change of parameter 

 implies that the stabilities of the lines of behaviour are changed. 

 For instance, consider the system 



dx/dt = — x -f ay, dy/dt = x — y -f 1 

 where x and y have been used for simplicity instead of x ± and x 2 . 

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

 fields shown in Figure 21/5/1. 



Figure 21/5/1 : Three fields of x and y when a has the values (left to 

 right) 0, 1, and 2. 



When a = there is a stable resting state at a? = 0, y = 1 ; 



when a = 1 there is no resting state ; 



when a = 2 there is an unstable resting state at x = — 2, 



y = -l. 

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



21/6. 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 group equations. 



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 

 ' joining ' two systems. A better method is to equate para- 

 meters in one system to variables in the other. When this is 



229 q 



