CHAPTER 21 



Parameters 



21/1. In the previous two chapters we have considered the 

 state-determiried system when it was isolated, with constant 

 condition? around it, or when no change came to its input. We 

 now turn to consider the state-determined system when it is 

 affected by changes in the conditions around it, when it is no 

 longer isolated, or when changes come to its input. We turn, in 

 other words, to consider the ' machine with input ' of /. to C, 

 Chapter 4. 



Experience has shown that this change corresponds to the 

 introduction of parameters into the canonical representations so 

 that they become of the form 



dxi/dt =fi{x v . . . , x n \ a ls a 2 , . . .) (i = 1, . . . , n) 



21/2. If the «'s are fixed at particular values the result is to make 

 the/'s a particular set of functions of the #'s and thus to specify 

 a particular state-determined system. From this it follows that 

 each particular set of values at the a's specifies a particular field. 

 In other words, the two sets: (1) the values at the a's and (2) the 

 fields that the system can show can be set in correspondence 

 — perhaps the most fundamental fact in the whole of this book. 

 (Figure 21/8/1 will illustrate it.) 



It should be noticed that the correspondence is not one-one but 



Figure 21/2/1. 

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