STABILITY 5/9 



5/8. In general, the results of repeated application of a trans- 

 formation to a state depend on what that state is. The outcome of 

 the test of finding what is 



lim T"(x) 



n—> CO 



will thus depend in general on which state is .r. Thus if there are 

 two disturbances available, D and E, and D takes a to b, while E 

 takes a to c (no order being implied between a, b and c) the Hmits 

 of T"D{a) and T"E(a) may be different. 



Thus the result of a test for stability, carried out in the manner 

 of S.5/6, may give different results according to whether the displace- 

 ment is D or E. The distinction is by no means physically un- 

 reasonable. Thus a pencil, balanced on its square-cut base, may 

 be stable to D, if Z) is a displacement of 1° from the vertical, but 

 may be unstable to E, if £" is a displacement of 5°. 



The representation given in S.5/6 thus accords with common 

 practice. A system can be said to be in stable equilibrium only if 

 some sufficiently definite set of displacements D is specified. If the 

 specification is explicit, then D is fully defined. Often D is not 

 given explicitly but is understood; thus if a radio circuit is said to 

 be "stable", one understands that D means any of the commonly 

 occurring voltage fluctuations, but it would usually be understood 

 to exclude the stroke of lightning. Often the system is understood 

 to be stable provided the disturbance hes within a certain range. 

 What is important here is that in unusual cases, in biological systems 

 for instance, precise specification of the disturbances D, and of the 

 state of equilibrium under discussion a, may be necessary if the dis- 

 cussion is to have exactness. 



5/9. The continuous system. In the previous sections, the states 

 considered were usually arbitrary. Real systems, however, often 

 show some continuity, so that the states have the natural relationship 

 amongst themselves (quite apart from any transformation imposed 

 by their belonging to a transducer) that two states can be "near" or 

 "far from" one another. 



With such systems, and a state of equilibrium a, D is usually 

 defined to be a displacement, from a, to one of the states "near" a. 

 If the states are defined by vectors with numerical components, i.e. 

 based on measurements, then D often has the effect of adding small 

 numerical quantities Si, §2^ • ■ ■, S,„ to the components, so that the 

 vector (xi, . . ., .y„) becomes the vector (.Yj + S,, . . ., .y„ + 8„). 



In this form, more specialised tests for stability become possible. 

 An introduction to the subject has been given in Design .... The 



79 



