CHAPTER 20 



Stability 



20/1. As will be seen in S. 21/14, the canonical representation 

 contains all the information that the real ' machine ' can give 

 relative to the selected system. By selecting a particular system 

 the experimenter has already acknowledged that he can obtain 

 only a finite amount of information from the infinite amount that 

 exists in the real ' machine ' ; yet even this reduction is often 

 insufficient, for the canonical representation of the behavioural 

 properties of x v ... , x n may still convey an unmanageably 

 large amount of information. Take the case, for instance, of the 

 cluster of 20,000 stars, about which the astronomer asks: will 

 the cluster condense to a ball, or will it disperse ? The canonical 

 representation can be set up (it has 120,000 variables), and it 

 contains the answer; but the labour of extracting it is so pro- 

 hibitively great that astronomers, and others in like position, 

 have looked for methods that do not use all the information 

 available in the canonical representation. Hence the introduc- 

 tion into science of statistical and topological methods, and the 

 use of concepts such as independence (S. 12/4) which may, if the 

 case is suitable, enable us to get a simple answer to a simple 

 question without the necessity for our going into every detail. 



Prominent among such concepts is that of stability. Its basic 

 elements have been given in /. to C, Chapter 5. Here we shall 

 treat it only in the form suitable for continuous systems, and only 

 with such rigour as is necessary for our main purpose. 



20/2. Given a state-determined system in unvarying conditions, 

 so that it has one field, and given a region in the field and a point in 

 the region, a line of behaviour from the point is stable, with respect 

 to that field and region and point, if it never leaves the region. 



20/3. If all the lines within a given region are stable from all 

 points within the region, and if all the lines meet at one point, 

 the system has normal stability. 



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