STABILITY 5/15 



5/15. The complex of ideas involved in "stability" can now be 

 summarised. 



First there is the state of equilibrium — the state that is unchanged 

 by the transformation. Then the state may become multiple, and 

 we get the stable set of states, of which the cycle and basin are 

 examples. 



Given such a state or set of states and some particular disturbance 

 we can ask whether, after a disturbance, the system will return to its 

 initial region. And if the system is continuous, we can ask whether 

 it is stable against all disturbances within a certain range of values. 



Clearly, the concept of stabiUty is essentially a compound one. 

 Only when every aspect of it has been specified can it be apphed 

 unambiguously to a particular case. Then if its use calls for so 

 much care, why should it be used at all? Its advantage is that, in 

 the suitable case, it can sum up various more or less intricate 

 possibilities briefly. As shorthand, when the phenomena are 

 suitably simple, words such as equilibrium and stability are of 

 great value and convenience. Nevertheless, it should be always 

 borne in mind that they are mere shorthand, and that the phenomena 

 will not always have the simplicity that these words presuppose. At 

 all times the user should be prepared to delete them and to substitute 

 the acutal facts, in terms of states and transformations and 

 trajectories, to which they refer. 



It is of interest to notice, to anticipate S.6/19, that the attempt to 

 say what is significant about a system by a reference to its stability 

 is an example of the "topological" method for describing a large 

 system. The question "what will this system do?", applied to, say, 

 an economic system, may require a full description of every detail 

 of its future behaviour, but it may be adequately answered by the 

 much simpler statement "It will return to its usual state" (or perhaps 

 "it will show ever increasing divergence"). Thus our treatment in 

 this chapter has been of the type required when dealing with the 

 very large system. 



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