CHAPTER 20 



Stability 



20/1. ' Stability ' is defined primarily as a relation between a 

 line of behaviour and a region in phase-space because only in 

 this way can we get a test that is unambiguous in all possible 

 cases. Given an absolute system and a region within its field, 

 a line of behaviour from a point within the region is stable if it 

 never leaves the region. 



20/2. 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. 



20/3. A resting state can be defined in several ways. In the 

 field it is a terminating point of a line of behaviour. In the 

 group equations of S. 19/10 the resting state X v . . . , X n is 

 given by the equations 



Xi = Lim Fi(x° ; t) [i = 1, . . . , n) . (1) 



t >-00 



if the n limits exist. In the canonical equations the values satisfy 



fi(X l3 . . ., X n )=0 (t = l n) . (2) 



A resting state is an invariant of the group, for a change of t 

 does not alter its value. 



m 



dxj 



be symbolised by J, is not identically zero, then there will be 

 isolated resting states. If J = 0, but not all its first minors are 

 zero, then the equations define a curve, every point of which 

 is a resting state. If J = and all first minors but not all second 

 minors are zero, then a two-way surface exists composed of 

 resting states ; and so on. 



If the Jacobian of the /'s, i.e. the determinant 



which will 



20/4. Theorem : If the /'s are continuous and differentiable, 

 an absolute system tends to the linear form (S. 19/27) in the 

 neighbourhood of a resting state. 



216 



