4/10 



STABILITY 



can leave the region; so the region is stable. On the other hand, 

 no such region can be marked around B (unless restricted to the 

 single point of B itself). 



The definition makes clear that change of either the field or the 

 region may change the result of the test. We cannot, in general, 

 say of a given system that it is stable (or unstable) unconditionally. 

 The field of Figure 4/5/1 showed this, and so does that of Figure 

 4/5/2. (In the latter, the regions restricted to any part of the 

 ?/-axis with the origin are stable; all others are unstable.) 



The examples above have been selected to test the definition 

 severely. Often the fields are simpler. In the field of the cube, 

 for instance, it is possible to draw many boundaries, all.oval, such 

 that the regions inside them are stable. The field of the Watt's 

 governor is also of this type. 



A field will be said to be stable if the whole region it fills is 

 stable ; the system that provided the field can then be called stable. 



4/9. Sometimes the conditions are even simpler. The system 

 may have only one state of equilibrium and the lines of behaviour 

 may all either converge in to it or all diverge from it. In such a 

 case the indication of which way the lines go may be given suffi- 

 ciently by the simple, unqualified, statement that ' it is stable ' 

 (or not). A system can be described adequately by such an 

 unqualified statement (without reference to the region) only when 

 its field, .i.e. its behaviour, is suitably simple. 



4/10. If a line of behaviour is re-entrant to itself, the system 

 undergoes a recurrent cycle. If 

 the cycle is wholly contained in a 

 given region, and the lines of be- 

 haviour lead into the cycle, the 

 cycle is stable. 



Such a cycle is commonly shown, 

 by thermostats which, after correct- 

 ing any gross displacement, settle 

 down to a steady oscillation. In 

 such a case the field will show, 

 not convergence to a point but 

 convergence to a cycle, such as 

 is shown exaggerated in Figure 4/10/1, 



49 



Figure 4/10/1. 



