STABILITY 5/6 



where it was pointed out that only when the set is stable can the 

 transformation proceed to all its higher powers unrestrictedly. 



Another reason is discussed more fully in S.10/4, where it is 

 shown that such stabiUty is intimately related to the idea of some 

 entity "surviving" some operation. 



Ex. 1 : What other sets are stable with respect to T1 



Ex. 2 : Is the set of states in a basin always stable ? 



Ex. 3 : Is the set of states in a cycle always stable ? 



Ex. 4: If a set of states is stable under T, and also under U, is it necessarily 

 stable under UTl 



DISTURBANCE 



5/6. In the cases considered so far, the equilibrium or stability 

 has been examined only at the particular state or states concerned. 

 Nothing has been said, or implied, about the behaviour at neighbour- 

 ing states. 



The elementary examples of equilibrium — a cube resting on its 

 face, a bilhard ball on a table, and a cone exactly balanced on its 

 point — all show a state that is one of equilibrium. Yet the cone is 

 obviously different, and in an important way, from the cube. The 

 difference is shown as soon as the two systems are displaced by 

 disturbance from their states of equilibrium to a neighbouring state. 

 How is this displacement, and its outcome, to be represented 

 generally ? 



A "disturbance" is simply that which displaces, that which 

 moves a system from one state to another. So, if defined accurately, 

 it will be represented by a transformation having the system's states 

 as operands. Suppose now that our dynamic system has transforma- 

 tion T, that a is a state of equilibrium under T, and that D is a given 

 displacement-operator. In plain English we say: "Displace the 

 system from its state of equilibrium and then let the system follow 

 its own laws for some time and see whether the system does or does 

 not come back to the same state". In algebraic form, we start 

 with a state of equilibrium a, displace the system to state D{a), and 

 then find TD{a), T~D(a), T^D{a), and so on; and we notice whether 

 this succession of states does or does not finish a.s a, a, a, .... 

 More compactly: the state of equilibrium a in the system with 

 transformation Tis stable under displacement D if and only if 



lim T"D(a) = a. 



n—>- CO 



Try this formulation with the three standard examples. With 

 the cube, a is the state with angle of tilt = 0". D displaces this 



77 



