THE ERROR-CONTROLLED REGULATOR 



12/11 



formation is not single-valued, more than one arrow can go from 

 each state. Thus the Markovian machine 



has the graph of Fig. 12/11/1, in which each arrow has a fraction 

 indicating the probability that that arrow will be traversed by the 

 representative point. 



Fig. 12/11/1 



In this particular example it can be seen that systems at c will all 

 sooner or later leave it, never to return. 



A Markovian machine has various forms of stability, which 

 correspond to those mentioned in Chapter 5. The stable region is 

 a set of states such that once the representative point has entered a 

 state in the set it can never leave the set. Thus a and b above form 

 a stable region. 



A state of equilibrium is simply the region shrunk to a single 

 state. Just as, in the determinate system, all machines started in a 

 basin will come to a state of equilibrium, if one exists, so too 

 do the Markovian; and the state of equilibrium is sometimes 

 called an absorbing state. The example of S.9/4 had no state of 

 equilibrium. It would have acquired one had we added the fourth 

 position "on a jBy-paper", whence the name. 



Around a state of equilibrium, the behaviour of a Markovian 

 machine differs clearly from that of a determinate. If the system 

 has a finite number of states, then if it is on a trajectory leading to a 

 state of equiUbrium, any individual determinate system must arrive 

 at the state of equilibrium after traversing a particular trajectory and 

 therefore after an exact number of steps. Thus, in the first graph 



229 



