CHANGE 2/17 



When the transformation becomes more complex an important 

 feature begins to show. Thus suppose the transformation is 



, ABCDEFGHIJKLMNPQ 

 ■^ D H D I QGQHAEENBA N E 



Its kinematic graph is: 



P C M~^B^H 



\ / 



N->A -^D K 



L I Ez^Q^G^F 



/ 

 J 



By starting at any state and following the chain of arrows we can 

 verify that, under repeated transformation, the representative point 

 always moves either to some state at which it stops, or to some cycle 

 around which it circulates indefinitely. Such a graph is like a map 

 of a country's water drainage, showing, if a drop of water or a 

 representative point starts at any place, to what region it will come 

 eventually. These separate regions are the graph's basins. These 

 matters obviously have some relation to what is meant by "stability", 

 to which we shall come in Chapter 5. 



Ex. 1 : Draw the kinematic graphs of the transformations of A and B in Ex. 2/4/3. 



Ex. 2: How can the graph of an identical transformation be recognised at a 

 glance ? 



Ex. 3 : Draw the graphs of some simple closed one-one transformations. What 

 is their characteristic feature ? 



Ex. 4: Draw the graph of the transformation Fin which n' is the third decimal 

 digit of logio(A? + 20) and the operands are the ten digits 0, 1, . . ., 9. 



Ex. 5: (Continued). From the graph of Kread off at once what is F(8), F2(4), 

 F4(6), F84(5). 



Ex. 6: If the transformation is one-one, can two arrows come to a single point? 



Ex. 1 : If the transformation is many-one, can two arrows come to a single point ? 



Ex. 8: Form some closed single-valued transformations like T, draw their 

 kinematic graphs, and notice their characteristic features. 



Ex. 9: If the transformation is single-valued, can one basin contain two cycles? 



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