21/10 PARAMETERS 



then a two-way interaction occurs. For instance, suppose we 

 start with the 2-variable system 



dx/dt = f 1 (x, y; a)' 

 dy/dt = f 2 {x, y) 



then the diagram of immediate effects is 



and the 1 -variable system dz/dt = <f>(z; b) 



> z 



If we make a some function of z, a = z say for simplicity, the new 

 system has the equations 



dx/dt =f 1 (x, y; zf\ 

 dy/dt =f 2 (x, y) I 

 dz/dt = <f>(z; b) J 



and the diagram of immediate effects becomes 



If a further join is made by putting b = y, the equations become 



dx/dt =f 1 (x, y; z) 

 dy/dt =f 2 {x, y) 

 dz/dt = 4(z; y) 



and the diagram of immediate effects becomes 



< y 



E 7 



In this method each linkage uses up one parameter. This is 

 reasonable; for the parameter used by the other system might 

 have been used by the experimenter for arbitrary control. So 

 the method simply exchanges the experimenter for another system. 



This method of joining does no violence to each system's 

 internal activities: these proceed as before except as modified by 

 the actions coming in through the variables which were once 

 parameters. 



21/10. Theorem: The whole made by joining parts is richer in 

 ways of behaving than the system obtained by leaving the parts 

 isolated. 



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