7. STATISTICAL PROPERTIES OF THE EPIGENETIC SYSTEM 



127 



Since in equations (60) and (61) we have expressions for Z^,,,, in the Hmits 

 at large and small |8, we can study the relations (83) in these limits of the talandic 

 temperature. When 6 is very small, we have 



77 



TT 



'Pi /'.• 



|8|//r2 ^Vifhih22-hn) 



whence 



-2 h 



22 



2|//|i8 





-Vl X2 



2\H\^ 



-Ihi 



2^\H\\ 



(85) 



The relation x] = xl gives us /jn = h^, but it is clear that the other relation 

 cannot be satisfied, since all the parameters are positive quantities. Therefore 

 when d is very small, entrainment between the oscillators cannot occur. This 

 is just what we expect, for in the limit of small 9 we have seen that the oscillators 

 are effectively linear and entrainment is a non-linear phenomenon. Another 

 conclusion from this result is that the condition h^ = hzz is not a sufficient 

 condition for entrainment, although it may define some other stable and sym- 

 metrical relation between the two variables .Vi and a-2. We cannot yet say what 

 this is likely to be. 



When the talandic temperature is large the non-hnearities in the system 

 are very marked and we may expect to find a definite possibility of entrainment. 

 In the limit (/3 very small) we use equation (61), viz. 



"pipt 



The quantities (84) are now 



—5 



.Y1X2 



1? 



■^2 



1_ 



1 



2j8|/fli/2/ 



1 



tan-i 





2m\ 



1/2 



f h 



22 



«12 



|//|l/2 



1 



h II tan 



-1 



H 



1/2' 



hi: 



'12 



tan 



//I/2 |H|l/2 



hi2 

 fhi hi 



2^3 1 // 1 1/2 I // 1 1/2 



/;97tan~i 



1/2 



H 



hi2 ) 



(86) 



