7. STATISTICAL PROPERTIES OF THE EPIGENETIC SYSTEM 



125 



behave identically. However, the condition is not sufficient, for it could equally 

 well imply a stable antiphase relation between the oscillators, or even some 

 other unfamiliar stable relationship between them. Since we are looking at 

 mean frequencies of zeros relative to a fixed reference line, i^ = 0, it is possible, 

 for example, that the oscillatory frequency of one variable is a fixed multiple of 

 the other. This last possibility could be investigated by the use of the mean 

 frequency functions directly, defined by 



O) 



%xi-v) = 



and 



CO 



%X2-V) = 



j3^e-i3(i//i//i..)v' 



CO 



J 



-"dt 



00 



(^hnyi^Z,^,^ 



I 



„-<» 



dt 



(0/lll)l'2[-pi+(/'l2/All)''] 



(78) 



which we obtain from equation (71) and the analogous expression for the 

 variable X2. Now since 



hn = 



knkzioc^ 



hii = 



^22^12a^ 



the condition (75) is simply h^ = hn- 



If under this constraint the oscillators are locked in synchrony or have a 

 stable antiphase relationship, then it should be true that 



a)\xi — v) = a)%X2 — v) 



(79) 



This would not be true, however, if the frequency of one variable was a multiple 

 of the other, in which case we would have 



oi'^{xi — v) = roi%X2 — v) 



r a rational fraction. 



For small /S and with /in = /^iz we get from (78) 



(xi'^{Xi — v) \xi 



where 



00 



(80) 



(81) 



as in equation (70). 



We now face a difficulty, for in order to evaluate these integrals in the limit 

 of small /3, for example, it is necessary to find a Fourier transform for a rather 

 unusual function and then determine an integral which has not yet yielded to 

 a closed expression in terms of which to study the parametric constraints 



