7. STATISTICAL PROPERTIES OF THE EPIGENETIC SYSTEM 1 29 



Since b < 1 this equation has roots. The equation x{= .Y1.V2 reduces to 



\HV'^i h22lh2 + h'n \_ \HV'^ 



tan 



We see that 



h22hn + hiih22 

 so with .Y = \H\^'^/h 12 this equation becomes 



tanc.Y = -Y 



1^22/ ^1: 



= c < \ 



(90) 



Equations (88), (89), and (90) cannot be satisfied simultaneously unless 

 a = b = c. These relations imply either /7i2 = or /?n = /?22 = 0, or/? ?2= ^11/^22 

 none of which is consistent with stable motion of the coupled oscillators. 

 However, by taking /^n = ^22 we satisfy equation (87) and this also gives b = c. 

 Therefore the three equations in (83) are simultaneously satisfied by taking 

 hu = /?22 and finding values of /?n and h^ which satisfy tan bx = x. 

 If now we make the substitution 



hi2 hu (^12 + ^22) 



into the expression for .y, in (86), we find ^ 



1 e 



X 



.v? = 



2K/hi + hn) 2{hn+hn) 



d 



d 



a2^i2(ai ^21 + ^2 ^22) 



remembering that ocikiik2i = a2^i2^22(^ii = ^^22)' 



This, then, is also the value of x^[xl and .Y2. 



The roots of equation (88) will give x] = xl in the limit of large 6, but we 

 have seen that these roots do not correspond to a condition of entrainment 

 between the coupled oscillators. It is not possible to say what type of inter- 

 action this implies, or whether the relationship is a stable one in the sense that 

 with the parameters fixed an ordered condition between the oscillatory motion 

 of .Yi and .Y2 will be reestablished after a disturbance. This question of stability 

 in the relationship between oscillators such as we are considering in this chapter 

 is an important one which will have to be investigated by a much more detailed 

 analysis than we have attempted in the present study. 



