7. STATISTICAL PROPERTIES OF THE EPIGENETIC SYSTEM 1 13 



This inequality shows us that stability of the strongly coupled oscillators 

 depends upon having the product of the self-interaction terms (kn and A'22) 

 larger than that of the cross-coupling terms (A'12 and ^21)- By controlling 

 the relative sizes of these parameters and the temporary levels of the variables 

 (thus establishing the proper "initial" conditions) it is possible to make the 

 system "switch" from one state to another, having either one of the compo- 

 nents (A'l, Yi) or (^^2, Y2) only, or both simultaneously. We call the disconti- 

 nuity involved when the inequality (62) changes sign, a topological disconti- 

 nuity, because in terms of the phase space of the coupled system the trajec- 

 tories undergo a qualitative change from elliptic to hyperbolic under the 

 change. In parameter space the surface /:ii^22 — ^'12^21 =0 (also a conic) 

 defines the "bifurcation values" of the parameters (Poincare, 1885). Such 

 topological discontinuities which depend upon microscopic parameter values, 

 are to be distinguished from statistical discontinuities which depend upon 

 macroscopic parameters. An example of the latter will be considered in the 

 next chapter. The possible significance of these discontinuities in relation to 

 induction and threshold phenomena in cells will be discussed in Chapter 8. 



We will assume from now on that \H\>0, so that the arctangent in 

 equation (61) is well-defined and has some value between and tt/I. The hmits 

 correspond to \H\=0 and \H\^^^/hi2=^ • The first limit we have seen to 

 define a discontinuity in the motion of the strongly coupled oscillators. The 

 second limit is given by /112 = 0, whence aia2^i2A:2i = 0. If either ku or k2i 

 is zero, than an uncoupling between components is involved and the equations 

 (23) are no longer integrable. If a^ or a2 is zero, then one or other of the 

 components is absent and we no longer have a coupled pair, the system reducing 

 to a single oscillator. Therefore we will also assume in the following that 

 hi2>0. 



The other two phase integrals are essentially the same as the corresponding 

 integrals for the system without strong interaction. We have, writing 

 >?i = l + vi/yi, 



00 



1= j e^^^y^^-^^-^^-q^^^^y^yid-qi 



"9 



AilQ 



AJQi 



Now with / = ^biyiTji, we get 



00 



Z^^ = 71 e^*. r. (^bi yi)-(^^ y+') [ e-' /^^ ^^ dt 



This is almost identical with the form of Z^. given in equation (35). The only 

 difference is that in place of jSZ?, we now have ^biyi, and there is a factor yi 

 in the integral. We can therefore use the asymptotic formulae for Z^. to get 



