7. STATISTICAL PROPERTIES OF THE EPIGENETIC SYSTEM 99 



The mean fraction of the time spent by this variable at values less than the 

 steady state is just the complement of this, which we write as 



T T Zj [2c J 



Now for very small 9 we know from equation (36) that 



Therefore we get 



so that also 



f 



^yfe)yo=2 



This shows us that when the talandic temperature of the system is very small 

 the oscillations are nearly symmetrical (in fact, nearly sinusoidal, as we will 

 see later) the variables spending about the same amount of time above as 

 below their steady state values. However, when d is large 



so that 



'■ ' y© 



^ « 1, — xO (47) 



Thus when the system is very excited (large 6) the oscillators spend most of 

 their time at values greater than the steady state, as we have already observed 

 in connection with equation (33). 



Information about the actual amplitudes of the oscillations above and 

 below the steady state values can also be obtained. Consider the function 



This is the mean amplitude of the variable x,, averaged over positive values 

 only. It reduces to 



00 



^+ (r+/r)z,J ''^ 







CO 



,IT)^c,zJ ^~'^' 



(TJT)^c 







1 d 



