112 TEMPORAL ORGANIZATION IN CELLS 



To evaluate this we introduce polar coordinates: 



^1 = rcosd, ^2 = I'sind 

 The angle between the lines 



|2=-P2 and ii-j^^i2= -(hn^y^Pi 

 IS (f) = tan-i^— ^ — 



"12 



SO that the integral now becomes 



<(l 00 







tan-i(|//|i^2//j^2) 



2i3|//|i/2 ^^*) 



The arctangent lies between and tt/I when the oscillatory motion in the 

 coupled system is stable. To see this, let us return to the integral, 



G(xiX2,yiy2) 



defined by equation (24). The part of this integral which reflects strong coup- 

 ling is the quadratic in Xi,X2. Taking j^i and y2 to be constants, the projection 

 of the surface (24) onto the (a-i,X2) plane is a conic defined by 



hiix\ + 2hi2X\X2 + h22X2 = constant 



This conic is an elhpse if and only if (A 11^^22 -^n) = |^| > 0. If |^| = 0, we 

 have a parabola, and if |// 1 < then we have a pair of hyperbolae. The latter 

 two possibilities correspond to unstable motion in the strongly coupled control 

 system in the sense that one or other of the pairs {X^, Y{) or {X2, Y2) is elimi- 

 nated from the system and the result is a single oscillator of the same kind as 

 that considered in the simple system defined by equation (18). Which of these 

 fails to "survive" in the system depends upon the initial conditions of the 

 oscillation as well as upon the parameter values. Thus, we may say that the 

 pair making up the strongly coupled oscillator (23) can coexist only if the 

 parameters of the system satisfy the inequality 



(Jinhi2-h]2) > 

 Substituting the original parameters in place of 7?^, we find that this inequality is 



/aia2^2i^'i2^'ii^'22 oqalk]2kly^ 



•)>o 



\ 4 4 



or, since all the parameters are positive quantities, 



(^hA-22- A 12^-21) > (62) 



