In the calculation of the normal form, it is more convevient to use the amplitude-phase
description of the system (Eqns. 9-10).
We will linearize the system around 
and 
. Our new equations are:
We now can easily calculate the Jacobian matrix for this system. Recall that the Jacobian for this system is a 2x2 matrix. The (11) entry is the partial derivative of 
with respect to . The (12) entry is the partial derivative of with respect to 
. The (21) entry is the partial derivative of with respect to . The (22) entry is the partial derivative of with respect to . For this system, the Jacobian is:
where a = 
and d = 
. The eigenvalues of this matrix are a and d. Refer to the previous section for explanation of 
as a bifurcation point. This analysis is identical to that in the previous section except that here, we use A and n variables to describe the system rather than what was used previously.