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.