..........................................(4.2.18)

............................. (4.2.19)

(4.2.20)

For steady state, . These non-linear algebraic equations can be solved numerically to obtain the fixed point response of the system.

By directly perturbing the reduced equations, one can study the stability of the non-trivial steady sate solution. But, as the reduced Eqs. (4.2.17-20) have the coupled terms and , the perturbed equations will not contain the perturbations or for trivial solutions and hence the stability of the trivial state cannot be studied by directly perturbing these equations. To circumvent this difficulty, normalization method can be adopted by introducing the transformation

.....................................................................(4.2.21)

into the reduced equations (4.2.17-4.2.20). Carrying out trigonometric manipulations, one arrives at the following normalized reduced equations or the Cartesian form of modulation equations:

.......................................(4.2.22)

.....................................(4.2.23)

........................................................(4.2.24)