............................................................................................................( 6.6.11 )

Now to include the mixed secular (or nearly secular or small divisor) term 1in Eq. (6.6.6) in this resonance condition one may write

.......................................................(6.6.12)

Now to eliminate the secular and near secular terms from Eq. (6.6.6) one can write

............................................( 6.6.13 )

Using in Eq.(6.6.13) and separating the real and imaginary parts following reduced equations are obtained.

....................................................................................(6.6.14)

................................................................(6.6.15)

Now to express the above equations in their autonomous form one may use the following transformation.

..................................................................................................................(6.6.16)

Hence, Eq. (6.6.14-15) can be written as

.................................................................................................. (6.6.17)

...................................................................(6.6.18)

By solving the above two equations, one can obtain a and γ , and then can write the solution of the system as

...................................................(6.6.19)

For steady state as the time derivative terms vanish, Eq. (6.6.17) and (6.6.18) can be written as