Prove that, for the system considered in Section 13.5.1, the minimum value of \(\omega^{2}\) for which the amplitude of subharmonic oscillations \(A\) will have a real value is given by

\[\omega_{\min }=\omega_{0}+\frac{21}{2048} \frac{F^{2}}{\omega_{0}^{5}}\]

Also, show that the minimum value of the amplitude, for stable subharmonic oscillations, is given by

\[A_{\min }=\frac{F}{16 \omega^{2}}\]