Consider the following Duffing equation,

\[
\ddot{x}+\alpha x+\varepsilon x^{3}=\Gamma \cos \omega t
\]

with \(\alpha, \Gamma, \omega>0\). The values of \(\omega\) for which the subharmonics occur are unknown. Expand \(x\) and \(\omega\) in a perturbation solution.