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For the directly forced cubic amplitude equation, a saddle-node bifurcation requires ${\Lambda }^2=1/s-1/(4s^4)$ and $\mu =s+1/(2s^2)$. The right side has its unique maximum $3/4$ at $s=1$, giving a cusp bifurcation at $(\Lambda ,\mu ,s)=(\sqrt{3}/2,3/2,1)$. The stability matrix has trace $2(\mu -2s)$ and determinant $(\mu -3s)(\mu -s)+{\Lambda }^2$. A Hopf bifurcation requires $\mu =2s$, $s(s^2+{\Lambda }^2)=1$ and ${\Lambda }^2-s^2>0$. These imply $\Lambda >2^{-1/3}$; equality is a double-zero linear degeneracy rather than an ordinary Hopf bifurcation.

Represent the leading real pattern by $A(T)e^{i{k}_{c}x}+\overline{A}(T)e^{-i{k}_{c}x}$. Before forcing, continuous translation symmetry acts as $A\mapsto e^{i\theta }A$, and spatial reflection acts as $A\mapsto \overline{A}$. Equivariance excludes quadratic terms and permits a linear term and $|A{|}^{2}A$ at cubic order. Symmetry under reflection makes their coefficients real. A supercritical bifurcation has a saturating cubic coefficient; rescaling amplitude and time makes it $-1$. The imperfection has a Fourier component at the same wavenumber, so its resonant projection contributes an additive real coefficient. On the spatially uniform wave envelope subspace the amplitude equation is therefore $A_T=\mu A-|A{|}^{2}A+\alpha$. Allowing modulations would additionally produce the spatial terms of the real Ginzburg–Landau equation.

For a forcing pattern moving in the positive $x$ direction, its laboratory-frame coefficient is $\alpha e^{-i\Omega T}$, where $\Omega$ is proportional to $k_{c}c$ after converting to slow time. The comoving amplitude is $B=e^{i\Omega T}A$, since the fast carrier becomes $e^{i{k}_c(x-ct)}$. Differentiation givesA constant phase choice makes $\alpha >0$, and complex conjugation reverses the sign of $\Omega$, so take $\Lambda =\Omega >0$. Put $B={\alpha }^{1/3}a$ and $\tau ={\alpha }^{2/3}T$. The rescaled linear coefficients are $\mu /{\alpha }^{2/3}$ and $\Lambda /{\alpha }^{2/3}$, and the forcing becomes one. Below the rescaled variables are renamed $A,T,\mu ,\Lambda$, giving the directly forced cubic amplitude equation.