Directly forced cubic amplitude equation ($A_T=(\mu +i\Lambda )A-|A{|}^{2}A+1$)

A forcing at the carrier wavenumber breaks continuous spatial translation symmetry and adds a constant to its complex amplitude equation. In a frame following a moving forcing pattern, its phase drift gives $i\Lambda A$. A nonzero forcing coefficient is normalized to one by scaling amplitude by its cube root and time by its two-thirds power. A steady intensity $s=|A{|}^2$ satisfies $s[(\mu -s{)}^2+{\Lambda }^2]=1$, and its phase is uniquely determined by the steady complex equation.

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.