= Steady–Hopf mode interaction
For a chosen orientation and phase-reduced steady amplitude $a$ and nonresonant oscillatory amplitude $z$, a schematic normal form is $\dot a=r_sa-g_sa^3-h_sa|z|^2$, $\dot z=(r_o+i\omega_0)z-g_o|z|^2z-h_oa^2z$. For $r_s,g_s>0$ the pure steady branch exists and is radially attracting; its oscillatory perturbation is damped when $r_o-h_{o,r}r_s/g_s<0$, where $h_{o,r}=\operatorname{Re}(h_o)$. For $r_o,g_{o,r}>0$, $g_{o,r}=\operatorname{Re}(g_o)$, the pure oscillatory branch exists and is radially attracting; its steady perturbation is damped when $r_s-h_sr_o/g_{o,r}<0$. The transverse inequalities are not existence or full stability tests by themselves. Mixed intensities $u=a^2>0$, $v=|z|^2>0$ solve $g_su+h_sv=r_s$, $h_{o,r}u+g_{o,r}v=r_o$. Their intensity <Jacobian matrix> has trace $-2(g_su+g_{o,r}v)$ and determinant $4uv(g_sg_{o,r}-h_sh_{o,r})$, giving attraction in intensities when the trace is negative and determinant positive. Resonant phases, oppositely travelling modes or zero-frequency limits need additional retained amplitudes; this schematic pair is not a universal complete normal form.
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