 Steady–Hopf mode interaction

For a chosen orientation and phase-reduced steady amplitude $a$ and nonresonant oscillatory amplitude $z$, a schematic normal form is $\dot{a}=r_{s}a-g_s{a}^3-h_{s}a|z{|}^2$, $\dot{z}=(r_o+i{\omega }_0)z-g_o|z{|}^{2}z-h_o{a}^{2}z$. 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 $h_{o,r}=\mathrm{Re}(h_o)$. For $r_o,g_{o,r}>0$, $g_{o,r}=\mathrm{Re}(g_o)$, the pure oscillatory branch exists and is radially attracting; its steady perturbation is damped when $r_s-h_s{r}_o/g_{o,r}<0$. The transverse inequalities are not existence or full stability tests by themselves. Mixed intensities $u=a^2>0$, $v=|z{|}^2>0$ solve $g_{s}u+h_{s}v=r_s$, $h_{o,r}u+g_{o,r}v=r_o$. Their intensity Jacobian matrix has trace $-2(g_{s}u+g_{o,r}v)$ and determinant $4uv(g_s{g}_{o,r}-h_s{h}_{o,r})$, 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.