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A codimension-two bifurcation requires two independent parameter conditions to hold simultaneously. Simultaneous stationary and Hopf bifurcation thresholds require both kinds of critical mode in the reduced amplitude equation. A crossing of modes at different wavenumbers is different from a zero-frequency double-zero degeneration of one fixed-wave-number cubic.

Near an oscillatory onset of rotating Rayleigh-Bénard convection, opposite travelling roll waves have coupled Hopf bifurcation amplitudes. Symmetry gives the displayed cubic amplitude equations, with generally complex self-coupling and cross-coupling coefficients. Write $a_s=\mathrm{Re}{g}_s$ and $a_c=\mathrm{Re}{g}_c$. For $r>0,a_s>0$, a single travelling wave has intensity $r/a_s$ and is stable against its counterpropagating competitor when $a_c>a_s$. Equal standing-wave intensities are $r/(a_s+a_c)$; with $r>0$ and positive denominator they are stable to intensity differences when $a_s>a_c$. Phase symmetries give orbital, rather than strict phase-decay, stability. Spatial modulation, mean flows and oblique convection rolls require additional stability tests.

oscillatory convection is time-dependent convection emerging through nonzero-frequency overstability or a Hopf bifurcation. At onset a complex-conjugate growth-rate pair crosses the imaginary axis. Travelling and standing convection rolls are distinct nonlinear possibilities, and their stability cannot be decided from the linear angular frequency alone.

For $s=i\omega$, real-imaginary separation of the rotating-convection cubic givesThe curve is admissible only if ${\omega }^2>0$, so $P<1$ and sufficient rotation are necessary. A formal minimum with nonpositive angular frequency squared is not a Hopf bifurcation. At zero angular frequency this curve meets the stationary curve for that fixed wavenumber in a double-zero limit.