= Cubic saturation of quasistatic magnetoconvection
{title2=$a^2=\frac{8(R-R_c)}{s^2+Q\pi^2}+O((R-R_c)^2)$}
The primary roll and <temperature> modes force only a horizontally uniform <temperature> mode proportional to $\sin(2\pi z)$ at quadratic order. Its amplitude is $c=-\alpha ab/(8\pi)$. Its feedback on the primary <temperature> mode saturates <magnetoconvection>, giving the displayed positive squared amplitude above threshold. Thus a simple roll onset is a supercritical <pitchfork bifurcation> for every $Q\ge0$. The three-mode <Fourier truncation> and full perturbation calculation agree through the cubic solvability condition because all forced quadratic modes are retained; this is not an assertion that the truncation gives an exact finite-amplitude PDE solution.
// Destination: dynamical-systems.bigb
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