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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 $Q\ge 0$. 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

A Fourier truncation retains a finite set of boundary-compatible Fourier modes and projects the PDE onto their span by the Galerkin method. Nonlinear products can generate unretained harmonics, so their coefficients are discarded by projection rather than declared zero pointwise. Agreement with a perturbation expansion through cubic order requires retention of every quadratically forced mode that feeds back on the critical mode at cubic order.

// Destination: dynamical-systems.bigb

The primary vorticity Jacobian determinant vanishes because ${\nabla }^2\psi =-s\psi$. The quadratic temperature Jacobian determinant isThe mean temperature correction therefore obeys $\dot{c}=-4{\pi }^{2}c-\alpha \pi ab/2$. Its Jacobian determinant with the primary streamfunction isProjecting on the retained primary modes gives, with $D=s+Q{\pi }^2/s$,For a nonzero steady roll, $b=(s^2+Q{\pi }^2)a/(R\alpha )$ and $c=-\alpha ab/(8\pi )$. Substitution in the second equation gives, within this truncation,The positive denominator ensures branches only on the supercritical side for every $Q\ge 0$. To verify dynamical stability as well as existence, the critical right eigenvector is $(1,\alpha /s)$; normalizing the left eigenvector gives linear eigenvalue derivativeQuadratic slaving gives $c=-{\alpha }^2{a}^2/(8\pi s)$, and projection of its feedback gives the cubic amplitude equationIts nonzero branches are stable in the center direction, while all other eigenvalues remain negative at a simple onset. Reflection about the mid-width sends $a\to -a$, so the bifurcation is a supercritical pitchfork bifurcation for all $Q$, in the generic case of a single critical roll mode.

The cubic saturation of quasistatic magnetoconvection result is identical in Fourier truncation and perturbation theory at the orders needed here. At quadratic order the only forced mode is precisely the retained horizontally uniform $\sin 2\pi z$ temperature mode. At cubic order its feedback projects on the primary mode with the same coefficient. The generated third vertical harmonic is orthogonal to that primary projection and contributes only to higher-order corrections. Thus both methods give the same linear threshold and cubic solvability condition; the truncated finite-amplitude branch is not an exact solution of the full PDE at all amplitudes.