 Three-mode porous magnetoconvection

A Galerkin method using one convective streamfunction mode and two Cartesian magnetic flux function modes yields the displayed system. With $\alpha =\pi /\sqrt{3}$, use $\psi =4\sqrt{2}a\sin (\alpha x)\sin (\pi z)$ and $A=(3\sqrt{2}/\pi )b\sin (\alpha x)\cos (\pi z)+(\sqrt{3}/\pi )c\sin (2\alpha x)$, time $\tau =(4{\pi }^2/3)t$, $q=9Q/(16{\pi }^2)$ and $r=9R/(64{\pi }^4)$. The origin's planar trace is $\sigma (r-1)-\zeta$ and determinant $\sigma \zeta (1+q-r)$; the third eigenvalue is $-\zeta$. The stationary threshold is $r=1+q$, and the oscillatory threshold is $r=1+\zeta /\sigma$ when $q>\zeta /\sigma$. They meet in a symmetric double-zero point at $(q,r)=(\zeta /\sigma ,1+\zeta /\sigma )$. Higher spatial harmonics generated by the quadratic terms are discarded, so this is a projected approximation rather than an exact invariant subspace of the original equations.