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Fix $\mu =r-1-q$ and $0<q<\zeta /\sigma$. The noncritical eigenvalues at $\mu =0$ are $-\zeta$ and $\sigma q-\zeta <0$. Include the constant parameter as an additional centre variable. Symmetry permits expansionswhere the omitted terms in the reduced equation have order ${\mu }^{2}a,\mu a^3,a^5$. At $\mu =0$, $a^{\prime }$ begins at cubic order, so $c^{\prime }=c_a{a}^{\prime }$ begins at fourth order. The $c$ equation therefore gives $c_2=3/{\zeta }^2$.

The invariance equation for $b$ must retain $b_a{a}^{\prime }=a^{\prime }/\zeta +\cdots$. From the first equation,Equating coefficients in $b_a{a}^{\prime }=a-\zeta b-ac$ givesHenceand the required reduced equation isBoth coefficients are positive. Thus the bifurcation is a subcritical pitchfork: the no-convection state is stable for $\mu <0$ and unstable for $\mu >0$. The small nonzero branches lie on the stable-origin side,and are unstable, since the derivative of the reduced vector field there is $-2C_1\mu >0$. Their two transverse directions remain stable, making them saddle equilibrium equilibria in the three-dimensional model. Setting $b^{\prime }=c^{\prime }=0$ without the invariance correction would incorrectly omit the common factor $(1-\sigma q/\zeta {)}^{-1}$. This is the cubic centre-manifold reduction of porous magnetoconvection.