= Square-pattern interaction with a sign-changing scalar mode
{title2=$M=\begin{pmatrix}p&-\alpha_1s\\4\gamma_1s&\mu_2\end{pmatrix}$}
A scalar mode that changes sign under a quarter-turn and is fixed by diagonal reflections couples quadratically through $AC,BC,A^2-B^2$. In the saturated three-mode <equivariant dynamical system>, a diagonal state $A=B=s$, $C=0$ has $s^2=\mu_1/(1+\lambda)$ and $p=2(\lambda-1)s^2$. The symmetric perturbation decays at rate $2\mu_1$; the remaining <stability matrix> is the displayed block. With $K=2\alpha_1\gamma_1/(\lambda-1)>0$, its <linear stability> interval is $-K<\mu_2<-p$, provided $p<K$. The endpoints are a symmetry-breaking <pitchfork bifurcation> and a <Hopf bifurcation>, respectively, away from degeneracies.
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