= Quadratic even-odd mode interaction
A <reflection> acting as $(A,B)\mapsto(A,-B)$ forces an <equivariant> <vector field> to have $\dot A$ even and $\dot B$ odd in $B$. With the zero solution preserved, its quadratic <normal form> is $\dot A=\lambda_1A+a_1A^2+a_2B^2$, $\dot B=\lambda_2B+a_3AB$. For positive coefficients, every mixed <equilibrium> with $B\ne0$ has <Jacobian determinant> $-2a_2a_3B^2<0$, hence is a <saddle equilibrium>. The two invariant-axis equilibria meet in a <transcritical bifurcation>, and the mixed branches meet them in <pitchfork bifurcations>.
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