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For a steady-state equivariant dynamical system, an axial subgroup has a one-dimensional real fixed-point subspace of a group action. A complex axial subgroup for a Hopf problem has a one-dimensional complex fixed-point subspace under the spatial group combined with the common phase action. Distinct conjugacy classes may yield distinct types of bifurcating branches.

Let $q_j$ denote the four critical wavevectors, and write $A_j=R_j{e}^{i{\theta }_j}$. A spatial translation by $(s,t)$ acts on the amplitudes asThe two independent translation directions in phase space are $(2,2,1,1)$ and $(1,-1,2,-2)$. Their common orthogonal complement is spanned by $(2,-2,-1,1)$ and $(1,1,-2,-2)$. Hence translation invariants of Fourier-mode phases can be chosen asAway from zero amplitudes, a translation-invariant function of the phases is a function of these two invariant angles. The four magnitudes are themselves invariant. Thus an equivariant dynamical system on the eight-real-dimensional centre manifold descends, after quotienting translation symmetry, to four magnitude equations and two invariant-phase equations. The phase coordinates are singular where an amplitude vanishes; the original complex-amplitude equations remain smooth there.

The wavevector selection rule for equivariant monomials proves the cubic regularity. For a monomial ${\prod }_j{A}_j^{p_j}{\overline{A}}_j^{q_j}$ in the first equation, put $d_j=p_j-q_j$. Translation covariance requiresAt total degree $N$, additionally $\sum |d_j|\le N$ and $N-\sum |d_j|$ is even. Conversely, any integer vector satisfying these conditions yields a monomial, with any remaining even degree supplied by factors $|A_j{|}^2$. This makes the degree test exhaustive. For example, eliminate the first two entries:Checking the integer pairs $|d_3|+|d_4|\le N$, retaining only integral $d_1,d_2$ with the norm and parity conditions, givesAt degree three the only possibility is therefore $A_1|A_j{|}^2$, $j=1,\dots ,4$. Square symmetry gives the same conclusion for every other equation. All cubic terms are regular.

Two generators of the square dihedral group act as follows. Reflection in the $x$ axis gives $(A_1,A_2,A_3,A_4)\mapsto (A_2,A_1,A_4,A_3)$, while interchange of $x,y$ gives $(A_3,{\overline{A}}_4,A_1,{\overline{A}}_2)$. Rotation by $\pi$ conjugates every amplitude, forcing the coefficients in a steady normal form of a dynamical system to be real. With real unfolding parameter $\sigma$ and real coefficients $c_1,c_2,c_3,c_4$, defineThe most general cubic normal form is ${\dot{A}}_j=A_j{G}_j+O(|A{|}^5)$. There are no further coefficient identifications: the square-group stabilizer of one of these oblique wavevectors is trivial, so its three couplings to the other magnitudes are independent. At cubic order ${\dot{R}}_j=R_j{G}_j$ and all phases, including ${\chi }_1,{\chi }_2$, are constant.

The third planform is fixed by a diagonal reflection and changes sign under a quarter-turn. Extend the preceding group representation byThe other diagonal reflection changes only the sign of $B$. Consequently the first two components of an equivariant dynamical system have factors $A$ and $B$, whereas its third component is even in each of $A,B$. The quarter-turn exchanges $A^2$ and $B^2$ while negating $C$. Its admissible quadratic coupling is therefore $A^2-B^2$ in the third equation and opposite $AC,BC$ couplings in the first two. At cubic order the only new terms are $A{C}^2,B{C}^2,C^3,C(A^2+B^2)$. The linear action on the two-dimensional irreducible mode has one coefficient ${\mu }_1$, while the distinct scalar representation has its own coefficient ${\mu }_2$. A small splitting of the two marginal linear modes supplies these two unfolding parameters. With real coefficients the resulting amplitude equations areThese equations follow by listing all equivariant monomials through degree three; the displayed minus signs are coefficient conventions, not consequences of symmetry.

In the specified reduced system put $A=B=s$, $C=0$. A nonzero branch exists for ${\mu }_1>0$, with $q=s^2={\mu }_1/(1+\lambda )$. Write perturbations as $u=(\delta A+\delta B)/2$, $v=(\delta A-\delta B)/2$, and $c=\delta C$. Linearization givesFor the square-pattern interaction with a sign-changing scalar mode, the remaining eigenvalues have sum $\tau =p+{\mu }_2$ and product $D=p{\mu }_2+4{\alpha }_1{\gamma }_{1}q=p({\mu }_2+K)$, where $K=2{\alpha }_1{\gamma }_1/(\lambda -1)>0$. A real two-dimensional stability matrix has both eigenvalues in the left half-plane exactly when its trace is negative and its determinant is positive. ThereforeIf ${\mu }_2<-K$, the block has negative determinant and one growing direction, explaining the instability at sufficiently small ${\mu }_2$. At the lower endpoint ${\mu }_2=-K$, one eigenvalue is zero and the other is $p-K<0$. The symmetry $(A,B,C)\mapsto (B,A,-C)$ fixes the diagonal state and negates its critical $(v,c)$ direction. Thus a generic nonlinear unfolding has a pitchfork bifurcation to unequal-amplitude states with nonzero $C$.

At the upper endpoint ${\mu }_2=-p$, the trace vanishes andThe symmetric eigenvalue remains negative. The critical conjugate pair crosses the imaginary axis with derivative $d\mathrm{Re}\sigma /d{\mu }_2=1/2$. If its cubic Hopf normal form coefficient is nonzero, this is a Hopf bifurcation and nearby periodic solutions exist on one side. Their stability and the side of existence depend on that coefficient; the linear calculation does not determine them. It does establish the generic possibility of oscillating square patterns. The equality $p=K$ is a double-zero degeneracy, outside this nondegenerate Hopf bifurcation argument.

Choose amplitude coordinates in which a quarter-turn and a diagonal reflection act as $r(A,B)=(B,-A)$ and $s(A,B)=(-A,B)$. These generate the natural two-dimensional group representation of the dihedral group $D_4$. An equivariant dynamical system satisfies $F(gv)=gF(v)$. The two independent sign changes make $F_A$ odd in $A$ and even in $B$, and $F_B$ odd in $B$ and even in $A$. The quarter-turn equates the two linear coefficients and the two sets of cubic coefficients. No quadratic monomial is allowed. Thus the square-symmetric cubic steady-state normal form isThe bifurcation parameter has been scaled so that the critical linear eigenvalue is $\mu$. Nondegeneracy at cubic order means $a\ne 0$, $a+b\ne 0$ and $a-b\ne 0$. The equilibrium equations factor as $A(\mu -a{A}^2-b{B}^2)=B(\mu -a{B}^2-b{A}^2)=0$. When both components are nonzero, subtracting the two bracketed expressions gives $(a-b)(A^2-B^2)=0$. Therefore the complete leading list of small steady branches is:

Each family is one orbit of the dihedral group. Each nonzero state retains one reflection fixing its amplitude line; its full spatial symmetry depends also on the chosen planforms. Higher-order terms perturb the branch amplitudes but preserve the nondegenerate local classification. For $\mu >0$, an axial branch is asymptotically stable precisely when $a>0$ and $b>a$, while a diagonal branch is asymptotically stable precisely when $a+b>0$ and $b<a$. Every cubic branch lying on $\mu <0$ has a positive radial eigenvalue $-2\mu$, so it is unstable. In the exceptional isotropic cubic case $a=b$, the cubic truncation instead has a circle of equilibria; higher-order anisotropy is then needed, which is why that case was excluded.

Let a finite or compact group $G$ act linearly on a real space $V$, and let $\dot{x}=F(x,\mu )$ be a smooth equivariant dynamical system: $F(gx,\mu )=gF(x,\mu )$. Assume $F(0,\mu )=0$. The representation is an absolutely irreducible real representation when its complexification is irreducible; equivalently, every real linear operator commuting with $G$ is a scalar multiple of the identity. Consequently the linearization at the trivial branch is $D_{x}F(0,\mu )=\alpha (\mu )I$. A generic steady-state crossing has $\alpha (0)=0$ and ${\alpha }^{\prime }(0)\ne 0$, with stable noncritical directions removed by a centre manifold if this is a reduction of a larger system.

The isotropy group of a vector $x$ is ${\Sigma }_x=\{g\in G:gx=x\}$. For a subgroup $\Sigma$, its fixed-point subspace of a group action is $\mathrm{Fix}(\Sigma )=\{x:\sigma x=x\text{ for every }\sigma \in \Sigma \}$. Its normaliser isThe normaliser preserves $\mathrm{Fix}(\Sigma )$, and $N_G(\Sigma )/\Sigma$ describes its residual symmetry. A steady-state axial subgroup is an actual isotropy subgroup with ${\dim }_{\mathbb{R}}\mathrm{Fix}(\Sigma )=1$.

The equivariant branching lemma states that, under the smoothness, equivariance, absolute irreducibility, and transverse scalar-eigenvalue crossing hypotheses above, each conjugacy class of axial isotropy subgroup gives a branch of nonzero equilibria in its fixed-point line through $(0,0)$. The generic leading nonlinear coefficient determines which parameter side the branch occupies and whether it is transcritical-like or pitchfork-like; the theorem does not assert that these branches are stable or exhaust every possible branch. The usual nondegeneracy condition excludes a vanishing leading nonlinear coefficient. Conjugate subgroups produce symmetry-related copies of a branch, rather than new isotropy types.

To explain the dynamical role of the fixed-point subspaces of a group action, if $x$ is fixed by $\Sigma$, then $\sigma F(x,\mu )=F(\sigma x,\mu )=F(x,\mu )$ for every $\sigma \in \Sigma$. Thus the vector field is tangent to that subspace, and uniqueness makes it flow invariant. On a fixed-point line one solves a scalar equilibrium equation and determines radial stability. Stability in the full space additionally requires all transverse eigenvalues to be stable. Higher-dimensional fixed-point spaces support their own restricted dynamics and may contain symmetry-constrained connecting orbits.

Finally, the normalizer action determines axial branch parity: if $N_G(\Sigma )/\Sigma$ contains an element acting as $x\mapsto -x$ on the fixed line, its scalar vector field must be odd, so a generic cubic term gives a pitchfork bifurcation. If that residual sign reversal is absent, a quadratic term is permitted and can give a transcritical-like branch. This is why fixed-space dimension alone does not determine either the branch type or its full stability.

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.

An equivariant dynamical system can contain the monomial ${\prod }_j{A}_j^{p_j}{\overline{A}}_j^{q_j}$ in its $\ell$th amplitude equation only if its total wavevector equals the output wavevector. Its degree is ${\sum }_j(p_j+q_j)$. Thus an integer vector $d=p-q$ contributes at degree $N$ exactly when the wavevector equation holds, $\Vert d{\Vert }_1\le N$, and $N-\Vert d{\Vert }_1$ is even. The extra even degree is supplied by factors $|A_j{|}^2$. This finite integer test distinguishes regular saturation terms from phase-sensitive resonance terms.