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For the directly forced cubic amplitude equation, a saddle-node bifurcation requires ${\Lambda }^2=1/s-1/(4s^4)$ and $\mu =s+1/(2s^2)$. The right side has its unique maximum $3/4$ at $s=1$, giving a cusp bifurcation at $(\Lambda ,\mu ,s)=(\sqrt{3}/2,3/2,1)$. The stability matrix has trace $2(\mu -2s)$ and determinant $(\mu -3s)(\mu -s)+{\Lambda }^2$. A Hopf bifurcation requires $\mu =2s$, $s(s^2+{\Lambda }^2)=1$ and ${\Lambda }^2-s^2>0$. These imply $\Lambda >2^{-1/3}$; equality is a double-zero linear degeneracy rather than an ordinary Hopf bifurcation.

A Hopf crossing of the transverse stability matrix requires zero trace and strictly positive determinant. Since ${\beta }_R<0$, the trace condition gives the positive thresholdThe determinant there isThus the required coefficient condition and the rotating-frame Hopf frequency areThe trace derivative $-2{\beta }_R$ is strictly positive, so with $d,\beta$ fixed the real parts cross zero with nonzero speed. Under this condition the determinant stays positive from $r=0$ to $r_H$; the pure branch first loses linear stability through this Hopf bifurcation. When $Q<Q_{\ast }$ its first instability is instead the steady one of part (a), and $D_H<0$ prevents a Hopf pair at the trace-zero point. Equality gives the already noted double-zero degeneracy.

Generically the new branch is periodic in the symmetry-reduced rotating frame, with a small oscillating odd-mode amplitude and a modulation of the even-mode amplitude. Reconstructing the common temporal phase combines the modulation frequency with the carrier frequency. The original amplitudes therefore describe modulated periodic oscillations, generically a two-frequency quasiperiodic state. If the two frequencies are rationally related, the reconstructed motion can be periodic instead. The spatial pattern no longer has a fixed mixture of the even and odd modes: their relative amplitudes and phase oscillate. As with the steady case, nonlinear nondegeneracy is needed for an ordinary Hopf branch, and its criticality or stability cannot be inferred from the trace and determinant alone.

The spatial reflection $x\mapsto -x$ acts on the two complex amplitudes as $(A,B)\mapsto (A,-B)$, because the first spatial eigenfunction is even and the second is odd. The first equation must therefore be even in $B,\overline{B}$, and the second odd. For a nonresonant double Hopf bifurcation, the temporal phases give independent actions $A\mapsto e^{i{\phi }_1}A$, $B\mapsto e^{i{\phi }_2}B$ in the cubic normal form of a dynamical system. A cubic monomial in the first equation must have net $A$ phase weight one and net $B$ phase weight zero. Only $A|A{|}^2,A|B{|}^2$ satisfy this; the second equation similarly has $B|B{|}^2,B|A{|}^2$. Parametrizing the independent cross-saturation coefficients givesAll coefficients can be complex: their real parts determine amplitude growth and their imaginary parts frequency shifts. These are cubic truncations of a family of reduced vector fields on the four-real-dimensional extended centre manifold, with the two unfolding parameters entering ${\nu }_1,{\nu }_2$.

The nonresonance assumption must be made explicit. Merely having ${\omega }_1\ne {\omega }_2$ does not imply this form: for ${\omega }_1=2{\omega }_2$, the quadratic terms $B^2$ in $\dot{A}$ and $A\overline{B}$ in $\dot{B}$ satisfy both temporal resonance and spatial parity. Thus the initial form is the generic nonresonant double-Hopf form, rather than a statement for every possible unequal frequency pair.

At the one-to-one reflection-symmetric one-to-one Hopf resonance, only the common temporal phase action remains. Cubic terms must then have total phase weight one. Spatial reflection retains, in the first equation, the three terms $A|A{|}^2,A|B{|}^2,B^2\overline{A}$; the second retains $B|B{|}^2,B|A{|}^2,A^2\overline{B}$. Common phase covariance excludes all quadratic terms. The additional resonant terms are consequently $-{\gamma }_1{B}^2\overline{A}$ and $-{\gamma }_2{A}^2\overline{B}$, with independent complex coefficients. This is the reflection-symmetric one-to-one Hopf resonance.

In the specified equal-coefficient specialization, put $r=R^2$. A nonzero pure-$A$ periodic branch exists for $\mathrm{Re}{\nu }_1>0$ and hasIts radial eigenvalue is $-2{\alpha }_{R}r<0$, and its phase is neutral. For a transverse $B$ perturbation the linear equation isIts time dependence can be removed exactly by $B=C{e}^{i\Omega t}$. Using ${\nu }_1-\alpha r=i\Omega$ and ${\nu }_2={\nu }_1-d$ givesWriting $C=u+iv$, $d=d_R+i{d}_I$, $\beta ={\beta }_R+i{\beta }_I$ yields the constant real stability matrixLet $Q=\mathrm{Re}(\beta \overline{d})={\beta }_R{d}_R+{\beta }_I{d}_I$. The transverse characteristic polynomial isBoth transverse eigenvalues have negative real part exactly whenThese hold at $r=0$ because $d_R>0$. Thus a sufficiently small pure branch is transversely stable; it is stable modulo its neutral temporal phase until one of the two inequalities fails. The secondary steady and Hopf thresholds of a resonant pure mode are derived in the two parts below. They refer to accessible points on the pure branch; in a multi-parameter problem $d$ may also vary, so a proposed parameter path must actually cross the corresponding surface.

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 the equilibrium be $A_{\ast }=R{e}^{i\theta }$ and write $\delta A=e^{i\theta }(u+iv)$. The linearization of the cubic term gives $\delta (|A{|}^{2}A)=2s\delta A+A_{\ast }^2\delta \overline{A}$. Therefore the real stability matrix isIts eigenvalues obeyThe square root may be imaginary; stability uses real parts. The determinant $D$ is exactly $F_s$, so a zero eigenvalue occurs only at the turning points derived above. Away from the cusp those are ordinary saddle-node bifurcations; the parameterization joining point has $D>0$.

At a candidate Hopf bifurcation, the trace vanishes: ${\mu }_H=2s_H$. Substitution into the steady cubic gives $s_H(s_H^2+{\Lambda }^2)=1$, which has one positive root. The pair is genuinely imaginary only ifThus $s_H<2^{-1/3}$ and, equivalently,This direct calculation proves the threshold without needing the additional monotonicity assumption. The equality gives a double-zero eigenvalue, not an ordinary Hopf bifurcation. Along a fixed-$\Lambda$ branch, implicit differentiation gives $ds/d\mu =-2s(\mu -s)/D$; at this crossing $d\mathrm{Re}\sigma /d\mu =1+4s_H^2/D_H>0$. A nonzero nonlinear Hopf coefficient then gives the usual Hopf bifurcation.

The fold and Hopf thresholds of a directly forced cubic amplitude divide the diagrams into three regimes. At noncritical points the two eigenvalues, counted with multiplicity by the signs of their real parts, are classified as follows:

At a fold one eigenvalue is zero; at a Hopf point the pair has zero real part. These nonhyperbolic points are marked separately in the sketches.

Let the uniform nonzero amplitude be $R$, so $\mu =R^2-\alpha R$ and $B=0$. Write $r=R(2R-\alpha )$. A perturbation with Fourier wavenumber $k=n\pi /L$ has stability matrixThe constant $B$ mode is disallowed because the conserved mean is prescribed. At $k=0$ the only allowed mode has eigenvalue $-r$, so homogeneous stability requires $r>0$. Then every nonzero-mode trace is negative, and its determinant is smallest in sign at the smallest allowed $k^2$. The exact periodic stability criterion for a conserved-field uniform amplitude isA zero in either inequality is a marginal case requiring nonlinear analysis. The zero-amplitude branch, separately, has amplitude growth rates $\mu -k^2$ and nonzero conserved-field growth rates $-\sigma k^2$, and is strictly stable when $\mu <0$.

For the long-domain interpretation one may take $\alpha >0$ by reversing the sign of $A$, and first consider the positive upper branch $R>\alpha /2$. It is homogeneously stable even though sufficiently long spatial modulations grow wheneverFor $0<\delta <1$, this givesAt a fixed finite $L$ the actual unstable part is smaller: it must also satisfy $(\pi /L{)}^2<\alpha R-2(1-\delta )R^2$. For $\delta \ge 1$, the right side is positive for every positive upper-branch $R$, so every such state is unstable to sufficiently long waves. The physical mechanism is feedback from the conserved field: an increase in $A^2$ reduces local $B$, and the term $-AB$ reinforces that increase.

There are two important limits on the wording of the requested stability conclusion. First,never vanishes at a finite regular point of this branch. At its fold the derivative is infinite. Thus a condition of zero $dR/d\mu$ cannot literally occur here. The meaningful contrast is that a positive-slope, homogeneously stable branch can still have a growing spatial mode.

Second, a long-wave conclusion cannot be asserted for every finite periodic domain. For example, $\alpha =1$, $\delta =2$, $\sigma =1$, $R=1$, $\mu =0$, $L=1$ give $r=1$ and $k^2+r-2\delta R^2=n^2{\pi }^2-3>0$ for every nonzero integer $n$. This state is strictly stable even though $\delta >1$. Nor does the unrestricted statement include every sign of $R$: $\alpha =1$, $\delta =2$, $R=-1/10$, $\mu =11/100$ give $r=3/25$ and $r-2\delta R^2=2/25>0$, hence stability for every period and every $\sigma >0$. These examples distinguish the exact stated real-variable periodic problem from its intended positive-amplitude, long-layer interpretation.

Put $y=\dot{x}$. The planar autonomous differential equation isIts equilibria have $y=0$, $x^2=p$. There are none for $p<0$, one at $p=0$, and two $x_{\pm }=\pm \sqrt{p}$ for $p>0$. At an equilibrium $x_{\ast }$ the stability matrix isWriting $s=\sqrt{p}>0$, $x_{+}=s$ is always a saddle equilibrium. At $x_{-}=-s$, the determinant is $2s>0$ and the trace is $s-q$: it is attracting for $q>s$ and repelling for $q<s$.

At $p=0$ and $q\ne 0$, the eigenvalues are $0,-q$; the nonzero quadratic term and parameter unfolding give a saddle-node bifurcation. At $p>0$, the trace crosses zero at $q=s$, with eigenvalues $\pm i\sqrt{2s}$, giving a Hopf bifurcation. It is nondegenerate: around $x=-s$ the equation is $\ddot{\xi }+(q-s)\dot{\xi }+2s\xi -{\xi }^2+\xi \dot{\xi }=0$. At $q=s$, a small oscillatory expansion of leading amplitude $r$ has $⟨\xi {\dot{\xi }}^2⟩=r^4/8+O(r^5)$, so the first nonlinear damping is positive and nonzero. For example, its second-order oscillatory expansion is $\xi =r\cos (\Omega t)+r^2/(2{\Omega }^2)-r^2\cos (2\Omega t)/(6{\Omega }^2)-r^2\sin (2\Omega t)/(6\Omega )$ with ${\Omega }^2=2s$. Thus the small attracting limit cycle is on the $q<s$ side. At $(p,q)=(0,0)$ the matrix has a double zero eigenvalue and a nontrivial Jordan block, the Bogdanov–Takens bifurcation organizing the two curves. The local bifurcation curves areA change between node and focus away from these curves is not itself a local bifurcation.

For the requested global exclusion, suppose $x(t)$ were periodic with period $T$. Integrate the original equation over a period. The integrals of $\ddot{x}$, $q\dot{x}$, and $x\dot{x}=\frac{1}{2}(d/dt)x^2$ vanish. ThereforeThe left side is nonnegative and the right side is negative when $p<0$. This contradiction proves there are no periodic orbits for $p<0$, independently of $q$.

With $y=\dot{x}$, the feedback system isFor $p=0$, $(x,y,z)=(0,0,0)$ is an equilibrium. Its stability matrix and characteristic polynomial areThus there is always a zero eigenvalue, with eigenvector $(1,0,0)$, so it is a nonhyperbolic equilibrium. If $q\ne -1$, the other two eigenvalues are off the imaginary axis: for $q>-1$ their real parts are negative, while for $q<-1$ their product is negative. At $q=-1$, all three eigenvalues are zero, explaining the exclusion in the next part.

A nonzero uniform equilibrium of the conserved-field real amplitude equation obeys $\mu =R^2-\alpha R$. Fixed zero mean excludes the constant $B$ perturbation. Its homogeneous amplitude eigenvalue is $-r$. For a nonzero Fourier wavenumber $k$, the stability matrix has trace $-r-(1+\sigma )k^2$ and determinant $\sigma k^2(k^2+r-2\delta R^2)$. The displayed conditions are necessary and sufficient for strict linear stability on a period $2L$. In an infinite-domain limit the nonzero-mode condition becomes $r\ge 2\delta R^2$; a long-wave instability need not occur in a sufficiently short domain.

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.

The scalar Wilson-Fisher fixed point is attractive within its critical surface, but has a relevant thermal direction. With $r=m^2{\Lambda }^{-2}$, $\lambda =g{\Lambda }^{-ϵ}$, and $K_D={\Omega }_D/(2\pi {)}^D$, its one-loop equations are $\dot{r}=2r+K_D\lambda /[2(1+r)]$ and $\dot{\lambda }=ϵ\lambda -3K_D{\lambda }^2/[2(1+r{)}^2]$. Their stability matrix at $r_{\ast }=-ϵ/6+O({ϵ}^2)$ and ${\lambda }_{\ast }=16{\pi }^2ϵ/3+O({ϵ}^2)$ has the displayed eigenvalues. Thus an unrestricted claim of infrared attraction is false: the bare mass must be tuned to the critical surface.