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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.

For a stationary finite-wavenumber instability, the relevant Landau–Ginzburg model is the real Ginzburg–Landau equation: its coefficients are real although its pattern amplitude is complex. The complex amplitude records both the strength and the position of the underlying pattern. Assume a simple critical pair of Fourier modes $\pm k_c$, a quadratic nondegenerate maximum of the linear growth rate, and no additional conserved or marginal mode. The last assumption is essential: the conserved field in the previous problem requires an additional evolution equation.

To derive the model, write a real evolution equation schematically as $u_t=L(r)u+N_2(u,u)+N_3(u,u,u)+\cdots$, with symmetric multilinear nonlinearities. Put $r={\epsilon }^2{r}_2$, $X=\epsilon x$, $T={\epsilon }^{2}t$ and use the method of multiple scales expansionHere $L(k_c;0)v=0$. At the next order the noncritical mean and second harmonic are slaved to the critical mode:where $L(0;0)w_0=-2N_2(v,\overline{v})$ and $L(2k_c;0)w_2=-N_2(v,v)$. These inverses exist under the no-other-critical-mode hypothesis. Let $v^{\ast }$ be an adjoint null vector normalized by $⟨v^{\ast },v⟩=1$. At order ${\epsilon }^3$, the Fredholm solvability condition requires the resonant forcing to be orthogonal to $v^{\ast }$. The cubic resonant contribution isThis shows why the cubic coefficient includes feedback from slaved harmonics as well as any original cubic nonlinearity. The resulting amplitude equation, after naming its linear control coefficient $\mu$, isHere $\lambda (k)$ is the critical linear dispersion relation for growth. The condition $g>0$ gives a supercritical bifurcation; $g<0$ calls for higher-order saturation, for example the quintic real Ginzburg-Landau equation.

A concrete check is the cubic Swift–Hohenberg equation $u_t={\epsilon }^2\mu u-(1+{\partial }_x^2{)}^{2}u-g_0{u}^3$. With $u=\epsilon (A{e}^{ix}+\overline{A}{e}^{-ix})+\cdots$, its linear operator contributes $4A_{XX}$ at resonant order, while the cubic resonance is $3|A{|}^{2}A$. Hence $A_T=\mu A+4A_{XX}-3g_0|A{|}^{2}A$. This directly realizes the perturbation-theory coefficients above.

Symmetry independently constrains the form. A fast spatial translation rotates $A$ by a constant phase, so every nondifferential monomial must transform like $A$. The leading allowed nonlinear term is $|A{|}^{2}A$, and a quadratic term is forbidden. A spatial reflection maps $A(X)$ to $\overline{A}(-X)$; it makes the leading coefficients real and rules out a drift term in a reflection-symmetric rest frame. The vanishing first derivative of the linear growth rate at $k_c$ is consistent with the absence of a leading first spatial derivative. These arguments constrain terms but do not calculate $\xi ,g$ or replace the solvability calculation. Breaking reflection or studying an oscillatory instability generally leads to the complex Ginzburg–Landau equation with genuinely complex coefficients.

The principal steady solutions are the zero state and the Ginzburg-Landau plane wavesTheir physical carrier wavenumber is $k_c+\epsilon q$. For $q=0$ the constant phase is arbitrary, giving a circle of uniform-amplitude states. The zero state has Fourier growth rate $\mu -\xi p^2$ and is stable for $\mu <0$. For $g>0$, each existing plane wave is stable to spatially homogeneous amplitude perturbations, with radial eigenvalue $-2g{R}^2$, and neutral to a constant phase shift.

Steady modulated states are also possible. Writing $A=a(X)e^{i\varphi (X)}$ gives the conserved spatial phase current $j=a^2{\varphi }_X$ andIts first integral is $\xi a_X^2/2+\mu a^2/2-g{a}^4/4+\xi j^2/(2a^2)=E$. Boundary conditions select nonuniform periodic amplitudes, fronts or defects from this spatial dynamics. In the invariant real subspace, for example,is a steady kink joining opposite real states. It is not stable in the unrestricted complex equation: the imaginary perturbation $v=\mathrm{sech}(\sqrt{\mu /(2\xi )}X)$ has eigenvalue $\mu /2>0$ under $v_T=\xi v_{XX}+(\mu -g{A}^2)v$. This illustrates why a real-amplitude calculation can miss a phase instability. With real coefficients the Lyapunov functionalsatisfies $A_T=-\delta \mathcal{F}/\delta \overline{A}$ and $d\mathcal{F}/dT=-2\int |A_T{|}^{2}dX$ under compatible boundary conditions. It supplies a variational structure absent from a generic complex-coefficient model.

To test full linear stability of a plane wave, set $A=e^{iqX}(R+u+iv)$ with real perturbations $u,v$. Let $b=g{R}^2=\mu -\xi q^2>0$. The linear equations areFor perturbation wavenumber $p$, the sideband spectrum of a real Ginzburg-Landau plane wave isThe phase branch at small $p$ isTherefore the robust stable band on the infinite line is $q^2<\mu /(3\xi )$, narrower than the existence band $q^2<\mu /\xi$. Crossing its boundary gives the longitudinal Eckhaus instability. At the boundary the quadratic phase-diffusion coefficient vanishes and the leading nonzero term is negative of order $p^4$. More exactly, a nonzero sideband grows only when $p^2<6q^2-2\mu /\xi$; a finite periodic domain for the wave envelope may exclude that interval. Constant-amplitude stability is understood modulo its neutral global phase, and the allowed perturbation wavenumbers must always be specified.

In a second spatial dimension, an anisotropic medium that selects one roll orientation permits an equation with ${\xi }_x{A}_{XX}+{\xi }_y{A}_{YY}$. For positive coefficients its plane-wave analysis is similar, with the second-dimensional contribution adding damping. But simply replacing $A_{XX}$ by an isotropic Laplacian is not a faithful extension for rotationally invariant stationary rolls. There the critical wavevectors form a circle. Near $(k_c,0)$,This forces the anisotropic transverse scaling of an isotropic roll envelope: $X=\epsilon x$, $Y={\epsilon }^{1/2}y$, $T={\epsilon }^{2}t$. For the positive carrier convention used here the Newell–Whitehead–Segel equation isThe opposite fast-carrier convention reverses the accompanying imaginary sign and detuning convention. For a transverse phase perturbation of $A=R{e}^{iqX}$ our convention gives $\sigma =-\xi q{p}^2/k_c-\xi p^4/(4k_c^2)$. Thus $q<0$ permits the transverse zigzag instability, in addition to the longitudinal Eckhaus instability. The fourth transverse derivative and mixed derivative in the square encode the critical-circle geometry.

One complex amplitude still describes only a narrow cone of nearby roll orientations. Competing well-separated wavevectors require coupled amplitude equations; resonant triples can admit quadratic interactions and produce the hexagonal convection amplitude equations when the underlying sign symmetry allows them. Strong disorder, defect cores with large gradients, far-from-onset patterns, boundaries, mean flows and conserved fields may all lie outside the single-amplitude approximation. The method of multiple scales gives controlled leading behavior for small amplitudes and slow modulations, rather than a universal model of all two-dimensional pattern selection.

Use the corrected same-node method identified above, with $\delta =1/M$, interior indices $1,\dots ,M-1$ and zero endpoint values. Its spatial matrix $L_{\delta }$ is real symmetric. Summation by parts gives, for a vector extended by zero at the endpoints,Hence $-L_{\delta }$ is positive definite, and ${\sum }_j(v_{j+1}-v_j{)}^2\le 4{\sum }_j{v}_j^2$ bounds its eigenvalues by $4a_{+}/{\delta }^2$. The Forward Euler method has amplification matrix $I+k{L}_{\delta }$. Its eigenvalues lie in $[1-4\mu a_{+},1]$, so their moduli are at most one whenSymmetry makes this a contraction in the mesh-weighted discrete L2 norm, uniformly over all grids and all coefficient samples satisfying the bounds. There is also a maximum norm proof: the update weights are $\mu a_{m-1/2}$, $1-\mu (a_{m-1/2}+a_{m+1/2})$, and $\mu a_{m+1/2}$. They are nonnegative and sum to one, giving a contraction after the boundary values are included.

This mesh-independent bound is sharp. For $a\equiv a_{+}$, the spatial eigenvectors are $\sin (j\pi m/M)$ and their amplification factors areIf $\mu >1/(2a_{+})$, sufficiently fine grids have a highest-frequency factor below $-1$, giving exponential growth in the step number.

On one fixed grid, rather than uniformly under refinement, the sharp universal stability of a numerical method bound in the discrete L2 norm is slightly larger:Indeed the quadratic form above is bounded by $a_{+}$ times the constant-coefficient difference form, whose largest eigenvalue is $4{\cos }^2(\pi /(2M))/{\delta }^2$; equality is realized by the constant coefficient. Its limit as $M\to \infty$ is the boxed bound. No such fixed-grid matrix conclusion repairs the ill-defined shifted-boundary update in the literal printed formula.

The circulant matrix is singular, so the meaning of convergence needs care. If $C$ is the cyclic shift, $A=C+C^{-1}-2I$ has Fourier eigenvalues $-4{\sin }^2(\pi j/n)$. Its null space consists of constants. Therefore $Ax=b$ is solvable exactly when ${\sum }_j{b}_j=0$, and any solution is determined only up to a constant. Neither iteration can converge to a unique solution for arbitrary right-hand sides.

For a compatible $b$, fix one solution $x_{\ast }$ and examine the error. The iteration matrix of the Jacobi method is $(C+C^{-1})/2$, with eigenvalues $\cos (2\pi j/n)$. The constant mode has eigenvalue one. For odd $n$ all other eigenvalues have modulus less than one, so the iterates converge to $x_{\ast }+c1$, with the constant selected by the initial mean. For even $n$, the alternating vector has eigenvalue $-1$, producing an undamped oscillation. With $b=0$ and alternating initial vector this is an explicit nonconvergent example. For even $n$ convergence occurs only when that error component vanishes. When $\sum b_j\ne 0$, the mean instead drifts by $-(\sum b_j)/(2n)$ at each step, so Jacobi cannot converge.

For Gauss-Seidel iteration, put $K=-A=D+L+U$, so $K$ is positive semidefinite with diagonal two. The homogeneous error iteration is $e^{+}=Be$, $B=-(D+L{)}^{-1}U$. Updating a coordinate exactly minimizes $E(e)=e^{T}Ke/2$ in that coordinate; completing the square gives an energy decrement $|\Delta e_j{|}^2$. Summing within a sweep givesThe same identity holds for the Hermitian energy of complex vectors. If $Be=\lambda e$ and $e$ is nonconstant, its energy is positive, so $|\lambda |\le 1$. Equality forces every update to be zero, whence $Ke=0$ and $e$ is constant, a contradiction. Thus all nonconstant eigenmodes have modulus strictly below one.

The eigenvalue one has only the constant eigenspace. To exclude a generalized eigenvector at one, observe thatIf $(B-I)v=1$, multiplication by $w^T$ would give $0=n$. Hence the constant eigenvalue is algebraically simple, andFor compatible $b$, Gauss-Seidel therefore converges from every start to some solution, for either parity of $n$. Its limit differs from $x_{\ast }$ by $1w^T(x^{(0)}-x_{\ast })/n$. For an incompatible $b$, the affine update gives $w^T{x}^{(r+1)}-w^T{x}^{(r)}=-{\sum }_j{b}_j$, so the iterates again cannot converge.

Thus semiconvergence of cyclic Poisson iterations gives the precise answer:Both have spectral radius one on the unrestricted space, so neither meets the strict stationary-iteration criterion for a unique limit independent of the starting constant. A spectral radius of one alone is not a proof that Gauss-Seidel's compatible iterates fail to converge. A normalization or projection removing the constant mode makes this distinction explicit.