Bifurcation theory

Bifurcation theory studies qualitative changes in equilibria and invariant sets as parameters vary.

Critical slowing down is the divergence of a relaxation time as a stable equilibrium point approaches loss of hyperbolicity. For a scalar ordinary differential equation it occurs when $f^{\prime }(N^{\ast })\uparrow 0$ and the recovery time of a stable equilibrium $-1/f^{\prime }(N^{\ast })$ diverges. At the threshold, nonlinear terms may replace exponential decay by algebraic decay; for example $N^{\prime }=-k{N}^3$ has $N(t)\sim (2kt{)}^{-1/2}$ for positive initial data.

A bifurcation parameter labels a family of dynamical systems in which a qualitative change of equilibria or other invariant sets is studied. Near a simple critical eigenvalue crossing, a nonzero derivative with respect to the physical control parameter permits using that eigenvalue itself as a local parameter. At a codimension-two bifurcation, two independent parameters are generally needed.

A global bifurcation changes an invariant set through a global orbit connection or collision that cannot be identified by a single equilibrium point's linearization alone. For example, a global bifurcation through a saddle homoclinic orbit occurs when an outgoing unstable separatrix returns to the same saddle equilibrium's stable manifold. A smooth global Poincaré return map then combines with a local passage map to determine which side contains a periodic orbit. A heteroclinic orbit instead joins distinct equilibrium points; an energy-balance or Melnikov integral can locate its breaking in a nearly Hamiltonian system.

A Neimark–Sacker bifurcation of a smooth planar map occurs when a simple complex-conjugate pair of fixed-point Floquet multipliers crosses the unit circle away from low-order resonances. If its radial normal form is $r^{\prime }=r(1+\eta +d{r}^2)+\cdots$ with $d\ne 0$, a small invariant closed curve has $r^2=-\eta /d+\cdots$. For $d<0$ it appears on $\eta >0$, where the fixed point has become unstable, and its radial Floquet multiplier is $1-2\eta +\cdots$, so the bifurcation is supercritical. For $d>0$ the curve lies on $\eta <0$ and is unstable, giving a subcritical bifurcation. Area-preserving maps have a vanishing cubic radial drift at an elliptic point and do not meet this dissipative nondegeneracy condition.

In a complex coordinate write $z^{\prime }=\lambda z+f_{20}{z}^2+f_{11}z\overline{z}+f_{02}{\overline{z}}^2$, with $|\lambda |=1$. Away from strong resonances of a planar map, the quadratic change $z=w+h_{20}{w}^2+h_{11}w\overline{w}+h_{02}{\overline{w}}^2$ has $h_{20}=f_{20}/({\lambda }^2-\lambda )$, $h_{11}=f_{11}/(1-\lambda )$ and $h_{02}=f_{02}/({\overline{\lambda }}^2-\lambda )$. Substituting and collecting $w^2\overline{w}$ gives $g_{21}=2f_{20}{h}_{11}+f_{11}{h}_{20}+f_{11}{\overline{h}}_{11}+2f_{02}{\overline{h}}_{02}$. The cubic radial coefficient is $\ell =\mathrm{Re}(\overline{\lambda }{g}_{21})$; $\ell <0$ gives a supercritical crossing when the multiplier modulus increases through one.

At a strong resonance of a planar map the Floquet multiplier of a fixed point is a root of unity of low order. The usual nonresonant Neimark–Sacker bifurcation requires ${\lambda }^k\ne 1$ for $k=1,2,3,4$. At a 1:3 resonance a quadratic term ${\overline{z}}^2$ cannot be removed from the normal form; at a 1:4 resonance a cubic term ${\overline{z}}^3$ remains. Consequently one cannot infer a quasiperiodic invariant curve from the unit-modulus crossing alone.

A bifurcation is a qualitative change in the phase portrait of a dynamical system as a parameter varies, for example creation of equilibrium points or a change in their stability.

A simplified local vector field obtained by smooth changes of state variables, and sometimes time or parameter, retaining the terms that cannot be removed at the required order. Near a simple zero eigenvalue, reduction to a centre manifold can give the transcritical form $\dot{v}=ϵv-v^2$ or a reflection-symmetric pitchfork form $\dot{x}=ϵx\pm x^3$. The nonzero leading coefficients and permitted coordinate changes are part of the classification.

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 $\dot{A}={\lambda }_{1}A+a_1{A}^2+a_2{B}^2$, $\dot{B}={\lambda }_{2}B+a_{3}AB$. For positive coefficients, every mixed equilibrium with $B\ne 0$ has Jacobian determinant $-2a_2{a}_3{B}^2<0$, hence is a saddle equilibrium. The two invariant-axis equilibria meet in a transcritical bifurcation, and the mixed branches meet them in pitchfork bifurcations.

A near-identity transformation is a local change of variables whose derivative at the reference point is the identity. In a normal form of a dynamical system, substituting $x=y+h(y)$ and solving $(I+Dh(y))\dot{y}=f(y+h(y))$ determines which nonlinear coefficients can be removed. The transformation is locally invertible by the inverse function theorem. Terms whose removal equations are singular are resonant and must be retained; a computed zero resonant coefficient cannot be assigned an arbitrary nonzero value by normalization.

A change in local branches or stability of equilibrium points of a dynamical system as a parameter varies. A zero Jacobian eigenvalue is necessary for the equilibrium-branch bifurcations of a smooth finite-dimensional system: without it, the implicit function theorem gives a unique nearby equilibrium branch. It is not sufficient by itself; nonzero coefficients in the reduced normal form of a dynamical system establish the bifurcation type. A Hopf crossing has nonzero imaginary eigenvalues and is a different local bifurcation.

A subcritical branch emerges on the parameter side where the original state remains stable. In the cubic Landau amplitude equation $\dot{A}=\mu A-g|A{|}^{2}A$ with $g<0$, the small nonzero branch lies at $\mu <0$ and is unstable to radial amplitude perturbations. Higher-order saturation can generate a finite-amplitude stable branch and hysteresis, but this does not follow from a cubic truncation alone.

A supercritical branch emerges on the parameter side where the original state loses stability. In the cubic Landau amplitude equation $\dot{A}=\mu A-g|A{|}^{2}A$ with $g>0$, the nonzero branch $|A{|}^2=\mu /g$ lies at $\mu >0$ and is stable to radial amplitude perturbations. Continuous phase symmetry still leaves a neutral phase direction. The direction of a branch and its stability should both be checked in a general problem of bifurcation theory.

A codimension-two bifurcation requires two independent parameter conditions to hold simultaneously. Simultaneous stationary and Hopf bifurcation thresholds require both kinds of critical mode in the reduced amplitude equation. A crossing of modes at different wavenumbers is different from a zero-frequency double-zero degeneration of one fixed-wave-number cubic.

A cusp is a two-parameter degeneracy at which two saddle-node bifurcation curves meet and the reduced steady equation has a triple root. In the displayed scalar normal form, simultaneous vanishing of the polynomial and its derivative gives ${\beta }_2=3x^2$, ${\beta }_1=-2x^3$. Thus the folds satisfy $4{\beta }_2^3=27{\beta }_1^2$, with ${\beta }_2\ge 0$. Between the folds there are three real equilibria, and outside them one. A transverse stable direction may be present in a higher-dimensional realization.

A fold-Hopf bifurcation combines a simple zero eigenvalue and a nonzero imaginary pair at an equilibrium point. In a three-dimensional centre manifold, a normal form of a dynamical system separates a real fold amplitude from a complex oscillatory amplitude. A rotationally symmetric normal form permits reduction to two real amplitude equations, while the oscillation phase continues to evolve. Secondary bifurcations of the amplitude flow can produce invariant tori in the full dynamical system.

The amplitude equations $u^{\prime }=-2uv$, $v^{\prime }={\lambda }_2+v^2+u^2$ satisfy $u^{\prime }=F_v$, $v^{\prime }=-F_u$, hence preserve the displayed first integral. For ${\lambda }_2=-a^2<0$ and $u\ge 0$, the centre equilibrium is $(a,0)$ with $F=2a^3/3$, while the two boundary saddle equilibria are $(0,\pm a)$ with $F=0$. The level $F=0$ contains the boundary connection and the arc $u^2=3(a^2-v^2)$ joining the two saddle equilibria. Levels $0<F<2a^3/3$ form closed periodic orbits. Adding an independent rotating phase turns these closed amplitude trajectories into invariant tori.

A saddle-node separatrix-loop point occurs when a saddle-node equilibrium simultaneously has a global returning separatrix at its strong stable manifold. In the local form $\dot{x}=x^2-\mu$, $\dot{y}=-\lambda y$, a global return to $y=h$ at $x=\nu$ makes the codimension-two point $\mu =\nu =0$. For $\mu >0$ a saddle loop occurs on $\nu =\sqrt{\mu }$. The saddle-node boundary $\mu =0$ has an invariant-circle segment on the $\nu <0$ side, while for $\nu >0$ a nearby periodic orbit avoids the bottleneck and survives the local equilibrium fold. The return geometry, rather than the local saddle-node normal form alone, determines the global orbit.

A Bogdanov–Takens bifurcation has a double zero eigenvalue with a nontrivial Jordan block and generic two-parameter unfolding. Its local diagram includes saddle-node bifurcations, Hopf bifurcations, and a curve of small homoclinic orbits. A weakly perturbed Hamiltonian scaling can locate the homoclinic curve through energy balance.

For $\lambda >0$, the equilibrium points are the origin and $(s,s^2)$ with $s=(1\pm \sqrt{1+4\mu })/2$. The nonzero branches have Jacobian trace $s-\lambda$ and determinant $\lambda s(2s-1)$. They have a saddle-node bifurcation at $\mu =-1/4$, $s=1/2$; the origin exchanges stability with the lower branch at the transcritical bifurcation $\mu =0$. A Hopf bifurcation occurs on the upper branch at $\mu =\lambda (\lambda -1)$ for $\lambda >1/2$. The point $(\mu ,\lambda )=(-1/4,1/2)$ has a double zero eigenvalue and a nontrivial Jordan block. Its generic unfolding includes a saddle-loop curve and the intervening stable periodic orbit region.

For $\dot{u}=u(\mu +u-v)$, $\dot{v}=\lambda u^2$ in $u>0$, the time change $d\tau /dt=u$ gives the affine linear system $u_{\tau }=\mu +u-v$, $v_{\tau }=\lambda u$. Shifting $w=v-\mu$ removes the forcing. Its eigenvalues are $(1\pm \sqrt{1-4\lambda })/2$. For $\lambda >1/4$, putting $\omega =\sqrt{4\lambda -1}/2$ givesFor fixed $\mu >0$ and initial $h,v_0$ tending to zero, the first return to small positive $u$ occurs after approximately $\pi /\omega =2\pi /\sqrt{4\lambda -1}$. This is a half-turn, not a full turn. At $\lambda =1/4$, $u=e^{\tau /2}[h+(\mu +h/2-v_0)\tau ]$ does not return and grows without bound. The loss of return diagnoses escape to infinity rather than a finite-amplitude homoclinic orbit. Physical time obeys $dt=d\tau /u$ and need not share the divergent desingularized travel time.

For $\lambda >0$, the equilibrium points are $x=\pm \sqrt{\lambda }$, $y=0$. The positive branch is a saddle; the negative branch has a subcritical Hopf bifurcation at $\mu =\sqrt{\lambda }$. A homoclinic orbit occurs at $\mu =(5/7)\sqrt{\lambda }+O(\lambda )$ near the double-zero point. Between this homoclinic curve and the Hopf curve, an unstable periodic orbit surrounds a stable equilibrium point. The factor $5/7$ follows from a Melnikov energy-balance method, using the integrals of $v^2$ and $u{v}^2$ along the leading conservative saddle loop.

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The sign $\sigma =\pm 1$ distinguishes an inverted quartic potential from a double-well potential. The origin has determinant $\lambda$ and trace $\kappa$. Nonzero equilibria obey $q^2=\lambda /\sigma$, with determinant $-2\lambda$ and trace $\kappa -2\lambda$. Its energy $p^2/2+\lambda q^2/2-\sigma q^4/4$ has derivative $(\kappa -2\sigma q^2)p^2$. The origin's Hopf bifurcation is supercritical for $\sigma =1$ and subcritical for $\sigma =-1$. In the latter case the nonzero equilibria have supercritical Hopf bifurcations at $\kappa =2\lambda <0$.

// Target: dynamical-systems.bigb

For a reflection-symmetric vector field $\dot{u}=v+f_{30}{u}^3+f_{21}{u}^{2}v+\cdots$, $\dot{v}=g_{30}{u}^3+g_{21}{u}^{2}v+\cdots$, introduce $x=u$, $y=\dot{u}$. Differentiating gives $\dot{x}=y$ and a cubic acceleration with $x^{2}y$ coefficient $b=g_{21}+3f_{30}$. The contribution $3f_{30}$ comes from differentiating the first equation's cubic term. Further cubic near-identity transformations can remove nonresonant terms but cannot turn $b=0$ into a nonzero coefficient by a nonsingular scaling. This is a useful check before assigning a generic reflection-symmetric Bogdanov–Takens bifurcation to a calculated Taylor expansion.

For a vector field with linear part $\dot{u}=v$, $\dot{v}=0$ and cubic terms $a_i{u}^3+b_i{u}^{2}v+c_{i}u{v}^2+d_i{v}^3$, write $(u,v)=(x,y)+(h_1,h_2)$ with homogeneous cubic $h_i$. The transformed cubic field is $f_3+Ah-DhA(x,y{)}^T$, where $A(x,y{)}^T=(y,0{)}^T$. Taking the coefficients of $h_2$ to be $(-a_1,c_2/2,d_2,0)$ and those of $h_1$ to be $((b_1+c_2/2)/3,(c_1+d_2)/2,d_1,0)$ eliminates every cubic term in $\dot{x}$ and the $x{y}^2,y^3$ terms in $\dot{y}$. The remaining normal form is $\dot{x}=y$, $\dot{y}=P{x}^3+Q{x}^{2}y$ through cubic order. The invariant damping coefficient contains the contribution $3a_1$ from differentiating the first equation.

Near a nilpotent origin with $\sigma =1$, $\mu =0$, the weighted scaling $u_{\mathrm{old}}={\epsilon }^{2}u$, $v_{\mathrm{old}}=\epsilon v$, $\tau =\epsilon t$, ${\mu }_{\mathrm{old}}={\epsilon }^2\mu$, $\sigma =1+{\epsilon }^{2}s$ makes the leading cubic confinement flow $u^{\prime }=-sv+v^3/2$, $v^{\prime }=2u$. Direct differentiation proves conservation of $H=u^2+s{v}^2/2-v^4/8$. For $s>0$ the origin is a centre and $(0,\pm \sqrt{2s})$ are saddles. Their heteroclinic connections have $H=s^2/2$ and $u=\pm (2s-v^2)/(2\sqrt{2})$ for $|v|\le \sqrt{2s}$. Positive lower energy levels surround the centre; higher levels escape the potential well.

For a chosen orientation and phase-reduced steady amplitude $a$ and nonresonant oscillatory amplitude $z$, a schematic normal form is $\dot{a}=r_{s}a-g_s{a}^3-h_{s}a|z{|}^2$, $\dot{z}=(r_o+i{\omega }_0)z-g_o|z{|}^{2}z-h_o{a}^{2}z$. For $r_s,g_s>0$ the pure steady branch exists and is radially attracting; its oscillatory perturbation is damped when $r_o-h_{o,r}{r}_s/g_s<0$, where $h_{o,r}=\mathrm{Re}(h_o)$. For $r_o,g_{o,r}>0$, $g_{o,r}=\mathrm{Re}(g_o)$, the pure oscillatory branch exists and is radially attracting; its steady perturbation is damped when $r_s-h_s{r}_o/g_{o,r}<0$. The transverse inequalities are not existence or full stability tests by themselves. Mixed intensities $u=a^2>0$, $v=|z{|}^2>0$ solve $g_{s}u+h_{s}v=r_s$, $h_{o,r}u+g_{o,r}v=r_o$. Their intensity Jacobian matrix has trace $-2(g_{s}u+g_{o,r}v)$ and determinant $4uv(g_s{g}_{o,r}-h_s{h}_{o,r})$, giving attraction in intensities when the trace is negative and determinant positive. Resonant phases, oppositely travelling modes or zero-frequency limits need additional retained amplitudes; this schematic pair is not a universal complete normal form.

At a stationary bifurcation, a real eigenvalue of the linearized system passes through zero. Nearby equilibria may then be created, destroyed, or exchange stability.

An amplitude equation describes the slow evolution of the coefficient of a critical mode near an instability threshold. Symmetry and solvability determine its leading nonlinear terms.

The partial differential equation $T_t=-\mu \Delta T-{\Delta }^{2}T+\nabla \cdot (|\nabla T{|}^2\nabla T)$ on a $2\pi$-periodic square conserves the spatial mean. On a fixed-mean subspace its first instability is at $\mu =1$ and has four real critical amplitudes. With $\mu =1+{\epsilon }^2\nu$ and $T=\epsilon (A{e}^{ix}+B{e}^{iy}+\text{complex conjugate})+\cdots$, projection onto the critical Fourier modes gives the displayed equation and its $A,B$ interchange. For $\nu >0$, squares have $|A{|}^2=|B{|}^2=\nu /5$, amplitude eigenvalues $-2\nu ,-2\nu /5$, and neutral translation phases. Rolls have a positive transverse growth rate $\nu /3$.

With periodic boundary conditions, the spatial mean of $B$ is a conserved quantity. For zero mean write $B=H_X$ with periodic zero-mean $H$. The functional $V=⟨\mu A^2/2+\alpha A^3/3-A^4/4-A_X^2/2-A^2{H}_X/2-H_X^2/(4\delta )⟩$ generates $A_T=\delta V/\delta A$, $H_T=2\delta \sigma \delta V/\delta H$. Thus $V_T=⟨A_T^2+H_T^2/(2\delta \sigma )⟩\ge 0$. Completing the square in $H_X$ bounds $V$ above by a coercive quartic polynomial when $0<\delta <1$, ruling out nonconstant recurrent motion on regular precompact trajectories.

For $0<\delta <1$ and $\alpha \ne 0$, a broad localized plateau in the conserved-field real amplitude equation has effective parameter $\nu =\mu -\delta ⟨A^2⟩$ and stationary equation $A^{\prime \prime }+\nu A+\alpha A^2-(1-\delta )A^3=0$. Equal endpoint values of its first integral give $R=2\alpha /[3(1-\delta )]$, $\nu =-2{\alpha }^2/[9(1-\delta )]$. If the plateau occupies fraction $h$ of a long periodic interval, $⟨A^2⟩\simeq h{R}^2$ supplies the displayed relation. It applies when both the plateau and exterior are long relative to the front width; $0<h<1$ gives its leading existence interval.

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 forcing at the carrier wavenumber breaks continuous spatial translation symmetry and adds a constant to its complex amplitude equation. In a frame following a moving forcing pattern, its phase drift gives $i\Lambda A$. A nonzero forcing coefficient is normalized to one by scaling amplitude by its cube root and time by its two-thirds power. A steady intensity $s=|A{|}^2$ satisfies $s[(\mu -s{)}^2+{\Lambda }^2]=1$, and its phase is uniquely determined by the steady complex equation.

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.

Two counterpropagating modes transform as $(A,B)\mapsto (e^{ik\delta }A,e^{-ik\delta }B)$ under spatial translation. Forcing at twice their natural frequency breaks continuous temporal phase symmetry but preserves simultaneous sign reversal. Hence $\overline{B}$ transforms like $A$ and $\overline{A}$ like $B$, allowing linear parametric coupling. An equal-amplitude phase-locked standing wave has $|A{|}^2=|B{|}^2=z>0$ with $|\beta +\gamma {|}^2{z}^2-2\mu \mathrm{Re}(\beta +\gamma )z+{\mu }^2-{\nu }^2=0$. A nonnegative discriminant is only an algebraic compatibility condition: a physical nonzero wave also requires a positive root.

Writing $U=\mathrm{Re}(\beta +\gamma )$ and $V=\mathrm{Im}(\beta +\gamma )$, the squared amplitude is $z_{\pm }=[\mu U\pm \sqrt{(U^2+V^2){\nu }^2-{\mu }^2{V}^2}]/(U^2+V^2)$. Reality of the square root does not guarantee positivity. For example, $U=1,V=0,\mu =-1,\nu =1/2$ gives two negative roots. Under the usual supercritical assumptions $\mu >0,U>0$, the plus root is positive whenever the discriminant is nonnegative.

// Destination: mathematical-biology.bigb

The Newell–Whitehead–Segel equation is an amplitude equation for slowly modulated stationary rolls in an isotropic pattern-forming medium. With the displayed sign convention, a roll $A=R{e}^{iqX}$ has $R^2=\mu -q^2$. A transverse phase modulation has linear growth rate $q{k}^2-k^4/4$, so long-wave transverse instability occurs for $q>0$. Reversing the carrier-wave convention reverses the sign attached to its detuning $q$; one must derive the stability sign from the operator actually used.

At a nonzero critical wavenumber in an isotropic medium, a small transverse wavevector changes its magnitude only at second order. Balancing radial detuning therefore requires the displayed unequal slow-space scales. With the positive carrier $e^{i{k}_{c}x}$, the leading Newell–Whitehead–Segel equation spatial term is $\xi ({\partial }_X-i{\partial }_Y^2/(2k_c){)}^{2}A$. A roll detuning $q$ has transverse phase growth $-\xi q{p}^2/k_c-\xi p^4/(4k_c^2)$; negative $q$ permits a zigzag instability. Changing the carrier convention changes the accompanying sign of $q$. Replacing this operator by an isotropic second-order Laplacian misses the critical circle's geometry.

Independent sign changes of three amplitudes and a cyclic coordinate permutation force each cubic component to be its own amplitude times a cyclic permutation of one quadratic polynomial. One convenient parametrization is $\dot{x}=x(\mu -aX-c{y}^2+e{z}^2)$ with cyclic companions and $X=x^2+y^2+z^2$. The axial and diagonal isotropy types give primary branches; two-coordinate branches can emerge in secondary pitchfork bifurcations when cross-couplings change sign.

When $c=e$ in the signed cyclic three-mode amplitude equations, $\dot{X}=2X(\mu -aX)$ and $\dot{V}=3V(\mu -aX)$ for $V=xyz$. Thus $V/X^{3/2}$ is conserved wherever $X>0$, precluding asymptotically stable isolated equilibria. On $X=\mu /a$, each octant contains closed contours of $V$ around the equal-magnitude equilibrium, bounded by heteroclinic arcs between coordinate axes if $c\ne 0$. For $c=0$ the sphere consists entirely of equilibria.

Three Fourier wavevectors sum to zero and form a hexagonal star. Translations give each amplitude the phase of its own wavevector; cyclic rotations permute them, and a half-turn conjugates them. These symmetries permit the resonant quadratic term and force its coefficients to be real. A spatial reflection would interchange the two cross-couplings and require $b=c$, but chiral patterns need not have that extra symmetry. The equations describe rolls, hexagons, phase locking and oscillatory changes in dominant orientation.

For equal positive amplitudes $r$, let $s=a+b+c$ and use $\mu +\alpha r-s{r}^2=0$. The radial Jacobian eigenvalue is $\alpha r-2s{r}^2$. The conjugate pair has real part $(b+c-2a)r^2-2\alpha r$ and imaginary parts $\pm \sqrt{3}(b-c)r^2$. When $b+c>2a>0$ and $b\ne c$, the upper hexagon branch loses amplitude stability through this pair at the displayed radius. The associated parameter is ${\mu }_H=2{\alpha }^2(4a+b+c)/(b+c-2a{)}^2$. Nonlinear terms decide the direction and stability of the resulting periodic branch.

At a positive equilibrium set $x_i=r_i^2$ and $p=r_1{r}_2{r}_3$. The equations imply $a{x}_i+b{x}_{i+1}+c{x}_{i+2}-\alpha p/x_i=\mu$. Suppose $b>a>c$ and relabel cyclically so $x_1$ is maximal. If $x_2\ge x_3$, subtracting equations one and two makes the cubic side positive unless all are equal, whereas $\alpha p(1/x_1-1/x_2)$ is nonpositive. If $x_2\le x_3$, subtracting equations two and three gives the opposite sign contradiction. Interchanging the orientation handles $c>a>b$. Thus all positive equilibria have equal amplitudes. With $\alpha \ne 0$, a state with exactly two nonzero amplitudes cannot be an equilibrium because it immediately forces the third.

For $A_i=r_i{e}^{i{\varphi }_i}$ with all $r_i>0$, the translation-invariant total phase is $\Phi ={\varphi }_1+{\varphi }_2+{\varphi }_3$. With $\alpha >0$, the displayed equation makes $\Phi =0$ stable and $\Phi =\pi$ unstable. A spatial translation can remove two phases, so the stable locked states have all amplitudes positive real in a suitable origin. This is a statement modulo translation, not that every translated representative is real; the zero solution and rolls belong to boundary strata.

A critical pattern Fourier mode transforms under a spatial translation as $A\mapsto e^{i\varphi }A$. A forcing at three times the critical wavenumber transforms as $F\mapsto e^{3i\varphi }F$. Their lowest resonant product with phase weight one is $F{\overline{A}}^2$. This determines the form of the leading weak-forcing amplitude equation, while projection onto an adjoint eigenfunction determines its coefficient. Symmetry permits this coupling but does not ensure that it is nonzero. Travelling forcing gives a time-dependent phase to $F$.

For small positive forcing-frame frequency $\omega$, use $\mu =\hat{\mu }{\omega }^2$, $C=\omega z$ and $\tau =\omega T$ in the three-to-one spatially forced amplitude equation. The leading system is $z_{\tau }+iz={\overline{z}}^2$. Writing $z=x+iy$ gives the Hamiltonian system $(x_{\tau },y_{\tau })=(H_y/2,-H_x/2)$ and the displayed first integral. The origin is a center equilibrium; three saddle equilibria at $(0,1)$ and $(\pm \sqrt{3}/2,-1/2)$ lie on $H=1/3$. The factorization $H-1/3=(1+2y)[x^2-(y-1{)}^2/3]$ reveals a triangular heteroclinic cycle, containing closed periodic orbits for every $0<H<1/3$.

The canonical three-to-one spatially forced amplitude equation has nonzero equilibrium points $C=R{e}^{i\theta }$ satisfying $\cos 3\theta =(R^2-\mu )/R$ and $\sin 3\theta =-\omega /R$. Thus $(R^2-\mu {)}^2+{\omega }^2=R^2$. Each positive amplitude has three phases separated by $2\pi /3$: two amplitude branches normally mean six complex equilibrium points. For $\mu >{\omega }^2-1/4$ the larger branch is asymptotically stable and the smaller consists of saddle equilibria, except that its zero-amplitude root at $(\mu ,\omega )=(0,0)$ is not a nonzero equilibrium point. Equality gives a saddle-node bifurcation. This phase locking breaks continuous translation symmetry down to threefold symmetry.

For the three-to-one spatially forced amplitude equation with forcing $e\exp [3i(\Omega T+\delta )]$, put $A=B\exp [i(\Omega T+\delta )]$. The real phase constant $\delta$ belongs inside the factor of $i$. If $c>0$ and the phase has been chosen so that $e>0$, the scaling $B=(e/c)C$, $\mathcal{T}=(e^2/c)T$ gives the displayed canonical amplitude equation, with $\mu =c\tilde{\mu }/e^2$ and $\omega =c\Omega /e^2$. A negative cubic saturation coefficient cannot give the same cubic sign under a forward-time normalization; zero forcing or zero cubic coefficient requires another scaling.

The real-coefficient Ginzburg–Landau amplitude equation has complex $A$ and real parameters, usually $\xi ,g>0$ near a supercritical stationary pattern onset. A detuned plane wave $A=\rho e^{iQX}$ has ${\rho }^2=(r-\xi Q^2)/g$. Its phase and amplitude sidebands determine the narrower Eckhaus instability stability band. It is a local envelope model; additional conserved fields or mean flows may need their own equations.

A cubic destabilizing term and quintic saturation describe a subcritical stationary pattern bifurcation. A plane wave $A=R{e}^{iQX}$ has squared amplitude $x=R^2$ satisfying $\mu -Q^2+\alpha x-x^2=0$. The nonzero branches meet at $x=\alpha /2$, $\mu =Q^2-{\alpha }^2/4$. The lower branch is unstable to amplitude perturbations; the upper branch can also undergo an Eckhaus instability of its phase.

For $A_T=\mu A+3\hat{s}A|A{|}^2-10A|A{|}^4+4A_{XX}$, a nonzero uniform real state with $B=A_0^2$ obeys $\mu =-3\hat{s}B+10B^2$. Real amplitude sidebands grow at $6\hat{s}B-40B^2-4{\ell }^2$, while phase sidebands grow at $-4{\ell }^2$. The maximum amplitude growth coefficient is $9{\hat{s}}^2/40$, attained at $B=3\hat{s}/40$. Thus nonzero uniform states cannot have modulation bifurcations if the smallest allowed slow wavenumber exceeds $3\hat{s}/(4\sqrt{10})$. For a physical domain of length $L$ and $X={\epsilon }^{2}x$, this gives $L<8\pi \sqrt{10}/(3s)$. The zero state is an exception: it has sideband thresholds $\mu =4{\ell }^2$ for every positive allowed wavenumber, so an assertion including all uniform states would be false.

For $r,\xi ,g>0$, longitudinal phase modulation of a detuned convection roll yields diffusion coefficient $D_{\varphi }=\xi (r-3\xi Q^2)/(r-\xi Q^2)$. The expression follows by eliminating the fast amplitude correction from the coupled linear amplitude-phase equations at small modulation wavenumber. Thus convection rolls become unstable for $Q^2>r/(3\xi )$ although they exist for $Q^2<r/\xi$. At the boundary the leading phase diffusion vanishes and higher spatial orders matter.

For a real Ginzburg–Landau equation plane wave with amplitude squared $b/g>0$, perturbing its amplitude and phase couples Fourier modes on either side of its carrier wavenumber. The displayed eigenvalues follow from the real-linear two-component perturbation equation. Their small-$p$ phase branch is $-\xi (\mu -3\xi q^2)p^2/(\mu -\xi q^2)+O(p^4)$, so the robust infinite-domain stable band is $q^2<\mu /(3\xi )$. On a finite periodic envelope domain only discrete $p$ are allowed; an instability requires an allowed $p$ with $p^2<6q^2-2\mu /\xi$. Uniform amplitude perturbations alone cannot determine this Eckhaus instability.

For the upper plane-wave branch of the quintic real Ginzburg-Landau equation, put $\Lambda =2R^2(2R^2-\alpha )>0$. A modulation of wavenumber $p$ has growth rates $-p^2-\Lambda /2\pm \sqrt{{\Lambda }^2/4+4Q^2{p}^2}$. Long-wave phase diffusion is positive when $4Q^2<\Lambda$, giving the displayed stability boundary. At equality the leading phase growth is $-p^4/\Lambda$, so the boundary is marginal in the diffusion coefficient but damped at the next spatial order. In the $(\mu ,Q)$ plane the boundary is $\mu =2Q^2-\alpha [\alpha +\sqrt{{\alpha }^2+16Q^2}]/8$.

The cubic Landau amplitude equationwith $g>0$ describes saturation through a supercritical pitchfork bifurcation when a reflection symmetry permits both signs of $A$.

A steady spatial forcing at twice the critical wavenumber permits coupling of the negative critical Fourier mode to the positive critical mode. A weakly nonlinear expansion then permits a term $\beta \overline{A}$ in the Landau amplitude equation. Its coefficient is found by projecting the resonant forcing-advection terms onto the adjoint eigenfunction. Reflection-symmetric forcing permits real coefficients. The symmetry permission does not prove a nonzero coefficient: it can vanish for a particular model or mode structure.