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The equilibria of $\dot{x}=(a^2-x)(a-y^2)$, $\dot{y}=x-y$ lie onfor every $a$, and on $x=y=\pm \sqrt{a}$ for $a\ge 0$. They meet in a subcritical pitchfork bifurcation at $a=0$; the positive square-root branch meets the $a^2$ branch and exchanges stability in a transcritical bifurcation at $a=1$.

For three-mode porous magnetoconvection, set $\mu =r-1-q$ and assume $0<q<\zeta /\sigma$. The transverse eigenvalues are negative. Centre-manifold invariance givesSubstitution into the first equation gives the displayed positive linear and cubic coefficients. Thus the bifurcation is a subcritical pitchfork bifurcation: the origin is stable for $\mu <0$ and unstable for $\mu >0$, while the small nonzero branches at $a^2=-{\zeta }^2\mu /(3q)$ are unstable. Simply setting $\dot{b}=0$ misses the time-derivative contribution responsible for the common factor $(1-\sigma q/\zeta {)}^{-1}$.

For $r>0$, the origin always exists. Its derivative $r-1$ makes it attracting for $0<r<1$ and repelling for $r>1$. Nonzero equilibria solveThey occur in symmetric pairs. On the first positive lobe $0<x<\pi$, the sinc function decreases from one to zero, giving one positive and one negative nonzero equilibrium for $0<r<1$. Both are repelling, since $f^{\prime }=\sin x/x-\cos x>0$ there. Near $r=1$,The unstable branches meet the stable origin in a subcritical pitchfork bifurcation. At $r=1$ the origin is still nonlinearly repelling because the leading term is $+x^3/6$. For $1<r\le 2$ it is the only equilibrium.

For each higher positive-sine lobe $2n\pi <x<(2n+1)\pi$, $n\ge 1$, the sinc function has a single positive maximum at $a_n$, whereIf $0<r<r_n$, that lobe has two roots: the smaller positive root is attracting and the larger is repelling, because $f^{\prime }=-xd(\sin x/x)/dx$. Their negative counterparts have the same stability. At $r=r_n$ the pair on each side merges in a saddle-node bifurcation; the double equilibria are semistable. For $r>r_n$ that lobe contributes none. The first such threshold is about $0.128375$.

For $n\gg 1$, put $b_n=(2n+1/2)\pi$; expanding near the sine maximum gives $a_n=b_n-b_n^{-1}+O(b_n^{-3})$ and $r_n\simeq 1/b_n$. Away from bifurcation values, the exact counting rule for $0<r<1$ and its small-$r$ estimate areAt a saddle-node threshold, count each merged positive or negative equilibrium once rather than twice. Infinitely many thresholds accumulate at $r=0$, consistent with the infinitely many equilibria in solution.

A simple zero eigenvalue occurs on $c=d$, except at $(1,1)$. The odd symmetry of the vector field gives a pitchfork bifurcation. Near this curve the critical eigenvalue isFor $0<d<1$ the other two eigenvalues are stable: this is a supercritical pitchfork bifurcation as $c$ decreases through $d$. For $d>1$ it is a subcritical pitchfork bifurcation in the center direction, with an already unstable transverse direction.

For a Hopf bifurcation, factor the characteristic polynomial as $(m+\tau )(m^2+{\omega }^2)$. Comparing coefficients givesThus the trivial branch has the local bifurcation curvesThe second curve is a Hopf bifurcation with a stable third eigenvalue. The Routh-Hurwitz stability criterion gives the stable trivial region $0<d<1$, $d<c<(d-2{)}^2$. Its Hopf bifurcation is subcritical. To check the sign, normalize the critical right eigenvector by $q_x=1$ and putThe second component of the normalized left projection is $G$. There is no quadratic term at the origin, and the cubic derivative in the second component is $C_2(q,q,\bar{q})=2$. The cubic Hopf coefficient is therefore $l_1=\mathrm{Re}G/\omega =4/[\omega ((4-d{)}^2+4{\omega }^2)]>0$. A small repelling periodic orbit lies on the stable-origin side. The diagram also includes the nonzero-branch information derived next.

Use local coordinates $q=\theta -\pi /2$, $p=\varphi$ at $P_1$. The Jacobian matrix, its trace and its determinant areConsequently $P_1$ is attracting when $T<0,D>0$, repelling when $T>0,D>0$, and a saddle equilibrium when $D<0$. A transition between a node and a focus at $T^2=4D$ changes the linear geometry but is not a bifurcation of the local topological phase portrait.

The stationary bifurcation curve is $\lambda +\kappa =2$. To identify it, write $C=\cos 2\varphi$, $S=\sin 2\varphi$. At a non-equatorial equilibrium point, eliminating $S$ givesNear $(3,-1)$, if $d=2-\lambda -\kappa >0$, these yield $p=\pm \sqrt{d}/2+O(d^{3/2})$ and $q=p+o(\sqrt{d})$. The two branches are related by the local reflection $(q,p)\mapsto (-q,-p)$. Their determinants are $-4d+o(d)$, so they are saddle equilibria. This is a pitchfork bifurcation; on the portion with $T<0$ the saddle branches lie on the side where $P_1$ is stable, giving a subcritical pitchfork bifurcation.

A Hopf bifurcation requires $T=0,D>0$, henceThe oscillation frequency is $\omega =\sqrt{-2(1+\kappa )}$. At the intersection $\boxed{(\lambda ,\kappa )=(3,-1)}$, $J\ne 0$ but $J^2=0$: there is a double zero eigenvalue with one eigenvector. The map from $(\lambda ,\kappa )$ to $(T,D)$ has nonzero Jacobian determinant, so two independent parameter conditions produce this codimension-two bifurcation. Reflection removes quadratic terms; the appropriate initial comparison is with a reflection-symmetric cubic double-zero normal form, subject to the cubic degeneracy checked below.

For completeness, the printed vector field has a subcritical, rather than supercritical, Hopf bifurcation. In the mechanical coordinates $x=p$, $y=\dot{p}$, at $\lambda =3$ its cubic acceleration has $x^{2}y$ coefficient $b=6(\kappa -3)(\kappa +1)/(\kappa -1{)}^2$ and $y^3$ coefficient $d_3=-2/(\kappa -1{)}^2$. For a small harmonic oscillation, the averaged cubic change of $H=(y^2+{\omega }^2{x}^2)/2$ is ${\omega }^2{A}^4(b+3d_3{\omega }^2)/8$, whereThus a small unstable periodic orbit lies on the attracting side of the Hopf bifurcation. This calculation matters when comparing the original equations with the later assumed normal form.

The truncated fields must be interpreted as a Galerkin method approximation: project each equation onto its retained spatial modes and discard orthogonal higher harmonics. They are not an exact pointwise solution of the full partial differential equations, because the nonlinear terms generate additional harmonics. Write $X=kx$, $Z=\pi z$, and abbreviate the printed normalizations byThen $\psi =Sa\sin X\sin Z$, $\theta =Tb\cos X\sin Z-c\sin 2Z/\pi$, and $\chi =Dd\sin X\cos Z+Ee\sin 2X$. All five amplitudes a,b,c,d,e depend on $\tau =pt$. Use the spatial Jacobian determinant $J(F,G)=F_x{G}_z-F_z{G}_x$ and the inner product over the stated rectangle. Orthogonality of the trigonometric modes makes projection equivalent to retaining their coefficients. In particular, ${\nabla }^2\psi =-p\psi$, so $J(\psi ,{\nabla }^2\psi )=0$.

For the temperature equation, direct products giveThe other temperature contribution is $-2kSac\cos X\sin Z\cos 2Z$. Since $\sin Z\cos 2Z=(\sin 3Z-\sin Z)/2$, its retained component is $kSac\cos X\sin Z$. Thus the projected temperature equation isHere the dot denotes $d/d\tau$. Since $kS/(pT)=1$ and $\phi =4{\pi }^2/p$, these equations give $\dot{b}=-b+a(1-c)$ and $\dot{c}=\phi (-c+ab)$.

Similarly, the two magnetic-advection products arewhile $J(\psi ,Ee\sin 2X)=-2\pi Sae\sin X\cos Z\cos 2X$ has retained component $\pi Sae\sin X\cos Z$. The projected induction equation is thereforeUsing $\pi S/(pD)=1$ and $4k^2/p=4-\phi$ gives $\dot{d}=-\zeta d+a(1-e)$ and $\dot{e}=-(4-\phi )\zeta e+\phi ad$.

It remains to project the Lorentz force in the vorticity equation. The linear term is $J(x,{\nabla }^2\chi )=p\pi Dd\sin X\sin Z$. The nonlinear cross term isThe Jacobian on the right is $2\pi kDEde\sin X\sin Z\cos 2X$. Its retained component is $-\pi kDEde\sin X\sin Z$, so the total retained magnetic term isMeanwhile the projected acceleration, viscous term and temperature forcing have coefficients $-p^{2}S\dot{a}$, $p^{2}Sa$, and $-RkTb$. Divide the projected vorticity equation by $-p^{2}S$ to obtain $\dot{a}=\sigma [-a+rb+\zeta qd\{(\phi -3)e-1\}]$, using $r=k^{2}R/p^3$ and $q={\pi }^{2}Q/p^2$. This derives the complete five-mode vertical-field magnetoconvection system:The Rayleigh number, Chandrasekhar number, Prandtl number and magnetic-to-thermal diffusivity ratio enter through r,q,$\sigma$ and $\zeta$ respectively.

Linearization at the zero-amplitude state leaves c and e with negative rates $-\phi$ and $-(4-\phi )\zeta$, while a,b,d have coefficient matrixA zero eigenvector satisfies $b=a$, $d=a/\zeta$ and $[-1+r-q]a=0$. Thus the stationary threshold is $\boxed{r_c=1+q}$. The reversal symmetry $(a,b,c,d,e)\mapsto (-a,-b,c,-d,e)$ makes the nontrivial steady states occur as a pair.

For these nonzero steady states put $A=a^2$. The temperature equations give $c=ab$ and $b=a(1-c)$, so $b=a/(1+A)$ and $c=A/(1+A)$. The induction equations give $\zeta d=a(1-e)$ and $(4-\phi )\zeta e=\phi ad$. With $\mu =(4-\phi ){\zeta }^2/\phi$, their solution isHere $k>0$ gives $0<\phi <4$, and positive diffusivity gives $\mu >0$. Substituting these amplitudes into the stationary a equation and dividing by a yields the stationary branch of five-mode magnetoconvection:At $a=0$ this meets $r=1+q$. Expansion in $A=a^2$ givesThe denominator of its initial slope is positive, and its numerator after multiplication by $(4-\phi ){\zeta }^2$ isTherefore $\mathcal{N}>0$ gives a supercritical pitchfork bifurcation direction, and $\mathcal{N}<0$ gives a subcritical pitchfork bifurcation direction. If $\mathcal{N}=0$, the leading slope vanishes and higher terms must be retained.

This classification describes which side of the stationary threshold contains the steady branch. It does not alone establish dynamical stability or rule out an earlier oscillatory instability. For example, at $r_c$ the characteristic polynomial factors asThe zero eigenvalue is simple unless the final coefficient vanishes; the other two eigenvalues need not both be stable for every positive parameter choice. The stationary branch formula and its direction criterion remain the requested algebraic conclusions.

At a fixed point $y=x$, and the first equation factors. The branches areThe three branches meet at $a=0$, and the positive square-root branch meets the quadratic branch at $a=1$. The Jacobian matrix at a general fixed point isOn $x=a^2$, its trace is $-1$ and determinant $a-a^4=a(1-a^3)$: the branch is attracting for $0<a<1$ and a saddle for $a<0$ or $a>1$. On $x=\pm \sqrt{a}$, the eigenvalues are $-1$ and $2a(1\mp a^{3/2})$. The negative branch is always a saddle for $a>0$; the positive branch is a saddle for $0<a<1$ and attracting for $a>1$. At the bifurcation values the zero eigenvalue requires the nonlinear analysis below.

For $a=1+\mu$, $u=x-1$, $v=y-x$, the equations areThe extended centre manifold has $v=h(u,\mu )$ with no linear terms. Since $h_u\dot{u}$ has degree at least three, its invariance equation to degree two is $0=-h_2-(2{\mu }^2-5\mu u+2u^2)$. ThusThe reduced quadratic factors as $(u-2\mu )(2u-\mu )$. Its two roots have reduced eigenvalues $3\mu$ and $-3\mu$, respectively, and the transverse eigenvalue remains negative. They exchange stability at $\mu =0$, a transcritical bifurcation. The exact branches have $u=2\mu +{\mu }^2$ and $u=\mu /2-{\mu }^2/8+\cdots$, consistent with this second-order evolution equation.

Near $a=0$, the slow equation has leading form $\dot{x}=x^3-ax$, so this is a subcritical pitchfork bifurcation with the parameter orientation displayed. For $a<0$ there is one nearby saddle, whose unstable direction is close to $y=x$ and stable direction close to vertical. For $a>0$ the central branch is attracting and the outer two equilibria near $\pm \sqrt{a}$ are saddles; their separatrices distinguish attraction toward the central equilibrium from departure outwards. The actual central location is $(a^2,a^2)$, not exactly the origin away from $a=0$.

Adding positive $\epsilon$ breaks the pitchfork symmetry. The leading equilibrium equation near zero is $x^3-ax+\epsilon =0$: the diagram unfolds into a single branch plus a saddle-node pair, with the latter appearing for positive $a$ of order ${\epsilon }^{2/3}$. More precisely the fold is at $x=(\epsilon /2{)}^{1/3}$, $a=3(\epsilon /2{)}^{2/3}$ at leading order. Near one the reduced equation becomes $2u^2-5\mu u+2{\mu }^2+\epsilon =0$. Its discriminant is $9{\mu }^2-8\epsilon$; the crossing is replaced by two saddle-node endpoints with a no-equilibrium gap for $|\mu |<\sqrt{8\epsilon }/3$ to leading order. Outside that gap, the lower reduced root is stable and the upper one unstable. These local approximations describe the sketch; the exact perturbed curves have higher-order corrections.

Assume $\hat{s}>0$, as required for the subcritical case. The zero state exists for every $\mu$. For a nonzero uniform real state, put $B=A_0^2>0$. Its equilibrium point condition is $\mu +3\hat{s}B-10B^2=0$, givingBoth are positive for $-9{\hat{s}}^2/40<\mu <0$. At $\mu =0$, the lower branch reaches zero and the upper has $B=3\hat{s}/10$. For $\mu >0$ only the upper branch is positive. Each positive $B$ represents real states $A_0=\pm \sqrt{B}$, as well as their constant complex wave phase rotations.

The two nonzero branches merge atThe radial linearization is ${\lambda }_a=\mu +9\hat{s}B-50B^2=2B(3\hat{s}-20B)$. It is negative on the upper branch and positive on the lower. The zero state is stable for $\mu <0$ and unstable for $\mu >0$. Thus a subcritical pitchfork bifurcation branch leaves zero towards negative $\mu$, and its fold connects to the stable finite-amplitude branch. Constant wave phase shifts are neutral for nonzero complex states; the stated attraction is in amplitude, with spatial wave phase perturbations damped except for that uniform symmetry mode.

Now $G(x)=x[{\mu }^2-(x^2-a{)}^2]$. The zero point exists for all parameters and has eigenvalue $G^{\prime }(0)={\mu }^2-a^2$, so it is stable for $|\mu |<|a|$ and unstable for $|\mu |>|a|$. At every nonzero fixed point,Both outer branches are stable and both inner branches unstable.

For $a>0$, the nonzero locations are as in part (i). At $(\mu ,x)=(\pm a,0)$ the local equation is $\dot{x}=({\mu }^2-a^2)x+2a{x}^3+O(x^5)$: these are subcritical pitchfork bifurcations, with the two unstable inner branches present on the side where zero is stable. At the bifurcation itself $\dot{x}\sim 2a{x}^3$, so zero is unstable. At $(0,\pm \sqrt{a})$ the inner and outer branches cross and exchange stability in transcritical bifurcations. The positive crossing is semistable, attracting from the right; the negative crossing is semistable, attracting from the left, because the additional factor $x$ reverses the local flow there.

For $a<0$, the outer stable pair exists only for $|\mu |>|a|$. The points $(\pm |a|,0)$ are supercritical pitchfork bifurcations, since the cubic coefficient $2a$ is negative. Zero loses stability as $|\mu |$ increases across $|a|$, and a stable symmetric pair appears. At the threshold zero remains nonlinearly stable, with $\dot{x}\sim 2a{x}^3$. There are no further bifurcations. The second row of the preceding figure gives the diagrams. The plots use $a=\pm 1$; the general locations are obtained from the formulas above.

The centre manifold theorem gives, near an equilibrium of a sufficiently smooth system, a local invariant manifold tangent to the generalized eigenspaces with zero-real-part eigenvalues, expressible as a graph of a function over them. Its reduced dynamics govern local equilibria and their stability in the presence of strictly stable transverse directions. The manifold need not be unique or analytic; finite Taylor jets can nevertheless be determined from invariance. Treating a parameter as a variable with $\dot{\mu }=0$ gives the extended centre manifold for a parameter.

At $r=1$ the linearized $(x,y)$ block has eigenvalues $0,-2$, and the $z$ direction has eigenvalue $-1$. Thus the origin is a nonhyperbolic equilibrium, with stable dimension two and non-extended centre dimension one. The extended centre dimension is two. The transformed equations areThe graph of a function satisfies $V(0,0)=Z(0,0)=0$ and $DV(0,0)=DZ(0,0)=0$, with invariance equations $V_u\dot{u}=\dot{v}$ and $Z_u\dot{u}=\dot{z}$. Symmetry permits $V$ odd and $Z$ even in $u$. Giving $u$ weight one and $\mu$ weight two, comparison yieldsHere $O_{\mathrm{w}}(j)$ means terms of weighted order at least $j$ under the stated scaling. If $a<1$, the pitchfork bifurcation is supercritical: stable branches $u\sim \pm \sqrt{\mu /(1-a)}$ appear for $\mu >0$ as the origin loses stability. If $a>1$, it is a subcritical pitchfork bifurcation: unstable branches occur for $\mu <0$, while the origin is stable there.

For $a=1$, assign $\mu$ weight four. The leading cubic terms in $\dot{u}$ cancel. The invariance equations now give $V=u^3/2+O_{\mathrm{w}}(5)$ and $Z=u^2+0u^4+O_{\mathrm{w}}(6)$: $Z_u\dot{u}$ first contributes at weight six and $v^2$ also first contributes there. Substitution then givesThe derivative along the nonzero branches is $-2\mu >0$, so they are unstable. The origin is stable for $\mu <0$, unstable for $\mu >0$, and nonlinearly unstable at $\mu =0$ because the leading term is $+u^5$. This is a degenerate subcritical pitchfork bifurcation.

A fixed point satisfiesThe branch with $y=0$ isEvery other fixed point can be parametrized by its $x$-coordinate:Equivalently,The $x_{-}$ pair exists for $0\le a\le 4$, and the $x_{+}$ pair for $3\le a\le 4$.

The Jacobian matrix isOn $P_0(a)$ its eigenvalues are $1$ and $a(3-a)$, so this branch has a nonhyperbolic equilibrium at $a=0$ and $a=3$. On a nonzero-$y$ branch,which vanishes at $x=2$, hence at $a=4$ and $y=\pm \sqrt{2}$. The three bifurcation values and locations are therefore

For the center manifold calculation, append $\dot{a}=0$ as in the extended centre manifold for a parameter.

At $(a,x,y)=(0,0,0)$ put $\mu =a$ and write the centre manifold as $x=h(y,\mu )$. Its centre-manifold invariance equation isIt gives $h=\mu -y^2+O(3)$, and henceThis is the pitchfork bifurcation normal form with stable nonzero centre branches for $\mu >0$: the bifurcation at $a=0$ is supercritical.

At $(a,x,y)=(3,3,0)$ put $u=x-3$ and $\mu =a-3$. ThenAgain $u=h(y,\mu )=\mu -y^2+O(3)$, soThe nonzero centre branches exist for $\mu >0$ and are unstable in the centre direction, whereas $y=0$ is centre-stable there. Thus the bifurcation at $a=3$ is a subcritical pitchfork bifurcation with reversed normal-form parameter.

Finally fix $y_0=\pm \sqrt{2}$ and putThe translated system isThe extended centre manifold hasand thereforeThis is the saddle-node bifurcation normal form. For $\mu <0$ there are two nearby fixed points, which coalesce and disappear at $\mu =0$. The complete reductions are recorded in the Extended centre-manifold reductions of the 2019 Cambridge reflection-symmetric system.

Suppose $a<0$. Near the pitchfork at $(\mu ,x)=(a,0)$, put $\lambda =\mu -a$. Keeping the lowest terms that distinguish the branches givesSince $-2a>0$, this is the pitchfork bifurcation normal form: the zero branch changes from stable to unstable and the stable branches $x=\pm \sqrt{\lambda }$ appear for $\lambda >0$.

Near $(\mu ,x)=(0,0)$, put $\lambda =\mu$. ThenHere $-a>0$, so for $\lambda >0$ the zero branch is stable and the two nearby nonzero branches are unstable. This confirms the parameter-reversed subcritical pitchfork bifurcation.

Finally fix either $x_0=\pm \sqrt{-2a}$ and putThe two vanishing factors have the leading expansionsWith the parameter-dependent coordinate $w=2x_{0}u-m$,This is the transcritical bifurcation normal form: the branches $w=0$ and $w=m$ cross and exchange stability.

PutThe equilibrium points of a dynamical system form the three branch familiesFor a simple equilibrium, fixed point stability for an autonomous differential equation is determined by the sign of $f_x$. On the three families,Thus solid branches in the figure are stable and dashed branches are unstable.

If $a<0$, the pair $x=\pm \sqrt{\mu -a}$ emerges stably from $x=0$ at $\mu =a$ in a pitchfork bifurcation. At $\mu =0$, the pair $x=\pm \sqrt{2\mu }$ meets $x=0$ with unstable nonzero branches; after reversing the parameter orientation this is a subcritical pitchfork bifurcation. At $\mu =-a$, the two positive branches cross and exchange stability, as do the two negative branches, giving two simultaneous transcritical bifurcations at $x=\pm \sqrt{-2a}$.

If $a=0$, every branch meets at $(\mu ,x)=(0,0)$. For $\mu <0$ only the stable trivial branch exists. For $\mu >0$, $x=0$ is stable, the inner branches $x=\pm \sqrt{\mu }$ are unstable, and the outer branches $x=\pm \sqrt{2\mu }$ are stable. This is a degenerate fifth-order bifurcation rather than a generic pitchfork.

If $a>0$, a supercritical pitchfork bifurcation occurs at $\mu =0$: the trivial branch loses stability and the stable branches $x=\pm \sqrt{2\mu }$ emerge. At $\mu =a$, the unstable branches $x=\pm \sqrt{\mu -a}$ meet the trivial branch, which regains stability; this is the parameter-reversed form of a subcritical pitchfork bifurcation.

These cases comprise the bifurcations of the 2020 Cambridge quintic system.

At an equilibrium, the second equation gives $x=y=:s$. The first equation then givesHence the fixed-point branches in the $(a,x)$ plane areandThe three branches meet at $a=0$. The branch $x=a^2$ meets the positive square-root branch again when $a^2=\sqrt{a}$, so the second bifurcation value is

The Jacobian matrix at a general point isOn the branch $P_0=(a^2,a^2)$ it is triangular, with eigenvaluesThus $P_0$ is unstable for $a<0$, asymptotically stable for $0<a<1$, and unstable for $a>1$.

On a square-root branch write $s=\pm \sqrt{a}$. Thenwhose trace is $-1$ and determinant is $2s^2(s^3-1)$. The negative branch $s=-\sqrt{a}$ is therefore a saddle equilibrium for every $a>0$. The positive branch is a saddle for $0<a<1$ and asymptotically stable for $a>1$; close to $a=1$ it is a stable node.

To resolve the nonhyperbolic point at $a=0$, make the prescribed substitutionand append $\dot{a}=0$. The equations becomeThe centre subspace is $Y=X$. Seek the extended centre manifold for a parameter asThe centre-manifold invariance equation isTo second order, $\dot{X}=-aX+O(3)$, so comparison of coefficients givesand hence $A=C=0$, $B=1$. ThereforeSubstitution into the $X$ equation gives the reduced flowIts central branch $X=0$ is unstable for $a<0$ and stable for $a>0$, while the two nonzero branches $X\sim \pm \sqrt{a}$ for $a>0$ are unstable. Thus the bifurcation at zero is a subcritical pitchfork bifurcation with reversed normal-form parameter $\mu =-a$, exactly as recorded by the extended centre manifold of the 2023 Cambridge quadratic-product system.

The complete bifurcation diagram is therefore as follows. For $a<0$, only $P_0$ exists and is unstable. For $0<a<1$, $P_0$ is stable while both $P_{+}$ and $P_{-}$ are saddles. For $a>1$, $P_{+}$ is stable while $P_0$ and $P_{-}$ are saddles. At $a=1$, $P_0$ and $P_{+}$ cross and exchange stability, so the bifurcation is transcritical. This gives the bifurcation diagram of the 2023 Cambridge quadratic-product system.

Finally consider the phase plane near $(x,y)=(1,1)$ with $|a-1|\ll 1$. If $a<1$, the lower equilibrium $(a^2,a^2)$ is a stable node and the upper equilibrium $(\sqrt{a},\sqrt{a})$ is a saddle. If $a>1$, the lower equilibrium $(\sqrt{a},\sqrt{a})$ is the stable node and the upper equilibrium $(a^2,a^2)$ is the saddle. In each case the saddle has one-dimensional stable and unstable manifolds; one unstable separatrix runs toward the nearby stable node, while the other runs out of the local neighbourhood. The two equilibrium branches and their local invariant manifolds exchange roles as $a$ passes through one, which is the standard local phase portrait of a transcritical bifurcation.

The Jacobian matrix at the origin and its characteristic polynomial areThe fixed-point multipliers areBy the Jury stability criterion, both lie strictly inside the unit circle exactly whenThe last condition always holds, since ${\mu }^2+\mu +2=(\mu +1/2{)}^2+7/4$. Therefore the origin is asymptotically stable exactly for $-1<\mu <0$ among the hyperbolic parameter values.

For completeness, its linear stability throughout the parameter line is as follows. When $\mu <-1$, the complex-conjugate Floquet multipliers have modulus $|\mu |>1$, so the origin repels. When $-1<\mu <-1/3$, their modulus is $|\mu |<1$, so it attracts with oscillation. At $\mu =-1/3$ the repeated Floquet multiplier is $1/3$, still attracting. For $-1/3<\mu <0$, both Floquet multipliers are real and in $(0,1)$. For $0<\mu <1$, $p(1)<0$ and $p(0)>0$, so one Floquet multiplier lies in $(0,1)$ and the other exceeds one: the origin is a saddle fixed point of a map. For $\mu >1$, the complex-conjugate Floquet multipliers have modulus $\mu >1$, so it repels.

Possible changes at the unit circle are exhausted by $p(1)=0$, $p(-1)=0$, or a nonreal pair with product one. They giveAt $\mu =0$ the Floquet multipliers are $0,1$; the nonzero fixed point branches in part (a) meet the origin, and part (c) identifies a subcritical pitchfork bifurcation. Indeed at $\mu =0$ the exact restricted map is $y\mapsto y+y^3$, which makes the origin unstable despite its neutral linear direction.

At $\mu =-1$ the Floquet multipliers are $i,-i$: this is a 1:4 strong resonance of a planar map, coinciding with the change from repelling to attracting as $\mu$ increases. A generic nonresonant Neimark–Sacker bifurcation conclusion would be unjustified. At $\mu =1$ there is a double Floquet multiplier $+1$; the matrix has a nontrivial Jordan block, and the two further nonzero fixed point branches meet the origin. Linear stability alone is inconclusive at these unit-modulus boundary values. The repeated Floquet multiplier at $\mu =-1/3$ is strictly inside the unit circle and gives no bifurcation.

Adjoin ${\mu }_{n+1}={\mu }_n$. At the origin the stable direction is the $x$ axis, with Floquet multiplier zero, while the $y$ and parameter directions have Floquet multiplier one. By the centre manifold theorem for a discrete dynamical system, write the extended centre manifold asOn it the scalar map is $Y=(1+\mu )y-\mu h(y,\mu )+y^3$. Its discrete centre-manifold invariance equation isPut $h=h_2+O({\rho }^3)$ with $\rho =|y|+|\mu |$. Since $Y=y+O({\rho }^2)$, the quadratic equation is simply $h_2(y,\mu )=\mu y$. ThusIn particular, setting the stable variable identically to zero would lose the leading parameter correction. The reduced map through total degree three isIn the natural pitchfork bifurcation scaling $\mu =O(y^2)$ this has leading form $y_{n+1}=y_n+\mu y_n+y_n^3$. Its nonzero fixed points satisfy $y^2\sim -\mu$, and their reduced fixed-point multipliers satisfy $1-2\mu +O({\mu }^2)>1$ for small $\mu <0$. The transverse Floquet multiplier remains inside the unit circle. Therefore the bifurcation at $\mu =0$ is a subcritical pitchfork bifurcation: the origin attracts for small negative $\mu$, each nonzero branch is a saddle fixed point of a map, and the origin is a saddle fixed point of a map for small positive $\mu$.