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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.

After adding $ϵ>0$, the equilibrium equation isThe constant term breaks the reflection symmetry $x\mapsto -x$. Consequently each symmetry-forced pitchfork becomes an imperfect pitchfork bifurcation, while each crossing that produced a transcritical bifurcation is disconnected. The remaining generic turning points are saddle-node bifurcations. The lower row shows this unfolding for a representative small positive $ϵ$; the upper row gives the unperturbed diagrams.

For $a<0$, both pitchforks and both transcritical crossings are destroyed. For $a=0$, the simultaneous fifth-order meeting splits into separated folds. For $a>0$, both pitchforks become imperfect. Hence, under the allowed constant perturbation,This is the structural stability of a bifurcation distinction between the nongeneric pitchfork and transcritical diagrams and the generic saddle-node diagrams left by their unfoldings.

At a non-axis fixed point the determinant and trace areThese formulas determine the solid stable and dashed unstable branches in the bifurcation diagram. The pair $x_{\pm }$ is born in a saddle-node bifurcation at $\mu =-(\beta +1{)}^2/4$. For $\beta \ne 1$, it exchanges branches with $x=1$ at $\mu =-\beta$ and with $x=0$ at $\mu =0$, both transcritical bifurcations. If $0<\beta <1$, the saddle-node lies to the left of $x=1$; if $\beta >1$, it lies to the right. For $\beta =1$, the saddle-node and the collision at $x=1$ combine into a pitchfork bifurcation at $\mu =-1$.

At an equilibrium, each product inmust vanish. This gives the two parameter-independent equilibriathe equilibria on $y=0$and the equilibria with both coordinates nonzerowhere matching signs are used.

The $y=0$ pair is born at $k=-9/4$, suggesting a saddle-node bifurcation. At $k=0$, one of these equilibria meets $(0,0)$, suggesting a transcritical bifurcation, while the two equilibria near $(1,2)$ are simultaneously born in a saddle-node. At $k=1$, the minus branch $(1-\sqrt{k},2-\sqrt{k})$ meets $(0,1)$, suggesting another transcritical bifurcation. At $k=4$, that branch meets the $y=0$ equilibrium $(-1,0)$, again suggesting a transcritical bifurcation. Thus the bifurcation parameter values are

There are two bifurcations in $0\le k\le 1/2$, both at $k=0$.

Near $(0,0,0)$, append $\dot{k}=0$. The plane $y=0$ is invariant and tangent to the extended center subspace, so it is an extended center manifold. The reduced equation is exactlyIts leading terms $-kx-3x^2$ have two branches, $x=0$ and $x=-k/3+O(k^2)$, which cross and exchange their center-direction stability. Hence this is a transcritical bifurcation.

For the bifurcation at $(1,2,0)$, setThe extended system becomesSolving the centre-manifold invariance equation for $v=h(X,k)$ givesSubstitution into the $X$ equation yieldsFor $k>0$ this has two nearby equilibria $X=\pm \sqrt{k}+O(k)$, while for $k<0$ it has none. It is therefore a saddle-node bifurcation. These reductions are collected in bifurcations of the 2022 Cambridge quadratic-cubic system.

With $s=0$, the equilibria satisfyBesides $g=0$, the two positive equilibria arewhere $g_{-}{g}_{+}=1$. Sincethe nonzero equilibrium relation $k=(1+g^2)/g$ givesThe fixed point stability for an autonomous differential equation therefore shows that $g=0$ and $g_{+}$ are stable, while $g_{-}$ is unstable. The graph of $f(g,0)$ starts at zero with negative slope, crosses upward at $g_{-}$, crosses downward at $g_{+}$, and tends to $-\infty$ as $g\to \infty$.

For a constant input $s$, write the equilibrium equation asThe low stable equilibrium and the intervening unstable equilibrium coalesce in a saddle-node bifurcation. More precisely, let $g_c$ be the first positive solution ofand defineEquivalently, $f(g_c,s_c)=0$ and ${\partial }_{g}f(g_c,s_c)=0$, with $g_c$ on the low-concentration branch. If $s_1>s_c(k)$, that branch no longer exists. Holding the input long enough carries the trajectory into the basin of attraction of the high state. When the input returns to zero, the concentration converges toThis is the saturating autocatalytic switch.

For $k\gg 1$, the threshold occurs at $g=O(k^{-1})$, soThe approximate double-root conditions areThus $g_c\simeq 1/(2k)$ andHence the constant in $s_c(k)\simeq C{k}^{-1}$ is $\boxed{C=1/4}$, as recorded by the strong-autocatalysis switching threshold.

Forthe zero-input stable states are $0$ and $(k+\sqrt{k^2-4})/2$. The switching threshold is the saddle-node bifurcation determined by $f(g_c,s_c)={\partial }_{g}f(g_c,s_c)=0$ on the low-concentration branch.

A bifurcation is structurally stable within a specified class of systems when every sufficiently small allowed perturbation preserves its local qualitative bifurcation diagram up to smooth changes of state and parameter coordinates. A generic saddle-node bifurcation is structurally stable in one-parameter families, whereas a pitchfork bifurcation requires a symmetry and a transcritical bifurcation requires intersecting equilibrium branches, so unrestricted perturbations destroy either one.

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.