Center manifold

A local center manifold is an invariant manifold tangent at a nonhyperbolic equilibrium to the generalized eigenspace whose eigenvalues have zero real part. Nearby stability reduces to the flow on this manifold when all transverse eigenvalues have negative real part.

For $\dot{z}=v/2$, $\dot{v}=-3v/2+3w$, $\dot{w}=-(z-v{)}^3/9$, the stable coordinate is $v-2w$ and the center manifold begins with $v=2w$. A cubic correction $h_3$ satisfies $w{\partial }_z{h}_3-(2/9)(z-2w{)}^3=-3h_3/2$, giving $h_3=4z^3/27-32z^{2}w/27+272z{w}^2/81-832w^3/243$. The derivative of the graph contributes at the same order as the cubic forcing. Simply setting $\dot{v}=0$ omits this lag and incorrectly changes the cubic damping coefficient of the reduced normal form.

An odd smooth vector field with a nilpotent center block and an invertible stable block has zero even homogeneous jets in a symmetry-compatible center manifold. At degree two its invariance equation has the form $(N-S)h_2=0$, where $N$ is the nilpotent differentiation operator generated by the center block and $S$ the invertible stable linear operator. Since their spectra are disjoint, this operator is invertible, so $h_2=0$. The same argument applies inductively to even orders after lower even jets vanish. This is a statement about local Taylor jets; it does not assert uniqueness of a global center manifold.

At $(x,y)=(\mu ,0)$ near $\mu =1$, use $X=x-\mu$, $Y=y$, $\nu =\mu -1$. The stable direction is the $X$ axis and the centre direction is the $Y$ axis. The extended centre manifold has $X=Y^2+O(\nu Y^2,Y^4)$ and reduced flow $\dot{Y}=-\nu Y-2Y^3+\cdots$, a supercritical pitchfork bifurcation in parameter $-\nu$. At $(0,1)$ near $\mu =-1$, use $X=x$, $Y=y-1$, $\nu =\mu +1$. The centre line is $Y=-X/2$, the stable line is $X=0$, and the reduced flow is $\dot{X}=\nu X-2X^2+\cdots$, a transcritical bifurcation. The branch $x=(\mu +1)/2$, $y=\sqrt{(1-\mu )/2}$ joins the two bifurcations and is stable for $-1<\mu <1$.

For the system in the 2024 Part II Paper 2 question at $r=1$, with
$v=(x+y)/2$ and $w=(x-y)/2$, the center manifold isIts reduced equation is

Forset $X=x-a^2$ and $Y=y-a^2$, and append $\dot{a}=0$. Near $(X,Y,a)=(0,0,0)$, the extended centre manifold and its reduced dynamics areThe reduced equation is a subcritical pitchfork with reversed parameter $\mu =-a$.

Forthe invariant center manifold near $(x,y,k)=(0,0,0)$ is $y=0$, with reduced equation $\dot{x}=x(-k-3x+x^2)$ and hence a transcritical bifurcation. Near $(x,y,k)=(1,2,0)$, put $X=x-1$ and $v=y-2-X$. The extended center manifold beginsand its reduced equation is $\dot{X}=2(X^2-k)+O(3)$, a saddle-node bifurcation.

Forthe extended centre manifold for a parameter gives the local reduced equationswhere $y_0=\pm \sqrt{2}$. Thus the first two points undergo respectively supercritical and subcritical symmetry-forced pitchfork bifurcations, while each of the last two points undergoes a saddle-node bifurcation.