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 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.