Critical manifold (${\mathcal{C}}_0=\{(x,y):F(x,y)=0\}$)

For a system $\epsilon \dot{x}=F(x,y)$, $\dot{y}=G(x,y)$ with fast variable $x$ and slow variable $y$, the critical manifold is the set of equilibria of the fast problem $dx/ds=F(x,y)$ with $y$ frozen. It is attracting where the fast Jacobian matrix has eigenvalues with negative real parts and repelling where those real parts are positive. A smooth graph can cease to be attracting at a fold point of a critical manifold.

For a critical manifold of a fast-slow system $\epsilon \dot{x}=F(x,y)$, $\dot{y}=G(x,y)$, normal hyperbolicity means that the fast linearization $D_{x}F$ has no eigenvalue with zero real part along the manifold. The slow variables are frozen for this fast subsystem. Uniform estimates on a compact portion require the real parts to remain bounded away from zero.

For $F(x,y)=y-x^2$, the critical manifold is $y=x^2$ and the fast derivative is $-2x$. Its positive-$x$ branch is attracting and its negative-$x$ branch repelling. At the fold $(0,0)$ the fast eigenvalue is zero, so normal hyperbolicity fails and the usual fast relaxation can break down.

In the broader dynamical-systems definition, contraction or expansion transverse to an invariant manifold must dominate the tangent dynamics. For a critical manifold of the limiting fast problem, tangent directions are equilibria, giving the stated nonzero-real-part test. A nonzero purely imaginary eigenvalue fails this test even though the Jacobian is invertible.

For one fast and one slow variable, a nondegenerate fold point of a critical manifold satisfies $F=F_x=0$ and $F_{xx}{F}_y\ne 0$. An attracting and a repelling branch of fast equilibria meet there. A slow trajectory reaching the fold generally leaves by a rapid jump when its slow drift crosses the fold transversely. The fold is defined relative to the fast direction: if fast trajectories are slanted in the plotted coordinates, their tangency to the critical manifold need not be an extremum of its plotted height.

Near a nondegenerate fold point of a critical manifold crossed with nonzero slow speed, the local balance can be written $\epsilon \dot{x}=y-x^2$, $\dot{y}=-1$ up to nonzero scalings and higher-order terms. A time interval $T$ gives $y=O(T)$; balancing $x^2$ with $y$ gives $x=O(T^{1/2})$, and balancing $\epsilon x/T$ with $T$ gives $T=O({\epsilon }^{2/3})$, $x=O({\epsilon }^{1/3})$. This layer delays departure from the fold relative to an instantaneous singular jump. It differs from the order-$\epsilon$ fast motion away from the fold.