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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.

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.

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.

Take $\gamma >{\gamma }_c$ and let $\epsilon \to 0^{+}$ with the other parameters fixed. The nullclines are the cubic polynomial graph $w=f(v)$ and the line $w=\gamma v$. In the phase plane, $v$ increases below the cubic polynomial graph and decreases above it; $w$ increases below the line and decreases above it. The fast ordinary differential equation, in time $s=t/\epsilon$, freezes $w$ and gives $dv/ds=f(v)-w$.

At $w=0$, the fast roots are $0,a,1$. Their linear stability follows from $f^{\prime }(0)=-a<0$, $f^{\prime }(a)=a(1-a)>0$ and $f^{\prime }(1)=a-1<0$. Thus a fixed $0<v^{\ast }<a$ falls towards the resting branch, whereas a fixed $a<v^{\ast }<1$ rises towards $v=1$. If $v^{\ast }>1$, the first fast segment instead settles down towards $1$, followed by the same recovery excursion. The threshold $a$ is the leading singular perturbation threshold; finite-$\epsilon$ corrections matter for initial values approaching $a$ with $\epsilon$.

LetThese are the roots of $f^{\prime }(v)=0$. The outer branches $v<v_L$ and $v>v_R$ of the critical manifold are attracting under fast motion; the middle branch is repelling. The full excursion has four stages.

For this cubic polynomial, the landing point is explicit. Since $f(v)-f(v_R)=-(v-v_R{)}^2(v-v_J)$, comparison of the quadratic coefficients givesThus a graph of $v(t)$ has a rapid rise, a slowly falling elevated plateau, a rapid fall through zero to a negative undershoot, and a slow approach back to zero. The figure shows both the phase plane and a computed time trace. The smaller initial perturbation stays near the resting branch and produces no elevated plateau.

The four-stage description concerns the singular limit. Close to a simple fold point of a critical manifold, balancing the local quadratic activation term with the slow drift gives $\Delta v=O({\epsilon }^{1/3})$ and $\Delta t=O({\epsilon }^{2/3})$ for fixed parameters. These layers shrink compared with the slow phases and do not alter the leading time integrals below.

On an attracting branch of the critical manifold, $w=f(v)$ to leading order. The chain rule therefore givesThe initial fast segment ends at $v=1$, and the first slow phase ends at the right fold point of a critical manifold, $v_R$. HenceOn this interval $f^{\prime }(v)<0$ and $\gamma v-f(v)>0$. The integrand is negative, but the upper limit is smaller than the lower limit, so $T_1>0$. The numerator tends to zero at the fold while the denominator remains positive; the leading integral is finite. Fast segments and the fold delay give vanishing corrections as $\epsilon \to 0^{+}$.