Slow manifold

In a fast-slow singular perturbation $ϵ\dot{x}=f(x,y)$, $\dot{y}=g(x,y)$, the critical manifold is ${\mathcal{C}}_0=\{f=0\}$. A slow manifold for positive $ϵ$ is a nearby locally invariant manifold on which the rapid transient has relaxed and the remaining motion occurs on the original slow time. These two manifolds generally differ: invariance must account for motion of $y$, not just set its instantaneous fast equilibrium.

For example, consider $ϵ\dot{x}=y-x$, $\dot{y}=-y$. A graph $x=h_{ϵ}(y)$ is invariant only ifFor $0<ϵ<1$, $h_{ϵ}(y)=y/(1-ϵ)$ satisfies this identity, whereas the critical manifold $x=y$ does not. The transverse displacement $z=x-y/(1-ϵ)$ obeys $\dot{z}=-z/ϵ$ and decays rapidly. Thus the slow graph is displaced by $O(ϵ)$ on bounded $y$ intervals and attracts with an explicitly separated time scale.

Persistence of a smooth slow graph requires suitable smoothness and uniform normal hyperbolicity on the region considered, usually a compact piece of the critical manifold. Negative real parts of the fast linearization give attraction; near a fold point of a critical manifold, this spectral condition fails and the simple reduction can break down.