Normal hyperbolicity

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.