Invariance equation for a graph
= Invariance equation for a graph
{title2=$Dh(x)F(x,h(x))=G(x,h(x))$}
For $\dot x=F(x,y)$ and $\dot y=G(x,y)$, a <differentiable> graph $y=h(x)$ is invariant precisely when $Dh(x)F(x,h(x))=G(x,h(x))$. This is the <chain rule> along a trajectory on the graph. The equation determines coefficients in a local power-series approximation of a <stable manifold> or <unstable manifold>; checking tangent directions alone does not ensure invariance.