An invariant manifold is a manifold preserved by a dynamical system: a trajectory beginning on it stays on it for as long as that trajectory is defined. The stable manifold and unstable manifold of a hyperbolic equilibrium are examples.
For and , a differentiable graph is invariant precisely when . 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.
