An invariant set is said to be a () invariant manifold if has the structure of a differentiable manifold (Wiggins 1990, p. 14).
When stable and unstable invariant manifolds intersect, they do so in a hyperbolic fixed point (saddle point). The invariant manifolds are then called separatrices. A hyperbolic fixed point is characterized by two ingoing stable manifolds and two outgoing unstable manifolds. In integrable systems, incoming and outgoing manifolds join up smoothly.