Invariant Manifold

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 $W^s$ and outgoing $W^u$ Manifolds all join up smoothly.

A stable invariant Manifold $W^s$ of a Fixed Point $Y^*$ is the set of all points $Y_0$ such that the trajectory passing through $Y_0$ tends to $Y^*$ as $j\to\infty$.

An unstable invariant Manifold $W^u$ of a Fixed Point $Y^*$ is the set of all points $Y_0$ such that the trajectory passing through $Y_0$ tends to $Y^*$ as $j\to -\infty$.

See also Homoclinic Point