Separatrix

A phase curve (i.e., an invariant manifold) which meets a hyperbolic fixed point (i.e., an intersection of a stable and an unstable invariant manifold) or connects the unstable and stable manifolds of a pair of hyperbolic or parabolic fixed points. A separatrix marks a boundary between phase curves with different properties.

For example, the separatrix in the equation of motion for the pendulum occurs at the angular momentum where oscillation gives way to rotation. There are also many systems that have pairs of connected fixed points, e.g., the flow in an open cavity, which has a separatrix that connects two parabolic points.