a saddle point

A saddle point is an equilibrium that attracts along one direction and repels along another — exactly like the seat of a horse saddle, which curves up front-to-back but down side-to-side. A marble placed precisely on the centre stays, but nudge it the wrong way and it rolls off. In the phase plane, trajectories sweep in toward a saddle along one special line, get close, then veer away along another, making the unmistakable crossing-hyperbola pattern.

For a linear system this is the case of two real eigenvalues with OPPOSITE signs, one negative and one positive. The eigenvector of the negative eigenvalue is the stable direction: along it, states are pulled straight in (this special incoming curve is the stable manifold). The eigenvector of the positive eigenvalue is the unstable direction: along it, states are flung straight out (the unstable manifold). Every other trajectory is a hyperbola-like curve that comes in following the stable direction, bends near the saddle, and leaves following the unstable direction. The two straight-line solutions are the only ones that actually reach the point.

A saddle is always unstable — almost every nearby state eventually departs — yet it is one of the most important features of a phase portrait, because its stable manifold acts as a watershed. Trajectories on one side of that incoming curve are routed to one fate, those on the other side to a different fate, so the saddle's stable directions often form the boundaries between basins of attraction. Saddles organise where everything else goes.