Saddle fixed point

A fixed point $p$ is a saddle if one eigenvalue satisfies $|{\lambda }_1|<1$ (stable direction) and another satisfies $|{\lambda }_2|>1$ (unstable direction).

In plain English, nearby points approach $p$ along one direction and are repelled along another.

The stable manifold is the set of points converging to $p$ under forward iteration. The unstable manifold is the set of points that approached $p$ under backward iteration. Together they create the characteristic saddle topology—stretching in one direction, contracting in another.

Saddles are fundamental to chaotic dynamics. They create sensitivity to initial conditions because small errors in the stable direction get amplified exponentially along the unstable direction.