Stable manifold theorem

In mathematics, especially in the study of dynamical systems and differential equations, the stable manifold theorem is an important result about the structure of the set of orbits approaching a given hyperbolic fixed point.

Let

be a smooth map with hyperbolic fixed point at ${\displaystyle p}$. We denote by ${\displaystyle W^{s}(p)}$ the stable set and by ${\displaystyle W^{u}(p)}$ the unstable set of ${\displaystyle p}$.

...
Wikipedia