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
f : U ⊂ R n → R n be a smooth map with hyperbolic fixed point at p . We denote by W s ( p ) the stable set and by W u ( p ) the unstable set of p .
The theorem states that
W s ( p ) is a smooth manifold and its tangent space has the same dimension as the stable space of the linearization of f at p . W u ( p ) is a smooth manifold and its tangent space has the same dimension as the unstable space of the linearization of f at p .Accordingly W s ( p ) is a stable manifold and W u ( p ) is an unstable manifold.
