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. It roughly states that the existence of a local diffeomorphism near a fixed point implies the existence of a local stable center manifold containing that fixed point. This manifold has dimension equal to the number of eigenvalues of the Jacobian matrix of the fixed point that are less than 1.
be a smooth map with hyperbolic fixed point at . We denote by the stable set and by the unstable set of .
The theorem states that
Accordingly is a stable manifold and is an unstable manifold.