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.^{[1]}

Let

be a smooth map with hyperbolic fixed point at . We denote by the stable set and by the unstable set of .

The theorem^{[2]}^{[3]}^{[4]} states that

- is a smooth manifold and its tangent space has the same dimension as the stable space of the linearization of at .
- is a smooth manifold and its tangent space has the same dimension as the unstable space of the linearization of at .

Accordingly is a **stable manifold** and is an **unstable manifold**.