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. 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.
Stable manifold theorem.
Let 
be a smooth map with hyperbolic fixed point at formula_2. We denote by formula_3 the stable set and by formula_4 the unstable set of formula_2.
The theorem states that
Accordingly formula_3 is a stable manifold and formula_4 is an unstable manifold.