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 the stable set the unstable set.

The theorem states that

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

• is a smooth manifold and its tangent space has the same dimension as the unstable space.

is a stable manifold is an unstable manifold.

See also

• Center manifold theorem

• Lyapunov exponent

Notes

References

External links