Stable manifold theorem

Stable manifold theorem

Let :f: U \subset \mathbb{R}^n \to \mathbb{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.

Notes

References

References

1. Shub, Michael. (1987). "Global Stability of Dynamical Systems". Springer.
2. Pesin, Ya B. (1977). "Characteristic Lyapunov Exponents and Smooth Ergodic Theory". [[Russian Mathematical Surveys]].
3. Ruelle, David. (1979). "Ergodic theory of differentiable dynamical systems". Publications Mathématiques de l'IHÉS.
4. Teschl. (2012). "Ordinary Differential Equations and Dynamical Systems". [[American Mathematical Society]].