Stable manifold theorem

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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.

Contents 1 Stable manifold theorem 2 See also 3 Notes 4 External links

[edit] Stable manifold theorem

Let

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^[1][2] 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.

[edit] See also

• Center manifold theorem
• Lyapunov exponent

[edit] Notes

1. ^ Pesin, Ya B (1977). "Characteristic Lyapunov Exponents and Smooth Ergodic Theory". Russ Math Surv 32 (4): 55–114. doi:10.1070/RM1977v032n04ABEH001639. 
2. ^ Ruelle, David (1979). "Ergodic theory of differentiable dynamical systems". Publications Mathématiques de l'IHÉS 50: 27–58. 

[edit] External links

• StableManifoldTheorem at PlanetMath.