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]
Stable manifold theorem
Let
\( {\displaystyle f:U\subset \mathbb {R} ^{n}\to \mathbb {R} ^{n}} \)
be a smooth map with hyperbolic fixed point at p {\displaystyle p} p. We denote by \( {\displaystyle W^{s}(p)} \) the stable set and by \( {\displaystyle W^{u}(p)} \) the unstable set of p.
The theorem[2][3][4] states that
\( {\displaystyle W^{s}(p)} \) is a smooth manifold and its tangent space has the same dimension as the stable space of the linearization of f at p.
\( {\displaystyle W^{u}(p)} \) is a smooth manifold and its tangent space has the same dimension as the unstable space of the linearization of f at p.
Accordingly \( {\displaystyle W^{s}(p)} \) is a stable manifold and \( {\displaystyle W^{u}(p)} \) is an unstable manifold.
See also
Center manifold theorem
Lyapunov exponent
Notes
Shub, Michael (1987). Global Stability of Dynamical Systems. Springer. pp. 65–66.
Pesin, Ya B (1977). "Characteristic Lyapunov Exponents and Smooth Ergodic Theory". Russian Mathematical Surveys. 32 (4): 55–114. Bibcode:1977RuMaS..32...55P. doi:10.1070/RM1977v032n04ABEH001639. Retrieved 2007-03-10.
Ruelle, David (1979). "Ergodic theory of differentiable dynamical systems". Publications Mathématiques de l'IHÉS. 50: 27–58. doi:10.1007/bf02684768. Retrieved 2007-03-10.
Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.
References
Perko, Lawrence (2001). Differential Equations and Dynamical Systems (Third ed.). New York: Springer. pp. 105–117. ISBN 0-387-95116-4.
Sritharan, S. S. (1990). Invariant Manifold Theory for Hydrodynamic Transition. John Wiley & Sons. ISBN 0-582-06781-2.
External links
StableManifoldTheorem at PlanetMath.org.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
