Stable manifold theorem explained
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
be a
smooth map with hyperbolic fixed point at
. We denote by
the
stable set and by
the
unstable set of
.
The theorem[2] [3] [4] states that
is a smooth manifold and its
tangent space has the same dimension as the
stable space of the
linearization of
at
.
is a smooth manifold and its tangent space has the same dimension as the
unstable space of the linearization of
at
.
Accordingly
is a
stable manifold and
is an
unstable manifold.
See also
Notes
- Book: Shub, Michael . Global Stability of Dynamical Systems . Springer . 1987 . 65–66 .
- Pesin. Ya B. Characteristic Lyapunov Exponents and Smooth Ergodic Theory. Russian Mathematical Surveys. 1977. 32. 4. 55–114. 10.1070/RM1977v032n04ABEH001639. 2007-03-10. 1977RuMaS..32...55P. 250877457 .
- Ruelle. David. Ergodic theory of differentiable dynamical systems. Publications Mathématiques de l'IHÉS. 1979. 50. 27–58. 2007-03-10. 10.1007/bf02684768. 56389695 .
- Book: Teschl. Gerald. Gerald Teschl. Ordinary Differential Equations and Dynamical Systems. American Mathematical Society. Providence. 2012. 978-0-8218-8328-0.
References
- Book: Perko, Lawrence . Differential Equations and Dynamical Systems . New York . Springer . Third . 2001 . 0-387-95116-4 . 105–117 .
- Book: Sritharan, S. S. . Invariant Manifold Theory for Hydrodynamic Transition . John Wiley & Sons . 1990 . 0-582-06781-2 .
