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

Accordingly is a stable manifold and is an unstable manifold.

See also

Notes

  1. ^ Shub, Michael (1987). Global Stability of Dynamical Systems. Springer. pp. 65–66.
  2. ^ 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.
  3. ^ 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.
  4. ^ Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.

References