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.

Stable manifold theorem[edit]

Let

be a smooth map with hyperbolic fixed point at . We denote by the stable set and by the unstable set of .

The theorem[1][2][3] 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[edit]

Notes[edit]

  1. ^ 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. 
  2. ^ 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. 
  3. ^ Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0. 

References[edit]

  • 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[edit]