Peixoto's theorem

In the theory of dynamical systems, Peixoto's theorem, proved by Maurício Peixoto, states that among all smooth flows on surfaces, i.e. compact two-dimensional manifolds, structurally stable systems may be characterized by the following properties:

  • The set of non-wandering points consists only of periodic orbits and fixed points.
  • The set of fixed points is finite and consists only of hyperbolic equilibrium points.
  • Finiteness of attracting or repelling periodic orbits.
  • Absence of saddle-to-saddle connections.

Moreover, they form an open set in the space of all flows endowed with the C1 topology.

See also

  • Andronov–Pontryagin criterion

References

  • Jacob Palis, W. de Melo, Geometric Theory of Dynamical Systems. Springer-Verlag, 1982


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