Peixoto's theorem explained

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 C¹ topology.

See also

• Andronov–Pontryagin criterion

References

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