Peixoto's theorem

Peixoto's theorem is a result in the theory of dynamical systems, specifically regarding the behavior of flow on two-dimensional manifolds. Named after the Brazilian mathematician M. Peixoto, the theorem classifies the vector fields on a two-dimensional manifold in terms of their qualitative behavior. The central idea of Peixoto's theorem is to provide conditions under which the local behavior of a dynamical system can be classified.