"2003 Colloquium on Differential Equations and Applications,
 Maracaibo, Venezuela.
Electronic Journal of Differential Equations,
Conference 13, 2005, pp. 29-34. 

Title: Critical points of the steady state of a Fokker-Planck equation

Authors: Jorge Guinez    (Univ. del Zulia, Maracaibo, Venezuela)
         Robert Quintero (Univ. del Zulia, Maracaibo, Venezuela)
	 Angel D. Rueda  (Univ. del Zulia, Maracaibo, Venezuela)
 
Abstract: 
 In this paper we consider a set of vector fields over the
 torus for which we can associate a positive function
 $v_{\epsilon }$ which define for some of them in a solution
 of the Fokker-Planck equation with $\epsilon $ diffusion:
 $$
 \epsilon \Delta v_{\epsilon }-\mathop{\rm div}(v_{\epsilon }X)=0\,.
 $$
 Within this class of vector fields we prove that $X$ is a
 gradient vector field if and only if at least one of the
 critical points of $v_{\epsilon }$ is a stationary point of
 $X$, for an $\epsilon >0$.  In particular we show a vector
 field which is stable in the sense of Zeeman but structurally
 unstable in the Andronov-Pontriaguin sense. A generalization
 of some results to other kind of compact manifolds is made.

Published May 30, 2005.
Math Subject Classifications: 58J60, 37C20.
Key Words: Almost gradient vector fields.