"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.