2003 Colloquium on Differential Equations and Applications, Maracaibo, Venezuela.
Electron. J. Diff. Eqns., Conference 13, 2005, pp. 29-34.

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

Jorge Guinez, Robert Quintero, Angel D. Rueda

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.

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Jorge Guíñez
Centro de Investigación de Matemática Aplicada (C.I.M.A.)
Facultad de Ingeniería, Universidad del Zulia
Apartado 10482, Maracaibo, Venezuela
email: jguinez@luz.edu.ve
Robert Quintero
Centro de Investigación de Matemática Aplicada (C.I.M.A.)
Facultad de Ingeniería, Universidad del Zulia
Apartado 10482, Maracaibo, Venezuela
email: rquintero@luz.edu.ve
Angel D. Rueda
Centro de Investigación de Matemática Aplicada (C.I.M.A.)
Facultad de Ingeniería, Universidad del Zulia
Apartado 10482, Maracaibo, Venezuela
email: ad-rueda@cantv.net

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