Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

A gradient vector field $X$ in $\mathbb{R}^n$ has two equilibria $x_1, x_2$. The vector field defines a cooperative dynamical system. The linearization about $x_1$ has one positive eigenvalues and n-1 negative, while about $x_2$, n negative eigenvalues. Let $x(t)$ be the heteroclinic connection of the two equilibria (the mountain pass).

My question is: is it true that the matrix $D_{x(t)}X$ of the linearization of the gradient system along $x(t)$, $\dot{\xi}=D_{x(t)}X\: \xi$, has a decreasing trace with respect to t?

Thank you.

share|cite|improve this question

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.