A gradient vector field $X=-\nabla(\varphi)$ 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?
