I'm working with a nonlinear autonomous system $x'=f(x)$. This system stays in $\mathbb{R}^n_+$ whenever it begins there, and it has a ray of equilibria, i.e. there is a positive vector $x_0$ so that for any $a \geq 0$, $f(ax_0)=0$. The linearization of the system at these equilibria has strictly negative real eigenvalues, except for a single zero eigenvalue. The eigenvector for this zero eigenvalue corresponds to the fact that a perturbation in the direction of $x_0$ leaves the system at another equilibrium.
From this can I conclude that the ray (excluding the origin, if need be) as a whole is asymptotically stable? That is, can I conclude that if I consider an initial condition which is a small perturbation from $bx_0$ for $b \geq 0$, the system will tend to $cx_0$ for some $c \geq 0$, possibly different from $b$? This seems intuitive, but I know that there can be subtle issues when the linearization has eigenvalues with zero real part.
The curve of equilibria is an example of an invariant manifold, for which the stable and unstable manifold theory applies. In your case, if you do not have eigenvalues with positive or zero real part other then the one, which is responsible for the pertrubations along your ray, hence locally the neighborhood of your ray is the stable manifold, and hence attracting.
The exact statement and a proof can be found in Hirsch, Pugh, Shub: "Invariant manifolds", Springer LNM 538 (1977), Theorem 4.1.