I am looking for advice concerning a specific situation related to centre manifold theory (compare Perko 2001).
The part which is known
Let's consider a differential equation in higher-dimensional Euclidean space which admits equilibrium points. Centre manifold theory yields the following nice results for individual equilibrium points, depending on whether the eigenvalues of the linearised differential equation (which in my case are real) vanish or not. More specifically:
Hartman-Grobman thm.: Let $x_0$ a equilibrium point of $\dot x=f(x)$ on $R^n$, ie $f(x_0)=0$, and $\dot y=Ay$ the linearised system in this point. Suppose that all eigenvalues of $A$ have non-zero real part. Then there is a homeomorphism of a neighbourhood of $x_0$ onto a neighbourhood of $0\in R^n$ such that solutions of the non-linear system are mapped onto solutions of the linearised system.
Furthermore, there are two manifolds with dimension corresponding to the number of eigenvalues with positive (resp negative) real part, the stable and unstable manifold. Every point one these manifolds converges to the equilibrium point under the action of the flow of the differential equation, for time $t$ tending to $+\infty$ resp $-\infty$. These two manifolds are unique.
In the case where $A$ also admits zero eigenvalues (one then calls $x_0$ a non-hyperbolic equilibrium point), one can additionally find a (non-unique) centre manifold: It is invariant under the flow and tangential to the space spanned by the eigenvectors to eigenvalue zero.
What I am looking for
Now back to my specific situation: I have a system with a curve consisting of equillibrium points, and I find one or two zero eigenvalues. In fact, along the curve one of the eigenvalues changes sign from positive to negative in a continuous way, all other eigenvalues have constant sign.
In the case of just one zero eigenvalue, I expect a similar result as the Hartman-Grobman thm., but the dimensions have to be adapted. In particular I would expect that the union of all stable manifolds is again a manifold, and it is stable in the sense that every point converges to some point on the curve.
There is mentioning of such results in the literature (Aulbach 1984, Bogojavlensky 1985), but I a still looking for an exact formulation and am complete proof. A hint on what can be shown and where I can find a reference would be appreciated.
edit 2014-04-16: corrected typo