Centre manifold theory for a curve of equilibrium points

Question

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

Answer 1

You will want to look for Normally Hyperbolic Invariant Manifolds or NHIMs. Briefly said, these are invariant submanifolds (i.e. a curve of equilibria is a special case) with hyperbolic dynamics in the normal directions: the spectrum (in this case: eigenvalues) in the normal directions must be bounded away from zero (or for eigenvalues: from the imaginary axis) and dominating the tangential spectrum. This definition thus does not apply at the points where you have more than one zero eigenvalue.

For a NHIM $M$, indeed there exist stable and unstable manifolds, which are foliations whose leaves are (un)stable manifolds of each single point $m \in M$.

Moreover, under more subtle conditions on the spectrum tangential and normal to $M$ there exist Hartman-Grobman like results as well. I think that in your case these should apply, since the equilibria generate trivial zero spectrum along $M$. You may also want to look for results in geometric singular perturbation theory since this can be rewritten such that the zero tangential spectrum condition holds there too, see also the third reference below.

Some references:

1. Hirsch, Pugh, Shub: "Invariant manifolds", Springer LNM 538 (1977)
2. Fenichel: "Persistence and smoothness of invariant manifolds for flows", Indiana Univ. Math. J. 21 (1971)
3. Fenichel: "Geometric singular perturbation theory for ordinary differential equations", J. Diff. Eq. 31 (1979)
4. Jones: "Geometric singular perturbation theory" in Dynamical systems (Montecatini Terme, 1994), Springer LNM 1609

Answer 2

I believe Jaap provided some great references for Normally Hyperbolic Invariant Manifolds (NHIM), but I want to address what happens in the special case when the NHIM is a manifold of equilibria, near points at which normal hyperbolicity of these manifolds is violated (i.e., you get a zero eigenvalue in a direction that is transverse to the manifold). In your case, this is when you get two zero eigenvalues.

In this case, qualitative changes of the local dynamics close to the manifold of equilibria occur at that point. These changes have many similarities with bifurcations, and the emerging theory is called bifurcation without parameters. Just as is the case with bifurcations, the qualitative changes fall into different categories, organized by normal forms.

Reference:

• Liebscher, Stefan. "Bifurcation without parameters". Berlin: Springer, 2015.