From what I can find, the proof of the Centre Manifold Theorem uses the Contracting Mapping Theorem which gives a unique solution. So my question is, where does the non-uniqueness come from in the proof? I am very well aware of the examples in which the Centre Manifold can be non-unique, but I don't see where it comes from.
Furthermore, where in the proof of the Hartman-Grobman Theorem does the fact that the equilibrium is hyperbolic enter the picture. More precisely, how would the Hartman-Grobman Theorem fail if the equilibrium is non-hyperbolic.
Thank you in advance.
Dear Jason:
1) the GLOBAL center manifold is unique. The proofs we all have in mind assure the uniqueness of the global center manifold.
2) The LOCAL center manifold may fail to be unique. This is due to the fact that to prove the existence of the local center manifold, one uses the existence of the global center manifold, along with "cut-off" functions. This choice of a particular "cut-off" function prevails the local center manifold from being unique.
3) Fortunately, this non-uniqueness causes no serious troubles: this is due to the fact that the restriction of the system you are interested in, on any choice of a local center manifold, is conjugate to the restriction of the same system, on any other choice of a center manifold. That is, they are the "same", up to a coordinate transformation.
4) In the Hartman-Grobman theorem, I am not sure which proof you have in mind. But, in most proofs, the hyperbolicity enters the picture in two ways: hyperbolicity guarantees that the (tangent space of the ) phase space may be written as the direct sum of two subspaces, in which subspaces the (linearization of the) system acts as a contraction and as a dilation respectively. Secondly, the hyperbolicity ensures the solvability of the functional equations, which make the "topological linearization" of the system possible.
5) Excellent references of these facts should include "Dynamical Systems" of Clark Robinson, and "Methods of Bifurcation Theory" of Chow and Hale, in case you wish to further study this subject.
Hope I helped,
Stavros.
Just a simple example for non-uniqueness. Consider the system x'=x^2 and y'=y, which is explicitly solvable.
The unstable manifold is obviously the y axis.
For any real c, a center manifold is given by y = c e^{-1/x}. They are all at the origin tangential to the center eigenspace and thus a center manifold.
Remark: I vaguely remember that for analytic systems there is only one center manifold with which is given by an analytic function (here y=0, i.e. c=0) but in general there might not be a version of the center manifold given by an analytic function.
In the (topologically) linearizable case the center manifold is completely determined by the linear counterpart. The case where the uniqueness can be lost is when we deal with non-linearizable systems. For instance, saddle nodes. Here is a quotation from the wikipedia entry of "Center manifold":
Consider the system \dot{x}=x^2 and \dot{y}=y. The unstable manifold at the origin is the y axis, and the stable manifold is the trivial set {(0, 0)}. Any orbit not on the stable manifold satisfies an equation on the form y=A\exp(-1/x) for some real constant A. It follows that for any real A, we can create a center manifold by piecing together the curve y=A\exp(-1/x) for x > 0 with the negative x axis (including the origin). Moreover, all center manifolds have this potential non-uniqueness, although often the non-uniqueness only occurs in unphysical complex values of the variables.
Not sure which proof of existence of a center manifold you think of.
The one I know constructs a local center-unstable manifold via the Hadamard Graph Transform and the contraction principle. Thus this manifold is unique and given by a graph over the center-unstable eigenspace. Then you pick one version the unstable manifold, by restricting the graph to the center eigenspace.
But this is only one possible choice. There might be many others. Only the whole center-unstable manifold is unique. But on that there might be many different center manifolds.
The only thing that just confused me is that in the example above (if you reverse time) one has y'=-y and x'= - x^2 and the center and the center-unstable manifold coincide, but the non-uniqueness should still hold. I still miss some argument here.
Dear Jason:
1) the GLOBAL center manifold is unique. The proofs we all have in mind assure the uniqueness of the global center manifold.
2) The LOCAL center manifold may fail to be unique. This is due to the fact that to prove the existence of the local center manifold, one uses the existence of the global center manifold, along with "cut-off" functions. This choice of a particular "cut-off" function prevails the local center manifold from being unique.
3) Fortunately, this non-uniqueness causes no serious troubles: this is due to the fact that the restriction of the system you are interested in, on any choice of a local center manifold, is conjugate to the restriction of the same system, on any other choice of a center manifold. That is, they are the "same", up to a coordinate transformation.
4) In the Hartman-Grobman theorem, I am not sure which proof you have in mind. But, in most proofs, the hyperbolicity enters the picture in two ways: hyperbolicity guarantees that the (tangent space of the ) phase space may be written as the direct sum of two subspaces, in which subspaces the (linearization of the) system acts as a contraction and as a dilation respectively. Secondly, the hyperbolicity ensures the solvability of the functional equations, which make the "topological linearization" of the system possible.
5) Excellent references of these facts should include "Dynamical Systems" of Clark Robinson, and "Methods of Bifurcation Theory" of Chow and Hale, in case you wish to further study this subject.
Hope I helped,
Stavros.
@Stravros,
I disagree with the uniqueness of globally defined center manifolds, because:
Consider the system x'=x^2 and y'=y.
Then for any real c, the graph y = c e^{-1/x} is by definition a globally defined center manifold. Obviously this is not unique.
Dear Prof. Blomker,
I am not quite sure what you do mean when you say "globally defined". How do you extend this curve at x=0?
We have to assume some kind of regularity for the center manifold. The exact version of the global center manifold theorem I have in mind is theorem 1.1 of "normal forms and bifurcations of planar vector fields" of Chow, Li and Wang. There the uniqueness of the global center manifold is proved, under the assumption that the vector field is Lipschitz continuous.
If you have another version of the (global) center manifold theorem in mind, I will be happy to hear more about it.
With my kind regards,
Stavros.
I am sorry. According to the book by Khalil, center manifold theorem is an existence theorem rather than a uniqueness theorem. If you follow the proof there (pp. 690--699), the manifold is unique within the set S, and the mapping from S to S is P that is defined by (C.61). In the derivation preceding this equation, we observe that the z(-\infty) must be assumed to be bounded in order for the (C.60) to be correct. That means that in the simple example that \dot{x}=-x^2; \dot{y}=-y, the y(-\infty) must be bounded. That gives the only possibility of $y \equiv 0$. Then, the proof continues to show that the operator P is a contraction mapping, etc. The key here is that the definition of the operator P avoid all the c\exp(-1/x) type of function from the sequence of (\eta_i)_{i=0}^\infty.
I hope that you get what I mean here. Happy reading.
Article Nonlinear systems / Hassan K. Khalil
@ Stavros
The curve y(x)=c e^{-1/x} with y(0)=0 is for every c\in\R a smooth infinitely often differentiable function, even at x=0. It is not analytic though (at least for c\not=0).
About Theorem 1.1 of "normal forms and bifurcations of planar vector fields" of Chow, Li and Wan:
For uniqueness they need the assumption that the Lipschitz-constant of the nonlinearity is globally sufficiently small!
In that situation the nonlinearity is globally only a tiny perturbation of the linear flow and the invariant manifolds of the linear system persists under this small perturbation. But this is not the interesting case.
For the more interesting cases, as you said, you need to apply a cut-off and you obtain non-uniqueness of the local-center manifold.
The 2D-toy example with the center manifolds y(x)=c e^{-1/x} was just to show that also the global center manifold might not be unique in the case when the Lipschitz constant of the nonlinearity is not globally small.
Dear Prof.Blomker,
1) I have done some calculations. Please correct me if I am wrong. y(x)=ce^{-1/x}, with y(0)=0, is not even a continuous curve at the origin (I calculate that lim_{x->0-}y(x)=\infty).
2) For the 2-d toy model you mention I think that the center manifolds are the curves
y(x)=0, x \leq 0 and y(x)=ce^{-1/x}, x>0.
I agree that these curves are center manifolds, which are smooth. Did you had these curves in mind?
3) Yes, the uniqueness of the center manifolds follows from the assumption that the Lipschitz constant should be globally small. I just disagree with you that this is not an interesting case for two reasons. Firstly, because such systems exist, and secondly, because any local center manifold of a system (and local manifolds we can compute in "difficult" systems) is the restriction in a neighbourhood of the singularity of the unique global center manifold of another system (global center manifolds we cannot compute in "difficult" systems, but is nice to know that the local objects are the restriction of a unique global object).
4) I admit though that what is interesting and what is not, is a matter of perspective..
With my kind regards,
Stavros.
Sorry, I stupidly just lost a sign. I thought about y = c e^{-1/ abs(x)} and simply forgot about x -x
Being interesting depends really on the perspective. For me small Lipschitz is just a small perturbation of the linear system, and thus the whole system is not really nonlinear in nature. But is surely important (and one of the few practical tools), if you want local theory at the fixed-point. Perfectly agree with you in that point.
When you instead of considering the original system, consider a system multiplied by a cup function, so the vector field coinsides with the initial vector fielld in a neighborhood of a fixed point but is different out of this vicinity. For each cup function you find unique central manifold but for different functions they could be different.
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