The centre manifold theorem and its proof

In my research I have often used centre manifolds but I have not thoroughly studied the proof of their existence. The standard reference I have quoted for this topic is the book of Carr. The basic statement is the following. Let $\dot x=f(x)$ be a dynamical system on $R^n$ and $x_0$ a point with $f(x_0)=0$ . Let $A=Df(x_0)$ . Then $R^n$ can be written as a direct sum of invariant subspaces of $A$ , $V_-\oplus V_c\oplus V_+$ , such that the real parts of the eigenvalues of the restrictions of $A$ to these subspaces are negative, zero and positive, respectively. $V_c$ is the centre subspace. The centre manifold theorem states that there exists an invariant manifold of the system passing through $x_0$ whose tangent space at $x_0$ is equal to $V_c$ . This manifold, which is in general not unique, is called a centre manifold for the system at $x_0$ . Theorem 1 on p. 16 of the book of Carr is a statement of this type. I want to make two comments on this theorem. The first is that Carr states and proves the theorem only in the case that the subspace $V_+$ is trivial although he states vaguely that this restriction is not necessary. The other concerns the issue of regularity. Carr assumes that the system is $C^2$ and states that the centre manifold obtained is also $C^2$ . In the book of Perko on dynamical systems the result is stated in the general case with the regularity $C^r$ for any $r\ge 1$ . No proof is given there. Perko cites a book of Guckenheimer and Holmes and one of Ruelle for this but as far as I can see neither of them contains a proof of this statement. Looking through the literature the situation of what order of differentiability is required to get a result and whether the regularity which comes out is equal to that which goes in or whether it is a bit less seems quite chaotic. Having been frustrated by this situation a trip to the library finally allowed me to find what I now see as the best source. This is a book called ‘Normal forms and bifurcations of planar vector fields’ by Chow, Li and Wang. Despite the title it offers an extensive treatment of the existence theory in any (finite) dimension and proves, among other things, the result stated by Perko. I feel grateful to those authors for their effort.

A general approach to proving the existence of a local centre manifold, which is what I am interested in here, is first to do a cut-off of the system and prove the existence of a global centre manifold for the cut-off system. It is unique and can be obtained as a fixed point of a contraction mapping. A suitable restriction of it is then the desired local centre manifold for the original system. Due to the arbitrariness involved in the cut-off the uniqueness gets lost in this process. A mapping whose fixed points correspond to (global) centre manifolds is described by Carr and is defined as follows. We look for the centre manifold as the graph of a function $y=h(x)$ . The cut-off is done only in the $x$ variables. If a suitable function $h$ is chosen then setting $y=h(x)$ gives a system of ODE for $x$ which we can solve with a prescribed initial value $x_0$ at $t=0$ . Substituting the solution into the nonlinearity in the evolution equation for $y$ defines a function of time. If this function were given we could solve the equation for $y$ by variation of constants. A special solution is singled out by requiring that it vanishes sufficiently fast as $t\to -\infty$ . This leads to an integral equation of the general form $y=I(h)$ . If $y=h$ , i.e. $h$ is a fixed point of the integral operator then the graph of $h$ is a centre manifold. It is shown that when certain parameters in the problem are chosen correctly (small enough) this mapping is a contraction in a suitable space of Lipschitz functions. Proving higher regularity of the manifold defined by the fixed point requires more work and this is not presented by Carr. As far as I can see the arguments he does present in the existence proof nowhere use that the system is $C^2$ and it would be enough to assume $C^1$ for them to work. It is only necessary to replace $O(|z|^2)$ by $o(|z|)$ in some places.

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