I've been working on the following exercise:
Prove that the system
$$
\begin{cases}
\dot{x} = -x^3,\\
\dot{y} = -y + x^2
\end{cases}
$$
has no analytic center manifold (supposed in the following way $y = h(x) = a_2x^2 + a_3x^3 + \cdots$, then $a_{2n+1} = 0, n \geq 1, a_2 = 1, a_{n+2} = na_n, n \geq 2$). Is the manifold $C^{\infty}$?
It's based on the example 2.5, page 315 of the book 'Methods in Bifurcation Theory' by Chow and Hale.
How should I find the center manifold or even prove there is none? The book says it is not difficult to see but I am struggling. Thank you for your help!
Best Answer
Hint: Here is one way to approach this: Denote by $\varphi_t(x,y)=(\varphi_t^1(x,y),\varphi_t^2(x,y))$ the unique solution of the ODE with the initial condition $(x,y)$ (one can of course compute closed forms in this particular case but this is not necessary for this discussion). Note that $\varphi_\bullet$ is a real analytic flow (see e.g. Is the flow of an analytic vector field also analytic?). The origin $(0,0)$ is the unique equilibrium point, and linearizing the vector field at the origin one obtains
$$\begin{pmatrix} 0 & 0 \\ 0 & -1 \end{pmatrix}, $$
which has eigenvalues $\lambda=-1$ and $\lambda=0$; signifying a stable and a center direction, respectively:
$$T_{(0,0)}\mathbb{R}^2 = S_{(0,0)}(\varphi)\oplus C_{(0,0)}(\varphi).$$
Here $S_{(0,0)}(\varphi)$ is the stable subspace which is the eigenspace for $\lambda=-1$, and $C_{(0,0)}(\varphi)$ is the stable subspace which is the eigenspace for $\lambda=0$. It's straightforward to see that these two correspond to the $y$- and $x$-axes, respectively.
It's also straightforward to verify that the $y$-axis is the real analytic global stable manifold $\mathcal{S}_{(0,0)}(\varphi)$ (see e.g. Why do we need the Hartman-Grobman theorem & the Stable Manifold Theorem to prove that any sink is asymptotically stable & source/saddle is unstable?) at the origin; as expected the global stable manifold is as regular as the flow, and any point on it converges to the origin exponentially fast under the flow.
The question is about the existence, uniqueness, and regularity of the local or global center manifold $\mathcal{C}_{(0,0)}(\varphi)$. The discussion preceding the cited example in the OP gives conditions when all three are guaranteed (the regularity in terms of the regularity of the flow, or the vector field that generates it).
Say there is indeed a (once continuously differentiable) (local or global) center manifold $\mathcal{C}_{(0,0)}(\varphi)$ at $(0,0)$. Then
By the first two items ($\dagger$) one can equivalently consider a function $\phi: (\mathbb{R},0)\to (\mathbb{R},0)$ (that is as regular as $\mathcal{C}_{(0,0)}(\varphi)$) such that
$$\mathcal{C}_{(0,0)}(\varphi) = \operatorname{graph}(\phi) = \{(x,y)\,|\, y=\phi(x)\}.$$
(For local center manifolds $\phi$ can be allowed to be defined near the origin only). Note that $\phi$ can not be merely the zero function, as the $x$-axis is not invariant under the flow.
Setting aside the regularity issues for now, let's consider the problem formally, so that we are looking for a function $\phi$ such that formally
$$\phi(x) = \sum_{n\geq0} \phi_n x^n, \phi'(x) = \sum_{n\geq 1} n\phi_n x^{n-1}$$
for a sequence $\phi_\bullet$ of real numbers that is yet to be determined. ($\dagger$: Actually the two items above can also be derived from this formal consideration.)
Let $(x,y)$ be a point on the center manifold. Then $\phi(x) = y$. Since the center manifold is $\varphi$-invariant, for $t$ a valid time parameter we should also have
$$\phi\circ \varphi^{1}_t(x,y) = \varphi^2_t(x,y).$$
Differentiating this equation at $t=0$, one obtains
$$\phi'(x) (-x^3) = -y+x^2 = -\phi(x)+x^2,$$
so that collecting terms on one side one obtains
$$-x^3\phi'(x)+\phi(x)-x^2 = 0.$$
Switching to formal power series one then obtains the formal vanishing:
$$\phi_0 + \phi_1 x + (\phi_2-1)x^2 + \sum_{n\geq 3} (\phi_n-(n-2)\phi_{n-2}) x^n = 0.$$
If $\phi$ were $C^q$ (i.e. if it were $q$ times continuously differentiable) for $q\in\mathbb{Z}_{\geq1}$, this final equation would be (actually, as opposed to formally) true modulo $o_{x\to0}(x^{p+1})$, then the first $q$ terms in this final formal power series would need to vanish; if $\phi$ were $C^\infty$, then all terms would need to vanish; if $\phi$ were $C^\omega$ (i.e. real analytic), then the power series for would also need to converge. The recursive relation for higher order terms implies that
$$\forall k\in\mathbb{Z}_{\geq1}: \phi_{2k} = 2^k (k-1)!,$$
whence there can be no such real analytic $\phi$, hence real analytic $\mathcal{C}_{(0,0)}(\varphi)$.
(This argument so far would not be sufficient to say anything about whether or not there are $C^\infty$ center manifolds.)