# Fredholm index vs. Limit cycle theory

Let $$A$$ be the algebra of all smooth functions $$f: \mathbb{R}^2 \to \mathbb{R}$$ such that $$f$$ is flat at the origin and is real analytic on $$\mathbb{R}^2 \setminus \{0\}$$.

Let $$B$$ be the subalgebra of $$A$$ consisting of all Schwartz functions in $$A$$.

For every polynomial vector field $$X = P(x,y) \partial_{x} + Q(x,y) \partial_{y}$$ we define the differential operator $$D_{X}(U)=PU_{x}+QU_{y}$$. Obviously $$A$$ and $$B$$ are invariant under this differential operator.

Let $$X=(y-(x^3 -x))\partial_{x}-x \partial_{y}$$ be the Van der Pol vector field.

What can be said about the codimension of the range of $$D_{X}:A \to A$$? Is it finite?

What can be said about the codimension of the range of $$D_{X}: B \to B$$? Is it finite?

We explain about the motivation for consideration of such $$A$$ and $$B$$:

We require the real analyticity on the punctured plane to avoid the obvious infinite codimension since if a limit cycle surrounds a non resonance singularity, using bump functions, one can show that the codimension is infinity, as we explained here. We require the flatness at the unique singularity at the origin to avoid some obstruction for existence of (even) formal power series solutions to the equation $$D_{X}.g=f$$ when $$X$$ is the Van der Pol equation or a more general algebraic vector field with degenerate singularity (vanishing some first Jets at the origin). The other reason for this flat requirement is that we would like to not engage with the problem of global analytic extension of a local real analytic solution (if it exists) to $$D_{X}.g=f$$. For definition of $$B$$, we require the Schwartz condition in order to apply the Fourier transform to convert a first order PDE, associated with a quadratic system, to higher order PDE to have a possible chance to work with an elliptic PDE. This is explained in the Remark 2 and its consecutive example of page 5 of the following note:

Finally, for the Van der Pol vector field $$X$$, what can be said about the codimension of the range of $$D_{X}$$ as an operator on the space $$C^{\omega}(\mathbb{R}^2)$$, the space of real analytic functions on the plane?

What about the codimension of the range of the operator $$L_X$$ when it acts on either $$\Omega^1(\mathbb{R}^2)$$ or $$\Omega^1(\mathbb{R}^2)/Z^1(\mathbb{R}^2)$$ where $$Z^1(\mathbb{R}^2)$$ is the space of closed $$1$$-forms? Do we have a possible chance for "finite codimension" in the latter quotient operator $$L_X:\Omega^1(\mathbb{R}^2)/(Z^1(\mathbb{R}^2) \to \Omega^1(\mathbb{R}^2)/(Z^1(\mathbb{R}^2)$$?

Note that the codimension of the range of $$L_X$$ is an upper bound for the number of closed orbits of $$X$$ The reason is written in the motivation part of the following post: Integral Separation of disjoint submanifolds of $\mathbb{R}^n$

## 1 Answer

The linearization of the vector field $X$ at the singular point zero is $$DX|_0 = \begin{pmatrix} 1 & 1\\ -1 & 0\end{pmatrix},$$ the eigenvalues of which are $$\lambda_{1, 2} = \frac{1}{2} \pm \frac{\sqrt{3}}{2} i,$$ hence both have positive real part. This allows to apply Thm. 4.1 in this paper, to conclude the following: Let $U$ be an open neighborhood of zero such that for each $x \in U$, $\Phi_t(x)$ converges to zero as $t \rightarrow - \infty$. Then given $v$ on $U$ which is flat at zero, the equation $$D_X u = v$$ has a unique solution $u$ in the space of smooth functions on $U$ that are flat at zero. I think that with some additional arguments, one could additionally show that $u$ is analytic, provided that $v$ is analyitic.

In any case, the trouble is that this only works locally around zero: One cannot choose $U = \mathbb{R}^n$, due to the existence of a limit cycle. The largest set one can choose for $U$ seems to be the interior of the limit cycle. Furthermore, it is not at all clear whether given a $v$ which is flat at zero extends continuously from the interior of the limit cycle to the limit cycle itself.

In a similar vein, I could believe that one can with similar methods get existence of solutions on the outside of the limit cycle. The problem now really is to glue these solutions together to form a smooth function on the limit cycle. This seems to be a very hard but nevertheless very interesting problem.

\Edit: I just noticed that the codimension of the range is infinite in any case: Let $x$ be in the limit cocycle. Then for any smooth function $u$ on $\mathbb{R}^2$, the function $t \mapsto u(\Phi_t(x))$ is periodic ($\Phi_t(x)$ being the flow of $X$). Hence $$D_X u\bigl(\Phi_t(x)\bigr) = \frac{\mathrm{d}}{\mathrm{d} t} u\bigl(\Phi_t(x)\bigr)$$ necessarily has a zero. Therefore, if $v$ is any function in the limit cocyle such that $v(x) \neq 0$ for all $x$ in the limit cycle, then it cannot be in the range.

Examples for such functions $v$ that lie in the algebra $B$ are $$v(x) = e^{-1/x^2 - x^2} x^k$$ for any $k \in \mathbb{N}_0$. This gives infinitely many linearly independent elements that are not in the range of $D_X$.

\Edit2: Indeed as Ali points out, some linear combination of these could be in the range, so this does not necessarily mean that the cokernel is infinite-dimensional.