In this question, we are interested in the number of limit cycles which appears in the following perturbational system:
\begin{equation}\cases{
x'=y -x^{2}+\epsilon P(x,y) \\
y'=-x+\epsilon Q(x,y) }
\end{equation}
where $P$ and $Q$ are polynomials of degree $n$. The unperturbed system (i.e, $\epsilon=0$) is not a Hamiltonian vector field, so we multiply the above vector field by the integrating factor
"$e^{-2y}$". After this multiplication, the (new) unperturbed system is a Hamiltonian vector field with Hamiltonian \begin{equation}
H(x,y)=e^{-2y}(y-x^{2}+1/2)
\end{equation}
The unperturbed system has a unique singularity at the origin, which is of center type. The region of closed orbits surrounding the center is $\{(x,y)|0\leq H(x,y) \leq 1/2\}$.
According to the theory of Abelian integrals and its relation to the second part of the Hilbert 16th problem, the number of limit cycles of the perturbed system is (equal to and ) closely related to the number of zeroes of the following integral function $I:[0,\;1/2]\to \mathbb{R}$:
\begin{equation}
I(c)=\int_{H^{-1}(c)} e^{-2y}(Pdy-Qdx)
\end{equation}
I have been motivated by a famous result of A. Varchenko about the finiteness of the number of the zeros of abelian integrals of polynomial perturbation of polynomial Hamiltonians (which is proved in [1]). Hence, I have the following two questions:
Questions:
- Assume that $P$ and $Q$ are fixed (given) polynomials. Is it true that either $I$ is identically zero, or it has only a finite number of zeros? Is there an explicit formula for the function $I(c)$? Or at least can we compute $\lim_{c \to 0} I(c)$?
- If the answer to the above question is affirmative, can we control the number of zeroes of $I$, in terms of the degree of polynomials $P,Q$?
Note 1: The proof of Varchenko is essentially based on algebraic geometry.
In fact, they consider the polynomial perturbation of polynomial Hamiltonian systems. (Please see page 5, section 3.5 of this paper. But in this particular question, the non-algebraic integrating factor $e^{-2y}$ destroys the algebro-geometric feature of the problem. So how can one remedy this problem?
Note 2: As a possible resolution to avoid the non-algebraic term $e^{-2y}$, we consider the following approach, which can be applied for every algebraic perturbation of algebraic vector fields, where the unperturbed system has a band of closed orbit (center). But note that it is not a Hamiltonian vector field, and we have a non-algebraic integrating factor.
Consider the polynomial vector field $$\begin{cases} x'=P+\epsilon A\\ y'=Q+\epsilon B \end{cases}$$
where the unperturbed system has a band of closed orbit.
We assume that there is a straight line $\ell$ parametrized by a real parameter $h\in \mathbb{R}$ with the following property:
Periodic orbits of the unperturbed system intersect $\ell$ transversally and $s'(h) \neq 0$ for all $h$ where $s(h)$ is the slope of the unperturbed vector field at point $h$. This is opposite to the concept of "isocline". For example, this is the case for the Liénard vector field
\begin{equation}\cases{
x'=y -x^{2}+\epsilon P(x,y) \\
y'=-x+\epsilon Q(x,y) }
\end{equation}
when $\ell$ is the $x$ axis. But it is not the case for the isocline $y$ axis.
We take a point $h \in \ell$. By $p(h)$, we mean the value of the Poincaré return map $p$ at $h$. Hence $p(h) \in \ell$ is the first return map for the orbit starting at $h \in \ell$. WLOG we may assume that the orientation of the solution curve is anti-clockwise. Consider the simple closed curve $\gamma$ consisting of the solution curve $h \to p(h)$ and the part of the straight line $p(h) \to h$. If $p(h) \neq h$, then the integral of $\int_{\gamma} \kappa_{g} \neq 2\pi$ , since two different points of the line $\ell$ has different slopes. (This is a consequence of the Gauss–Bonnet theorem). So for $\epsilon$ sufficiently small, the above curvature integral is $2 \pi$ if and only if $p(h)=h$.
Now we compute the integral of the curvature:
$$T(h)=\int_{\tilde{\gamma}}(\frac{(P+\epsilon A)(Q_{x}+\epsilon B_{x})+(Q+\epsilon B)(Q_{y}+\epsilon B_{y})}{P^2+Q^2})dx -(\frac{(Q+\epsilon B)(P_{y}+\epsilon A_{y})+(P+\epsilon A)(P_{x}+\epsilon A_{x})}{P^2+Q^2})dy$$
where $\tilde{\gamma}$ is the same as $\gamma$, but we remove the straight line $p(h) \to h$. (Similar to the orbit in this picture, which starts from $r_{0}$ and ends to the $p(r_{0})$).
The above integral is equal to $2\pi +\epsilon c(h) + O(\epsilon^2)$, where $c(h)$ is the following:
$$c(\alpha)= \int_{\alpha} (\frac{(AQ_{x}+PB_{x}+BQ_{y}+QB_{y})(P^2+Q^2)+2AP+2BQ}{{(P^2+Q^2)}^2})dx-\int_{\alpha}(\frac{(BP_{y}+QA_{y}+AP_{x}+PA_{x})(P^2+Q^2)+2AP+2BQ}{{(P^2+Q^2)}^2})dy$$
where $\alpha $ is the closed orbit of the unperturbed system starting at $h$
Now in the integral $c(h)$, we do not have any non-algebraic term.
So this would possibly imply that the zero set of $c(h)$ is equal to the zero set of $I(h)$ where $I(h) $ is the standard abelian integral $I(h)= \int_{\alpha(h)} Ady-Bdx$. Because both zero sets are corresponding points of generating limit cycles.
Is the latter statement true? Is the computation of $c(h)$ correct? Does this situation make facilities for computations of abelian integrals to count the number of generating limit cycles?
Reference
[1] Arnol’d, V. I.; Gusejn-Zade, S. M.; Varchenko, A. N., Singularities of differentiable maps. Volume II: Monodromy and asymptotics of integrals Translated from the Russian by Hugh Porteous, (English) Monographs in Mathematics, 83. Boston, MA etc.: Birkhäuser Verlag, pp. viii+492 (1988), MR0966191, Zbl 0659.58002.
Best Answer
This paper contains a conjecture and a partial result about the abelian integral under discussion :"Computer-assisted techniques for the verification of the Chebyshev property of Abelian integrals"