Zeros of Abelian integrals for a quartic Hamiltonian with figure-of-eight loop through a nilpotent saddle
Consider the perturbed polynomial system \[ \frac{dx}{dt} = \frac{\partial H}{\partial y} + \varepsilon f(x,y),\quad \frac{dy}{dt}=-\frac{\partial H}{\partial x} + \varepsilon g(x,y) , \] where \(H(x,y)\) is a polynomial function of degree \(m+1\), \(f(x,y)\) and \(g(x,y)\) are polynomial functions of degree \(n\), and \(\varepsilon\) is a small parameter. Suppose that for \(\varepsilon =0\), the system has a family of closed orbits \(\Gamma_{h}\) define by \(H(x,y)=h\). Let \(Z(m,n)\) be the least upper bound for the number of isolated zeros of the abelian integral \[ I(h) = \oint_{\Gamma_{h}}^{} g(x,y) dx -f(x,y) dy . \] The weakened Hilbert's 16th problem is to find \(Z(m,n)\) in terms of \(m\) and \(n\). There are many results for certain values of \(m\) and \(n\). The authors consider the Hamiltonian \[ H(x,y) = -x^{2} +x^{4} + y^{4} + r\,x^{2}y^{2},\,\,r \geq 0, r\neq 2. \] The unperturbed system has three singular points \[ S(0,0),C_{1} (-\frac{\sqrt{2}}{2}), C_{2}(\frac{\sqrt{2}}{2},0). \] The period annulus around \(C_{1}\) (\(C_{2}\)) is denoted by \(\Gamma_{h}^{L}\) (\(\Gamma_{h}^{R}\)). Let \(B_{1}(n)\) be the value of \(Z\) corresponding to \(\Gamma_{h}^{L}\) or \(\Gamma_{h}^{R}\). Let \(B_{2}\) be the value of \(Z\) corresponding to the unbounded period annulus. Using a known procedure, the authors prove that \[ \begin{aligned} B_{1} (n) &\leq 40n +6 \left[ \frac{n-1}{4} \right] + 6 \left[ \frac{n-2}{4} \right] + 11,\\ B_{2}(n) &\leq 3n + 3 \left[ \frac{n-1}{4} \right] + 13.\end{aligned} \]
scientific article; zbMATH DE number 6523606
scientific article