Let \(H(x,y)\) be a real polynomial in \(x\) and \(y\) of degree \(n+1\) and consider the Hamiltonian system \[ \dot{x} = \frac{\partial H}{\partial y}, \qquad \dot{y} = -\frac{\partial H}{\partial x}, \] where the dot denotes derivation with respect to an independent variable. Assume that this system has a period annulus. One bifurcation problem is to study which of these periodic orbits remain as limit cycles after a polynomial perturbation of the system. Consider a perturbation of the form: \[ \dot{x} = \frac{\partial H}{\partial y} + \varepsilon f(x,y), \qquad \dot{y} = -\frac{\partial H}{\partial x} + \varepsilon g(x,y), \] where \(f(x,y)\) and \(g(x,y)\) are polynomials in \(x\) and \(y\) of degree at most \(n\). This bifurcation problem can be generically solved by studying the number of isolated zeroes of the following Abelian integral: \[ I(h) = \oint_{L(h)} f(x,y)dx+g(x,y)dy, \] where \(L(h)\) is one of the orbits in the considered period annulus of the Hamiltonian system which continuously depends on \(h \in J\) with \(H(x,y)=h\) and \(J\) and open interval in \(\mathbb{R}\).
The authors consider a Hamiltonial of the form \(H(x,y)=y^2+P_7(x)\) where \(P_7(x)\) is the following polynomial of degree \(7\): \[ \begin{array}{lll} P_7(x) & = & \frac{x^7}{7} - (\alpha+\beta+2\gamma+1) \frac{x^6}{6} \\ & & + (\alpha \beta +2(\alpha+\beta+1)\gamma+\gamma^2+\alpha+\beta) \frac{x^5}{5} \\ & & - (2 \alpha \beta \gamma+(\alpha+\beta)\gamma^2+\alpha \beta+2(\alpha+\beta)\gamma+\gamma^2) \frac{x^4}{4} \\ & & + (\alpha \beta \gamma^2+2\alpha \beta \gamma +(\alpha+\beta)\gamma^2) \frac{x^3}{3} - \alpha \beta \gamma^2 \frac{x^2}{2}, \end{array} \] with \(\alpha\), \(\beta\) and \(\gamma\) real parameters such that \(0\leq \alpha \leq \beta \leq \gamma \leq 1\). The corresponding Hamiltonian system is \[ \dot{x} = 2y, \quad \dot{y} = x(x-\alpha)(x-\beta)(x-\gamma)^2(x-1). \] The authors study the Abelian integrals \[ I_0(h) = \oint_{L(h)} ydx, \quad I_1(h) = \oint_{L(h)} xydx. \] In particular the authors deal with the monotonicity of the ratio \[ P(h) = \frac{I_1(h)}{I_0(h)}. \] First the authors provide the bifurcation diagram of the Hamiltonian system and then study the monotonicity of \(P(h)\) in each different phase portrait.