The monotonicity of the ratio of two abelian integrals (Q2847203)

The authors establish a criterion for the monotonicity of the ratio NEWLINE\[NEWLINE\int_{\Gamma_h}xydx/\int_{\Gamma_h}ydx,NEWLINE\]NEWLINE where \(\Gamma_h\) belongs to a continuous set of ovals (period annulus) contained in the level sets of the function \(H(x,y)=h\), where \(H=y^2+\Psi(x)\). It is assumed that \(\Psi(x)\) is an analytic function in a neighborhood of some point \(a\in{\mathbb R}\) which satisfies (H1) \(\Psi'(x)(x-a)>0\), and (H2) \(\Psi'(x)\sim (x-a)^{2k-1}\), \(k\) natural, as \(x\to a\). The criterion is applied to determine the cases for which the small perturbation NEWLINE\[NEWLINE\begin{aligned} \dot{x}&=-H_y,\\ \dot{y}&=H_x+\varepsilon (\beta_0+\beta_1x)y, \end{aligned}NEWLINE\]NEWLINE where \(H\) has the potential NEWLINE\[NEWLINE\Psi(x)=-\frac{uv}{2}x^2+\frac{u+v+uv}{3}x^3-\frac{1+u+v}{4}x^4+\frac15x^5NEWLINE\]NEWLINE with either \(0\leq u\leq v\leq 1\) or \(u=\bar{v}\in {\mathbb C}\setminus {\mathbb R}\), produces a unique limit cycle born from the period annulus under consideration. The authors compare their results to those obtained by \textit{L. Gavrilov} and \textit{I. D. Iliev} [Trans. Am. Math. Soc. 356, No. 3, 1185--1207 (2004; Zbl 1043.34031)].