The cyclicity of a class of quadratic reversible centers defining elliptic curves

Abstract

<p style='text-indent:20px;'>In this paper, the cyclicity of period annulus of an one-parameter family quadratic reversible system under quadratic perturbations is studied which is equivalent to the number of zeros of any nontrivial linear combination of three Abelian integrals. By the criteria established in [<xref ref-type="bibr" rid="b28">28</xref>] and the asymptotic expansions of Abelian integrals, we obtain that the cyclicity is two when the parameter in <inline-formula><tex-math id="M1">\begin{document}$ (-\infty,-2)\cup[-\frac{8}{5},+\infty) $\end{document}</tex-math></inline-formula>. Moreover, we develop new criteria which combined with the asymptotic expansions of Abelian integrals show that the cyclicity is three when the parameter belongs to <inline-formula><tex-math id="M2">\begin{document}$ (-2,-\frac{8}{5}) $\end{document}</tex-math></inline-formula>.</p>