On the Number of Limit Cycles Bifurcating from a Quartic Reversible Center

This paper deals with the bifurcation of limit cycles from a quartic reversible and non-Hamiltonian system. By using the averaging theory and some mathematical technique on estimating the zeros of the function, we show that under small polynomial perturbation of degree 3n+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3n+1$$\end{document}, at most 3n-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3n-3$$\end{document} limit cycles bifurcate from the period annulus of the unperturbed system for n>3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n>3$$\end{document}, while at most 2n limit cycles appear from the period annulus of the unperturbed system for n=1,2,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=1, 2, 3$$\end{document}. And the upper bound for the latter case is sharp.