Limit Cycles for a Class of Three-Dimensional Polynomial Differential Systems

Perturbing the system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\dot{x} = - y{\left( {1 + x} \right)},\,\dot{y} = x{\left( {1 + x} \right)},\,\dot{z} = 0$\end{document} inside the family of polynomial differential systems of degree n in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{3}$\end{document}, we obtain at most n2 limit cycles using the first-order averaging theory. Moreover, there exist such perturbed systems having at least n2 limit cycles.