An estimation for the number of limit cycles in a Liénard-like perturbation of a quadratic nonlinear center

The number of limit cycles which bifurcates from periodic orbits of a differential system with a center has been extensively studied recently using many distinct tools. This problem was proposed by Hilbert in 1900, and it is a difficult problem, so only particular families of such systems were considered. In this paper, we study the maximum number of limit cycles that can bifurcate from an integrable nonlinear quadratic isochronous center, when perturbed inside a class of Liénard-like polynomial differential systems of arbitrary degree n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}. We apply the averaging theory of first order to this class of Liénard-like polynomial differential systems, and we estimate that the number of limit cycles is 2[(n-2)/2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2[(n-2)/2]$$\end{document}, where [.]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[.]$$\end{document} denotes the integer part function.