Limit Cycles in a Class of Planar Discontinuous Piecewise Quadratic Differential Systems with a Non-regular Line of Discontinuity (II)

In our previous work, we have studied the limit cycles for a class of discontinuous piecewise quadratic polynomial differential systems with a non-regular line of discontinuity, which is formed by two rays starting from the origin and forming an angle alpha=pi/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = \pi /2$$\end{document}. The unperturbed system is the quadratic uniform isochronous center x(center dot)=-y+xy\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{x} = -y + x y$$\end{document}, y(center dot)=x+y2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{y} = x + y<^>2$$\end{document} with a family of periodic orbits surrounding the origin. In this paper, we continue to investigate this kind of piecewise differential systems, but now the angle between the two rays is alpha is an element of(0,pi/2)boolean OR[3 pi/2,2 pi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,\pi /2)\cup [3\pi /2,2\pi )$$\end{document}. Using the Chebyshev theory, we prove that the maximum number of hyperbolic limit cycles that can bifurcate from these periodic orbits using the averaging theory of first order is exactly 8 for alpha is an element of(0,pi/2)boolean OR[3 pi/2,2 pi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,\pi /2)\cup [3\pi /2,2\pi )$$\end{document}. Together with our previous work, which concerns on the case of alpha=pi/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =\pi /2$$\end{document}, we can conclude that using the averaging theory of first order the maximum number of hyperbolic limit cycles is exactly 8, when this quadratic center is perturbed inside the above-mentioned classes separated by a non-regular line of discontinuity with alpha is an element of(0,pi/2]boolean OR[3 pi/2,2 pi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,\pi /2]\cup [3\pi /2,2\pi )$$\end{document}.