Transcritical and zero-Hopf bifurcations in the Genesio system

In this paper we study the existence of transcritical and zero-Hopf bifurcations of the third-order ordinary differential equation x⃛+ax¨+bx˙+cx-x2=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dddot{x}} + a {\ddot{x}} + b {\dot{x}} + c x - x^2 = 0$$\end{document}, called the Genesio equation, which has a unique quadratic nonlinear term and three real parameters. More precisely, writing this differential equation as a first-order differential system in R3\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 prove: first that the system exhibits a transcritical bifurcation at the equilibrium point located at the origin of coordinates when c=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=0$$\end{document} and the parameters (a, b) are in the set {(a,b)∈R2:b≠0}\{(0,b)∈R2:b>0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{(a,b) \in \mathbb {R}^2 : b \ne 0\} {\setminus } \{(0,b) \in \mathbb {R}^2 : b > 0\}$$\end{document}, and second that the system has a zero-Hopf bifurcation also at the equilibrium point located at the origin when a=c=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=c=0$$\end{document} and b>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b>0$$\end{document}.