Dynamics of two Einstein–Friedmann cosmological models

We describe completely the dynamics of the two Einstein–Friedmann cosmological models, which can be characterized by the Hamiltonians H=12(py2-px2)+e2xV(y),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} H = \frac{1}{2} (p_{y}^2 - p_{x}^2) + e^{2 x} V(y), \end{aligned}$$\end{document}with the cosmological potentials V(y)=eλy\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(y)=e^{\lambda y}$$\end{document}, or V(y)=(a+by)ey\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(y)=(a+by)e^{y}$$\end{document} with λab≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda a b\ne 0$$\end{document}.