Dynamics of two Einstein–Friedmann cosmological models

We describe completely the dynamics of two Einstein–Friedmann cosmological models, which can be characterized by Hamiltonians $$\begin{aligned} H = \frac{1}{2} (p_{y}^2 - p_{x}^2) + e^{2 x} V(y), \end{aligned}$$with potentials \(V(y)=e^{\lambda y}\), or \(V(y)=(a+by)e^{y}\) with \(\lambda a b\ne 0\).