Dynamical system analysis of ΛCDM

$\Lambda\mathrm{CDM}$ is the simplest model that is broadly consistent with the observations. Written in terms of the density parameters of matter, radiation and the cosmological constant, the system is

$$\begin{aligned} \Omega'_{\mathrm{m}} &= \Omega_{\mathrm{m}}\,(\Omega_{\mathrm{r}} - 3\Omega_\Lambda) \\[1ex] \Omega'_{\mathrm{r}} &= \Omega_{\mathrm{r}}\,(\Omega_{\mathrm{r}} - 3\Omega_\Lambda - 1) \\[1ex] \Omega'_\Lambda &= \Omega_\Lambda\,(\Omega_{\mathrm{r}} - 3\Omega_\Lambda + 3) \end{aligned}$$

The constraint equation $\Omega_{\mathrm{m}} + \Omega_{\mathrm{r}} + \Omega_\Lambda = 1$ removes one of the three. Eliminating $\Omega_\Lambda$ reduces the system to two dimensions:

$$\begin{aligned} \Omega'_{\mathrm{m}} &= \Omega_{\mathrm{m}}\,(3\Omega_{\mathrm{m}} + 4\Omega_{\mathrm{r}} - 3) \\[1ex] \Omega'_{\mathrm{r}} &= \Omega_{\mathrm{r}}\,(3\Omega_{\mathrm{m}} + 4\Omega_{\mathrm{r}} - 4) \end{aligned}$$

Its phase portrait shows the familiar history: trajectories leave the radiation-dominated point, pass close to matter domination and end on the de Sitter point where $\Omega_\Lambda = 1$.