Cosmological dynamics of spatially flat Einstein-Gauss-Bonnet models in various dimensions: Low-dimensional $\Lambda$-term case

In this paper we perform a systematic study of spatially flat [(3+D)+1]-dimensional Einstein-Gauss-Bonnet cosmological models with $\Lambda$-term. We consider models that topologically are the product of two flat isotropic subspaces with different scale factors. One of these subspaces is three-dimensional and represents our space and the other is D-dimensional and represents extra dimensions. We consider no {\it Ansatz} on the scale factors, which makes our results quite general. With both Einstein-Hilbert and Gauss-Bonnet contributions in play, the cases with $D=1$ and $D=2$ have different dynamics due to the different structure of the equations of motion. We analytically study equations of motion in both cases and describe all possible regimes. It is demonstrated that $D=1$ case does not have physically viable regimes while $D=2$ has smooth transition from high-energy Kasner to anisotropic exponential regime. This transition occurs for two ranges of $\alpha$ and $\Lambda$: $\alpha > 0$, $\Lambda > 0$ with $\alpha \Lambda \leqslant 1/2$ and $\alpha < 0$, $\Lambda > 0$ with $\alpha\Lambda < -3/2$. For the latter case if $\alpha\Lambda = -3/2$, extra dimensional part has $h\to 0$ and so the size of extra dimensions (in the sense of the scale factor) is reaching constant value. We report substantial differences between $D=1$ and $D=2$ cases and between these cases and their vacuum counterparts, describe features of the cases under study and discuss the origin of the differences.