Cosmological dynamics of spatially flat Einstein-Gauss-Bonnet models in
  various dimensions. Vacuum case
  In this paper we perform a systematic study of vacuum spatially flat
((3+D)+1)-dimensional Einstein-Gauss-Bonnet cosmological models. We consider
models which topologically are the product of two flat isotropic subspaces with
different scale factors. One of these subspaces is 3D and represents our space
and the other is D-dimensional and represents extra dimensions. We consider no
ansatz of the scale factors, which makes our results quite general. With both
Einstein-Hilbert and Gauss-Bonnet contributions in play, the cases with D=1,
D=2, D=3 and $D\geqslant 4$ have different dynamics due to different structure
of the equations of motion. We analytically study equations of motion in all
cases and describe all possible regimes. It appears that the only regimes with
nonsingular future asymptotes are the Kasner regime in GR as well as
exponential regimes. As of the past asymptotes, for a smooth transition only
Kasner regime in Gauss-Bonnet is an option. With that at hand, we are down only
to two viable regimes -- "pure" Kasner regime (transition from high- to
low-energy Kasner regime) and a transition from high-energy Kasner to
anisotropic exponential solution. It appears that these regimes take place for
different signs of the Gauss-Bonnet coupling $\alpha$: "pure" Kasner regime
occur for $\alpha > 0$ at low D and $\alpha < 0$ for high D; anisotropic
exponential regime is reached only for $\alpha > 0$. So if we restrain
ourselves with $\alpha > 0$ solutions, the only late-time regimes are Kasner
for D=1, 2 and anisotropic exponential for $D\geqslant 2$. Also, low-energy
Kasner regimes ($a(t)\propto t^p$) have expansion rates for (3+1)-dimensional
subspace ("our Universe") ranging from p=0.5 (D=1) to $p=1/\sqrt{3} \approx
0.577$ ($D\to\infty$), which contradicts with dust-dominated Friedmann
prediction (p=2/3).
