Cosmological dynamics of spatially flat Einstein-Gauss-Bonnet models in
  various dimensions: High-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} of the scale
factors, which makes our results quite general. With both Einstein-Hilbert and
Gauss-Bonnet contributions in play, $D=3$ and the general $D\geqslant 4$ cases
have slightly 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 with special interest on the realistic
regimes. Our analysis suggests that the only realistic regime is the transition
from high-energy (Gauss-Bonnet) Kasner regime, which is the standard
cosmological singularity in that case, to the anisotropic exponential regime
with expanding three and contracting extra dimensions. Availability of this
regime allows us to put constraint on the value of Gauss-Bonnet coupling
$\alpha$ and the $\Lambda$-term -- this regime appears in two regions on
$(\alpha, \Lambda)$ plane: $\alpha < 0$, $\Lambda > 0$, $\alpha\Lambda
\leqslant 1/2$ and $\alpha > 0$, $\alpha\Lambda \leqslant (3D^2 - 7D +
6)/(4D(D-1))$, including entire $\Lambda < 0$ region. The obtained bounds are
confronted with the restrictions on $\alpha$ and $\Lambda$ from other
considerations, like causality, entropy-to-viscosity ratio in AdS/CFT and
others. Joint analysis constraints ($\alpha$, $\Lambda$) even further: $\alpha
> 0$, $D \geqslant 2$ with $(3D^2 - 7D + 6)/(4D(D-1)) \geqslant \alpha \Lambda
\geqslant - (D+2)(D+3)(D^2 + 5D + 12)/(8(D^2 + 3D + 6)^2)$.
