  In this paper we perform a systematic classification of the regimes of
cosmological dynamics in Einstein-Gauss-Bonnet gravity with generic values of
the coupling constants. We consider a manifold which is a warped product of a
four dimensional Friedmann-Robertson-Walker space-time with a $D$-dimensional
Euclidean compact constant curvature space with two independent scale factors.
A numerical analysis of the time evolution as function of the coupling
constants and of the curvatures of the spatial section and of the extra
dimension is performed. We describe the distribution of the regimes over the
initial conditions space and the coupling constants. The analysis is performed
for two values of the number of extra dimensions ($D\geqslant 6$ both) which
allows us to describe the effect of the number of the extra dimensions as well.
