behave as in certain homogeneous models. For instance, for $D=4$ pure gravity this guiding homogeneous model is the Bianchi IX model [@BKL; @misner69], while for $D=11$ supergravity it is its naive one-dimensional reduction involving space-independent metric and three-forms [@dh2].

Whether it is the first or the second behaviour that is relevant depends on: (i) the spacetime dimension $D$, (ii) the field content (presence or absence of the dilaton, types of $p$-forms), and (iii) the values of the various dilaton couplings $\lambda_j$. Previous work reached the following conclusions:

-   The oscillatory behaviour is general for pure gravity in spacetime dimension $4$ [@BKL], in fact, in all spacetime dimensions $4 \leq D \leq 10$, but is replaced by a Kasner-like behaviour in spacetime dimensions $ D \geq 11$ [@DHS]. (The sense in which we use “general" will be made precise below.)

-   The Kasner-like behaviour is general for the gravity-dilaton system in all spacetime dimensions $D \geq 3$ (see [@BK2; @AR] for $D=4$).

-   The oscillatory behaviour is general for gravity coupled to $p$-forms, in absence of a dilaton or of a dual $(D-2)$-form ($0<p<D-2$) [@dh2]. In particular, the bosonic sector of $11$-dimensional supergravity is oscillatory [@dh1]. Particular instances of this case 