fields $A^{(j)}$’s on the evolution of $g_{\mu \nu}$ and $\phi$ can be asymptotically neglected. In that regime the spatial curvature terms can be also neglected with respect to the leading order part of the extrinsic curvature terms. In other words, as emphasized by BKL, time derivatives asymptotically dominate over space derivatives so that one sometimes uses the terminology “velocity-dominated" behaviour [@Eardley], instead of “generalized Kasner behaviour”. We shall use both terminologies indifferently in this paper, recalling that in the presence of $p$-forms, which act as potentials for the evolution of the spatial metric and the dilaton (as do the spatial curvature terms), “velocity-dominance" means not only that the spatial curvature terms can be neglected, but also that the $p$-forms can be neglected in the Einstein-dilaton evolution equations.[^3]

2.  \[mixmasterbehaviour\] The second regime, known as [*“oscillatory”*]{} [@BKL], or [*“generalized mixmaster”*]{} [@misner69] behaviour, is more complicated. It can be described as the succession of an infinite number of increasingly shorter Kasner regimes as one goes to the singularity, one following the other according to a well-defined “collision" law. This asymptotic evolution is presumably strongly chaotic. It is expected that, at each spatial point, the scale factors of a general inhomogeneous solution essentially 