coupling terms among the $p$-forms which can be either of the Yang-Mills type ($1$-forms), Chern-Simons type [@jdt] or Chapline-Manton type [@cm; @pvnetal]. The degrees of the $p$-forms are restricted to be smaller than or equal to $D-2$ since a $(D-1)$-form (or $D$-form) gauge field carries no local degree of freedom. In particular, scalars ($n_j=0$) are allowed among the $A^{(j)}$’s but we then require that the corresponding dilaton coupling $\lambda_j$ be non-zero, so that there is only one “dilaton”. Similarly, we require $\lambda_j \not=0$ for the $(D-2)$-forms (if any), since these are “dual”[^1] to scalars. This restriction to a single dilaton is mostly done for notational convenience: if there were other dilatons among the $0$-forms, then, these must be explicitly treated on the same footing as $\phi$ and separated off from the $p$-forms because they play a distinct rôle. In particular, they would appear explicitly in the generalized Kasner conditions given below and in the determination of what we call the subcritical domain. The discussion would proceed otherwise in the same qualitative way.

The main motivation for studying actions of the class (\[001\]) is that these arise as bosonic sectors of supergravity theories related to superstring or M-theory. In fact, in 