  In this paper we consider $(n+1)$-dimensional cosmological model with scalar
field and antisymmetric $(p+2)$-form. Using an electric composite $Sp$-brane
ansatz the field equations for the original system reduce to the equations for
a Toda-like system with $n(n-1)/2$ quadratic constraints on the charge
densities. For certain odd dimensions ($D = 4m+1 = 5, 9, 13, ...$) and
$(p+2)$-forms ($p = 2m-1 = 1, 3, 5, ...$) these algebraic constraints can be
satisfied with the maximal number of charged branes (i.e.} all the branes have
non-zero charge densities). These solutions are characterized by self-dual or
anti-self-dual charge density forms $Q$ (of rank $2m$). For these algebraic
solutions with the particular $D$, $p$, $Q$ and non-exceptional dilatonic
coupling constant $\lambda$ we obtain general cosmological solutions to the
field equations and some properties of these solutions are examined. In
particuilar Kasner-like behavior, the existence of attractor solutions.
