  We investigate the dynamics of a flat isotropic brane Universe with
two-component matter source: perfect fluid with the equation of state
$p=(\gamma-1) \rho$ and a scalar field with a power-law potential $V \sim
\phi^{\alpha}$. The index $\alpha$ can be either positive or negative. We
describe solutions for which the scalar field energy density scales as a
power-law of the scale factor (so called scaling solutions). In the nonstandard
brane regime when the brane is driven by energy density square term these
solutions are rather different from their analogs in the standard cosmology. A
particular attention is paid to the inverse square potential. Its dynamical
properties in the nonstandard brane regime are in some sense analogous to those
of the exponential potential in the standard cosmology. Stability analysis of
the scaling solutions are provided. We also describe solutions existing in
regions of the parameter space where the scaling solutions are unstable or do
not exist.
