  We study the evolution of maximally symmetric $p$-branes with a
$S_{p-i}\otimes \mathbbm{R}^i$ topology in flat expanding or collapsing
homogeneous and isotropic universes with $N+1$ dimensions (with $N \ge 3$, $p <
N$, $0 \le i < p$). We find the corresponding equations of motion and compute
new analytical solutions for the trajectories in phase space. For a constant
Hubble parameter, $H$, and $i=0$ we show that all initially static solutions
with a physical radius below a certain critical value, $r_c^0$, are periodic
while those with a larger initial radius become frozen in comoving coordinates
at late times. We find a stationary solution with constant velocity and
physical radius, $r_c$, and compute the root mean square velocity of the
periodic $p$-brane solutions and the corresponding (average) equation of state
of the $p$-brane gas. We also investigate the $p$-brane dynamics for $H \neq
{\rm constant}$ in models where the evolution of the universe is driven by a
perfect fluid with constant equation of state parameter, $w={\cal P}_p/\rho_p$,
and show that a critical radius, $r_c$, can still be defined for $ -1 \le w <
w_c$ with $w_c=(2-N)/N$. We further show that for $w \sim w_c$ the critical
radius is given approximately by $r_c H \propto (w_c-w)^{\gamma_c}$ with
$\gamma_c=-1/2$ ($r_c H \to \infty$ when $w \to w_c$). Finally, we discuss the
impact that the large scale dynamics of the universe can have on the
macroscopic evolution of very small loops.
