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 \geq 3 , p < N , 0 \leq 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 { 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 \leq 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 .