Quasi-homologous spherically symmetric branes and their symmetry   breaking
  We revisit the dynamical system based approach of spherically symmetric vacuum braneworlds, pointing out and studying the existence of a transcritical bifurcation as the dark pressure parameter changes its sign, we analyze some consequences of not discard the brane cosmological constant. For instance, it is noteworthy that the existence of an isothermal state equation between the dark fluid parameters cannot be obtained via the requirement of a quasi-homologous symmetry of the vacuum. 
M. C. B. Abdalla, P. Carlesso, and J. M. Hoff da Silva
gr-qc hep-th