Localizing branes with bifurcating bulks

Abstract We study the problem of evolution bulk 5-fluids having an embedded braneworld with a flat, de Sitter, or anti-de Sitter geometry. introduce new variables to express Einstein equations as dynamical system that depends on equation state parameter $$\gamma $$ γ and exponent $$\lambda λ . For linear fluids (i.e., =1$$ = 1 ), our formulation leads partial decoupling thus exact solution. find such fluid develops transcritical bifurcation around value =-1/2$$ - / 2 , how this behaviour affects stability solutions. nonlinear fluids, situation is more diverse. overall attractor at =1/2$$ draw enough phase portraits exhibit in detail dynamics. show =3/2$$ 3 structurally unstable typical for other forms Consequently, we observe noticeable dependence qualitative solutions different ‘polytropic’ bulk. In addition, prove existence Dulac function signifying impossibility closed orbits certain subsets space. also provide ample numerical evidence gravity localizing brane which satisfy all energy conditions.