A Degenerate Bogdanov-Takens Normal Form for FLRW Cosmologies

In this paper, we first show that the Einstein field equations for all perfect-fluid FLRW cosmologies can be written as a planar dynamical system with the equation of state parameter $w$ and cosmological constant $\mathrm{\Lambda }$ as parameters. An important equilibrium point of this dynamical system is the origin which represents Minkowski spacetime. It is shown that the Einstein field equations in a neigbourhood of this point are equivalent to a degenerate Bogdanov-Takens normal form. This normal form admits a set of equilibrium points that describes a set of solutions to the Einstein field equations that have a constant rate of expansion, negative spatial curvature, zero cosmological constant and dust.