Dynamics of a closed viscous universe

Abstract

We use a dynamical systems approach based on the method of orthonormal frames to study the dynamics of a non-tilted Bianchi Type IX cosmological model with a bulk and shear viscous fluid source. We begin by completing a detailed fix-point analysis which give the local sinks, sources and saddles of the dynamical system. We then analyze the global dynamics by finding the $\alpha$-and $\omega$-limit sets which give an idea of the past and future asymptotic behavior of the system. The fixed points were found to be a flat Friedmann-LeMa\^{i}tre-Robertson-Walker (FLRW) solution, Bianchi Type $II$ solution, Kasner circle, Jacobs disc, Bianchi Type $VII_{0}$ solutions, and several closed FLRW solutions in addition to the Einstein static universe solution. Each equilibrium point was described in both its expanding and contracting epochs. We conclude the paper with some numerical experiments that shed light on the global dynamics of the system along with its heteroclinic orbits. With respect to past asymptotic states, we were able to conclude that the Jacobs disc in the expanding epoch was a source of the system along with the flat FLRW solution in a contracting epoch. With respect to future asymptotic states, we were able to show that the flat FLRW solution in an expanding epoch along with the Jacobs disc in the contracting epoch were sinks of the system. We were also able to demonstrate a new result with respect to the Einstein static universe. Namely, we gave certain conditions on the parameter space such that the Einstein static universe has an associated stable subspace. We were however, not able to conclusively say anything about whether a closed FLRW model could be a past or future asymptotic state of the model.