Asymptotic states of magnetic Bianchi I cosmologies

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Abstract
The Einstein - Maxwell field equations for the class of hypersurface-orthogonal Bianchi I cosmologies with a -law perfect fluid and a pure, homogeneous source-free magnetic field are written as an autonomous differential equation in terms of expansion-normalized variables. The equilibrium points of the associated dynamical system correspond to transitively self-similar cosmologies, some of which appear to be previously undiscovered. It is proven that for each value of the equation of state parameter , there is a unique self-similar cosmology which acts as a late asymptotic state for a set of models of non-zero measure in this class. It is then shown that there exists a flow-invariant compact subset of phase space which generates oscillations on the Kasner circle of equilibrium points. Monotonic functions and numerical evidence are used to support the conjecture that this set is the attractor into the past for generic models.

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