Average rate of separation of trajectories near the singularity in mixmaster models

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Abstract
The system of equations for a mixmaster cosmological model is reduced to a geodesic flow on a pseudo-Riemannian manifold. This geodesic flow is, on the average, locally unstable in the first and second Belinskij-Khalatnikov-Lifshitz (BKL) approximations. In the geometrized model of dynamics we define an average rate of separation of nearby trajectories with the help of a geodesic deviation equation in a Fermi basis. It turns out that the standard indicator for detecting chaotic behavior, a principal Lyapunov exponent, can be obtained from a normal separation vector. We also show that the principal Lyapunov exponents are always positive in the first and second BKL approximations. If the period of oscillations in the long phase (the second BKL approximation) is infinite, the principal Lyapunov exponent tends to zero.

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