COMMENTS ON THE COMPUTATION OF LIAPUNOV EXPONENTS FOR THE MIXMASTER UNIVERSE

To explain the discrepancy between recently computed vanishing Liapunov exponents for the evolution of Mixmaster universes and the positive Liapunov exponent for the associated 1-dimensional map first discussed by Belinskii, Khalatnikov, and Lifshitz, the Liapunov exponents computed from a numerical universe evolution are compared using several time variables. Previous numerical results of vanishing Liapunov exponents were obtained with time variables which increased roughly exponentially in each epoch. Here it is found that minisuperspace proper time, which increases by a fixed amount during each epoch, yields non-vanishing Liapunov exponents within the limited number of epochs numerically accessible. The map parameter u as measured along the trajectory attains the values predicted by the map to very high accuracy (except near the maximum of expansion) even though the metric coefficients deviate in some cases from idealized Mixmaster behavior. The number of consecutive single epoch eras is shown to be related to the presence of u in an interval bounded by ratios of Fibonacci numbers.