Non-integrability of the mixmaster universe

Details Non-integrability of the mixmaster universe Non-integrability of the mixmaster universe

1994-06-13

arxiv.org/abs/solv-int/9406002v2

J.Phys. A28 (1995) 657-668

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

11 pages LaTeX in J.Phys.A style (ioplppt.sty) + 6 PostScript figures compressed and uuencoded with uufiles. Revised version t

Scientific paper

10.1088/0305-4470/28/3/019

We comment on an analysis by Contopoulos et al. which demonstrates that the governing six-dimensional Einstein equations for the mixmaster space-time metric pass the ARS or reduced Painlev\'{e} test. We note that this is the case irrespective of the value, $I$, of the generating Hamiltonian which is a constant of motion. For $I < 0$ we find numerous closed orbits with two unstable eigenvalues strongly indicating that there cannot exist two additional first integrals apart from the Hamiltonian and thus that the system, at least for this case, is very likely not integrable. In addition, we present numerical evidence that the average Lyapunov exponent nevertheless vanishes. The model is thus a very interesting example of a Hamiltonian dynamical system, which is likely non-integrable yet passes the reduced Painlev\'{e} test.

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