dorsal/arxiv
View SchemaNon-integrability of the mixmaster universe
| Authors | Freddy Christiansen, Hans Henrik Rugh, Svend Erik Rugh |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9406002 |
| URL | https://arxiv.org/abs/solv-int/9406002 |
| DOI | 10.1088/0305-4470/28/3/019 |
| Journal | J.Phys. A28 (1995) 657-668 |
Abstract
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.
{
"annotation_id": "b9813d94-a78e-43e4-ba64-5ffcd3f0d2ca",
"date_created": "2026-03-02T18:02:48.244000Z",
"date_modified": "2026-03-02T18:02:48.244000Z",
"file_hash": "0d133e352d1f232bd4d94c001c8827137fd1beb1804f82ad32824819fcd45295",
"private": false,
"record": {
"abstract": "We comment on an analysis by Contopoulos et al. which demonstrates that the\ngoverning six-dimensional Einstein equations for the mixmaster space-time\nmetric pass the ARS or reduced Painlev\\\u0027{e} test. We note that this is the case\nirrespective of the value, $I$, of the generating Hamiltonian which is a\nconstant of motion. For $I \u003c 0$ we find numerous closed orbits with two\nunstable eigenvalues strongly indicating that there cannot exist two additional\nfirst integrals apart from the Hamiltonian and thus that the system, at least\nfor this case, is very likely not integrable. In addition, we present numerical\nevidence that the average Lyapunov exponent nevertheless vanishes. The model is\nthus a very interesting example of a Hamiltonian dynamical system, which is\nlikely non-integrable yet passes the reduced Painlev\\\u0027{e} test.",
"arxiv_id": "solv-int/9406002",
"authors": [
"Freddy Christiansen",
"Hans Henrik Rugh",
"Svend Erik Rugh"
],
"categories": [
"solv-int",
"chao-dyn",
"gr-qc",
"nlin.CD",
"nlin.SI"
],
"doi": "10.1088/0305-4470/28/3/019",
"journal_ref": "J.Phys. A28 (1995) 657-668",
"title": "Non-integrability of the mixmaster universe",
"url": "https://arxiv.org/abs/solv-int/9406002"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "c4568ef7-edcb-4cb9-8c8a-4bc2d949d0a0",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}