{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:31:32Z","timestamp":1787232692228,"version":"build-2736575974"},"reference-count":21,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>We analyze the dynamics of a class of cosmological solutions of the Einstein\u2013Vlasov equations. These equations describe an ensemble of collisionless particles (which represent galaxies or clusters of galaxies) that interact gravitatively through Einstein's equations of general relativity. The cosmological models we consider are spatially homogeneous, of Bianchi type IX, and locally rotationally symmetric (LRS). We prove that generic solutions exhibit an oscillatory behavior close to the singularities (the \u201cbig bang\u201d in the past and the \u201cbig crunch\u201d in the future); this is in contrast to the behavior of Einstein-vacuum or Einstein\u2013Euler solutions. To establish this result we make use of dynamical systems theory; we introduce dimensionless dynamical variables that are defined on a compact state space. In this formulation the oscillatory behavior of generic solutions is represented by the existence of heteroclinic cycles on the boundary of the state space.<\/jats:p>","DOI":"10.1137\/100782590","type":"journal-article","created":{"date-parts":[[2010,11,5]],"date-time":"2010-11-05T20:13:40Z","timestamp":1288988020000},"page":"1244-1262","source":"Crossref","is-referenced-by-count":13,"title":["Oscillations toward the Singularity of Locally Rotationally Symmetric Bianchi Type IX Cosmological Models with Vlasov Matter"],"prefix":"10.1137","volume":"9","author":[{"given":"Simone","family":"Calogero","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J. Mark","family":"Heinzle","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,11,4]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"H. Andr\u00e9asson,\n                      The Einstein-Vlasov system\/Kinetic theory\n                      , Living Rev. Relativity, 8 (2005), article 2; available online at http:\/\/www.livingreviews.org\/lrr-2005-2.","DOI":"10.12942\/lrr-2005-2"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2008.05.010"},{"key":"R3","unstructured":"J. Binney and S. Tremaine,\n                      Galactic Dynamics\n                      , Princeton University Press, Princeton, NJ, 1987."},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1080\/00018737000101171"},{"key":"R5","unstructured":"S. Calogero and J. M. 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Ellis,\n                      The Large Scale Structure of Space-Time\n                      , Cambridge University Press, Cambridge, UK, 1973.","DOI":"10.1017\/CBO9780511524646"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1088\/0264-9381\/23\/10\/016"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevD.41.2444"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1063\/1.526713"},{"key":"R14","doi-asserted-by":"crossref","unstructured":"L. Perko,\n                      Differential Equations and Dynamical Systems\n                      , Springer, New York, 2001.","DOI":"10.1007\/978-1-4613-0003-8"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/BF02096962"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1063\/1.531400"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1006\/aphy.1994.1061"},{"key":"R18","doi-asserted-by":"crossref","unstructured":"A. D. Rendall,\n                      Partial Differential Equations in General Relativity\n                      , Oxford University Press, Oxford, UK, 2008.","DOI":"10.1093\/oso\/9780199215409.001.0001"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1088\/0264-9381\/16\/6\/305"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1088\/0264-9381\/17\/22\/310"},{"key":"R21","doi-asserted-by":"crossref","unstructured":"J. Wainwright and G. F. R. Ellis,\n                      Dynamical Systems in Cosmology\n                      , Cambridge University Press, Cambridge, UK, 1997.","DOI":"10.1017\/CBO9780511524660"}],"container-title":["SIAM Journal on Applied Dynamical Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/100782590","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:16:18Z","timestamp":1787231778000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/100782590"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":21,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/100782590"],"URL":"https:\/\/doi.org\/10.1137\/100782590","relation":{},"ISSN":["1536-0040"],"issn-type":[{"value":"1536-0040","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}