{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T10:17:23Z","timestamp":1773656243043,"version":"3.50.1"},"reference-count":16,"publisher":"World Scientific Pub Co Pte Lt","issue":"11","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2014,11]]},"abstract":"<jats:p> In this paper, we examine the localization problem of compact invariant sets of Raychaudhuri equations with nonzero parameters. The main attention is attracted to the localization of periodic\/homoclinic orbits and homoclinic cycles: we prove that there are neither periodic\/homoclinic orbits nor homoclinic cycles; we find heteroclinic orbits connecting distinct equilibrium points. We describe some unbounded domain such that nonescaping to infinity positive semitrajectories which are contained in this domain have the omega-limit set located in the boundary of this domain. We find a locus of other types of compact invariant sets respecting three-dimensional and two-dimensional invariant planes. Besides, we describe the phase portrait of the system obtained from the Raychaudhuri equations by the restriction on the two-dimensional invariant plane. <\/jats:p>","DOI":"10.1142\/s0218127414501363","type":"journal-article","created":{"date-parts":[[2014,11,28]],"date-time":"2014-11-28T11:50:32Z","timestamp":1417175432000},"page":"1450136","source":"Crossref","is-referenced-by-count":9,"title":["Dynamical Analysis of Raychaudhuri Equations Based on the Localization Method of Compact Invariant Sets"],"prefix":"10.1142","volume":"24","author":[{"given":"Alexander P.","family":"Krishchenko","sequence":"first","affiliation":[{"name":"Bauman Moscow State Technical University, 2-aya Baumanskaya ul., 5, 105005, Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Konstantin E.","family":"Starkov","sequence":"additional","affiliation":[{"name":"Instituto Politecnico Nacional \u2013 CITEDI, Av. IPN 1310, Mesa de Otay, Tijuana 22510, B.C., Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2014,11,28]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1063\/1.3243455"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevD.79.124004"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1142\/S0219887809003746"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1134\/S1064562410020444"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1142\/S021812741102963X"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1007\/s12043-007-0110-9"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1016\/S0898-1221(97)00130-2"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1007\/s10625-006-0003-6"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1016\/j.physleta.2005.12.104"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127409023457"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1016\/j.physleta.2010.02.035"},{"key":"rf13","doi-asserted-by":"publisher","DOI":"10.1016\/j.physleta.2011.06.064"},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.1142\/S021812741350020X"},{"key":"rf15","doi-asserted-by":"publisher","DOI":"10.1063\/1.3559065"},{"key":"rf16","doi-asserted-by":"publisher","DOI":"10.1063\/1.3651477"},{"key":"rf17","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-011-1321-y"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127414501363","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T23:40:15Z","timestamp":1565134815000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127414501363"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,11]]},"references-count":16,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2014,11,28]]},"published-print":{"date-parts":[[2014,11]]}},"alternative-id":["10.1142\/S0218127414501363"],"URL":"https:\/\/doi.org\/10.1142\/s0218127414501363","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,11]]}}}