{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T15:30:57Z","timestamp":1787239857623,"version":"build-2736575974"},"reference-count":31,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2013,1]]},"abstract":"<jats:p>We consider a family of 2-dimensional ODEs of the form $\\Delta_\\xi(z)z' = f_\\xi(z)$ depending on a real parameter $\\xi$ which was investigated by Vladimirov [Rep. Math. Phys., 61 (2008), pp. 381--400]. In this system, there exist stationary points $p_\\xi$ which belong to the set of zeros of $\\Delta_\\xi$. We prove, using rigorous numerics, the existence of a homoclinic orbit to $p_\\xi$ for some parameter value $\\xi = \\xi_h$. Due to the singularity of the system it takes a finite time to travel along this orbit, and this property gives rise to a compacton-like traveling wave in some hydrodynamic system describing relaxing media. Our approach could be used to prove similar results in other singular systems as well.<\/jats:p>","DOI":"10.1137\/120901271","type":"journal-article","created":{"date-parts":[[2013,9,11]],"date-time":"2013-09-11T04:30:07Z","timestamp":1378873807000},"page":"1541-1565","source":"Crossref","is-referenced-by-count":11,"title":["A Homoclinic Orbit in a Planar Singular ODE---A Computer Assisted Proof"],"prefix":"10.1137","volume":"12","author":[{"given":"Robert","family":"Szczelina","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Piotr","family":"Zgliczy\u0144ski","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2013,9,10]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/12087654X"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1137\/100812008"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/10.3.379"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2005.07.016"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1512\/iumj.2003.52.2245"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1512\/iumj.2003.52.2407"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2004.12.003"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/120873960"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127494000587"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/26\/23\/047"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(89)90153-2"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1137\/0728042"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1007\/BF01063734"},{"key":"atypb15","unstructured":"J. Guckenheimer and P. Holmes,\n                      Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields\n                      , Springer-Verlag, New York, 2002."},{"key":"atypb16","first-page":"11","author":"Hiraoka Y.","year":"2007","journal-title":"RIMS Kokyuroku Bessatsu B3, Res. Inst. Math. Sci. (RIMS), Kyoto, Japan"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1007\/s12346-011-0040-2"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/20\/9\/008"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2006.09.016"},{"key":"atypb20","first-page":"193","volume":"4","author":"Oishi S.","year":"1998","journal-title":"J. Univers. Comput. 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Szczelina,\n                      Author Home Page\n                      , http:\/\/www.ii.uj.edu.pl\/$\\sim$szczelir\/p\/papers (2012)."},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1007\/s102080010025"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2008.12.019"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2004.03.013"}],"container-title":["SIAM Journal on Applied Dynamical Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/120901271","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T14:55:30Z","timestamp":1787237730000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/120901271"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,1]]},"references-count":31,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2013,1]]}},"alternative-id":["10.1137\/120901271"],"URL":"https:\/\/doi.org\/10.1137\/120901271","relation":{},"ISSN":["1536-0040"],"issn-type":[{"value":"1536-0040","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,1]]}}}