{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,11,16]],"date-time":"2022-11-16T11:26:38Z","timestamp":1668597998597},"reference-count":19,"publisher":"World Scientific Pub Co Pte Lt","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2005,3]]},"abstract":"<jats:p> Given a system of two autonomous ordinary differential equations whose right-hand sides are polynomials, it is very hard to tell if any nonsingular trajectories of the system are contained in algebraic curves. We present an effective method of deciding whether a given system has an invariant algebraic curve of a given degree. The method also allows the construction of examples of polynomial systems with invariant algebraic curves of a given degree. We present the first known example of a degree 6 algebraic saddle-loop for polynomial system of degree 2, which has been found using the described method. We also present some new examples of invariant algebraic curves of degrees 4 and 5 with an interesting geometry. <\/jats:p>","DOI":"10.1142\/s0218127405012442","type":"journal-article","created":{"date-parts":[[2005,5,31]],"date-time":"2005-05-31T12:20:40Z","timestamp":1117542040000},"page":"1033-1044","source":"Crossref","is-referenced-by-count":5,"title":["AN ALGORITHM FOR FINDING INVARIANT ALGEBRAIC CURVES OF A GIVEN DEGREE FOR POLYNOMIAL PLANAR VECTOR FIELDS"],"prefix":"10.1142","volume":"15","author":[{"given":"GRZEGORZ","family":"\u015aWIRSZCZ","sequence":"first","affiliation":[{"name":"IBM Watson Research Center, Yorktown Heights NY 10598, USA"},{"name":"Institute of Mathematics, University of Warsaw, 02\u2013097 Warsaw, Banacha 2, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1006\/jmaa.2001.7476"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1016\/S0893-9659(03)90004-8"},{"key":"rf4","author":"Chavarriga J.","journal-title":"Bull. Sci. Math."},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-01-06253-0"},{"key":"rf6","author":"Christopher C.","journal-title":"J. Math. Anal. Appl."},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-6911-1"},{"key":"rf8","first-page":"60","volume":"2","author":"Darboux G.","journal-title":"Bull. Sci. Math. 2\u00e8me s\u00e9rie"},{"key":"rf9","first-page":"1349","volume":"6","author":"Evdokimenco R. M.","journal-title":"Diff. Eqs."},{"key":"rf10","first-page":"1095","volume":"9","author":"Evdokimenco R. M.","journal-title":"Diff. Eqs."},{"key":"rf11","first-page":"215","volume":"15","author":"Evdokimenco R. M.","journal-title":"Diff. Eqs."},{"key":"rf12","first-page":"983","volume":"9","author":"Filiptsov V. F.","journal-title":"Diff. Eqs."},{"key":"rf13","first-page":"1021","author":"Goldstine H. H.","journal-title":"Bull. AMS"},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-1982-15008-X"},{"key":"rf16","volume-title":"The Fractal Geometry of Nature","author":"Mandelbrot B.","year":"1982"},{"key":"rf18","first-page":"23","volume":"8","author":"Qin Y.-X.","journal-title":"Acta Math. Sin."},{"key":"rf19","first-page":"433","volume":"49","author":"Turing A.","journal-title":"Mind"},{"key":"rf22","volume-title":"The Computer and the Brain","author":"von Neumann J.","year":"1958"},{"key":"rf23","first-page":"335","volume":"2","author":"Yablonskii A. I.","journal-title":"Diff. Eqs."},{"key":"rf24","first-page":"1681","volume":"4","author":"\u017bo\u0142adek H.","journal-title":"Trans. Amer. Math. Soc."}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127405012442","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T00:07:05Z","timestamp":1565136425000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127405012442"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,3]]},"references-count":19,"journal-issue":{"issue":"03","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2005,3]]}},"alternative-id":["10.1142\/S0218127405012442"],"URL":"https:\/\/doi.org\/10.1142\/s0218127405012442","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,3]]}}}