{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,16]],"date-time":"2026-05-16T03:50:57Z","timestamp":1778903457699,"version":"3.51.4"},"reference-count":23,"publisher":"World Scientific Pub Co Pte Ltd","issue":"06","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2007,6]]},"abstract":"<jats:p>In this paper we study the nonchaotic and chaotic behavior of all 3D conservative quadratic ODE systems with five terms on the right-hand side and one nonlinear term (5-1 systems). We prove a theorem which provides sufficient conditions for solutions in 3D autonomous systems being nonchaotic. We show that all but five of these systems: (15a, 15b), (18b), (41)(A = \u22131), (43b), and (49a, 49b) are nonchaotic. Numerical simulations show that only one of the five systems, (43b), really appears to be chaotic. If proved to be true, it will be the simplest ODE system having chaos.<\/jats:p>","DOI":"10.1142\/s021812740701821x","type":"journal-article","created":{"date-parts":[[2007,9,4]],"date-time":"2007-09-04T11:33:08Z","timestamp":1188905588000},"page":"2049-2072","source":"Crossref","is-referenced-by-count":19,"title":["NONCHAOTIC AND CHAOTIC BEHAVIOR IN THREE-DIMENSIONAL QUADRATIC SYSTEMS: FIVE-ONE CONSERVATIVE CASES"],"prefix":"10.1142","volume":"17","author":[{"given":"JACK","family":"HEIDEL","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Nebraska at Omaha, Omaha, NE 68182, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"FU","family":"ZHANG","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Tennessee Tech University, Cookeville, TN 38505, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1006\/jdeq.2002.4166"},{"key":"rf2","doi-asserted-by":"crossref","DOI":"10.1007\/b97589","volume-title":"CHAOS: An Introduction to Dynamical Systems","author":"Alligood K. T.","year":"1996"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127496000023"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1119\/1.18980"},{"key":"rf6","volume-title":"Ordinary Differential Equations","author":"Hale J. K.","year":"1980"},{"key":"rf7","volume-title":"Ordinary Differential Equations","author":"Hartman P.","year":"1964"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1006\/jdeq.1996.0060"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/12\/3\/012"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(92)90143-B"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1119\/1.18594"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1175\/1520-0469(1963)020<0130:DNF>2.0.CO;2"},{"key":"rf13","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/27\/13\/025"},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.1016\/0375-9601(76)90101-8"},{"key":"rf15","volume-title":"Real and Complex Analysis","author":"Rudin W.","year":"1987"},{"key":"rf16","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.50.R647"},{"key":"rf17","doi-asserted-by":"publisher","DOI":"10.1119\/1.18585"},{"key":"rf18","doi-asserted-by":"publisher","DOI":"10.1016\/S0375-9601(97)00088-1"},{"key":"rf19","doi-asserted-by":"publisher","DOI":"10.1119\/1.19538"},{"key":"rf20","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198508397.001.0001","volume-title":"Chaos and Time Series Analysis","author":"Sprott J. C.","year":"2003"},{"key":"rf21","doi-asserted-by":"publisher","DOI":"10.1017\/S0308210500026664"},{"key":"rf22","doi-asserted-by":"publisher","DOI":"10.1016\/S0960-0779(99)00095-8"},{"key":"rf23","first-page":"27","volume":"4","author":"Yang X. S.","journal-title":"Far East J. Dyn. Syst."},{"key":"rf24","first-page":"1289","volume":"10","author":"Zhang F.","journal-title":"Nonlinearity"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S021812740701821X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,17]],"date-time":"2024-02-17T18:51:42Z","timestamp":1708195902000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S021812740701821X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,6]]},"references-count":23,"journal-issue":{"issue":"06","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2007,6]]}},"alternative-id":["10.1142\/S021812740701821X"],"URL":"https:\/\/doi.org\/10.1142\/s021812740701821x","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,6]]}}}