{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T12:38:59Z","timestamp":1773405539656,"version":"3.50.1"},"reference-count":17,"publisher":"Hindawi Limited","license":[{"start":{"date-parts":[[2013,1,1]],"date-time":"2013-01-01T00:00:00Z","timestamp":1356998400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"name":"Initiation Project for PHD of Guangdong Province","award":["S2012040008074"],"award-info":[{"award-number":["S2012040008074"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2013]]},"abstract":"<jats:p>The traditional \u0160i'lnikov theorems provide analytic criteria for proving the existence of chaos in high-dimensional autonomous systems. We have established one extended version of the \u0160i'lnikov homoclinic theorem and have given a set of sufficient conditions under which the system generates chaos in the sense of Smale horseshoes. In this paper, the extension questions of the \u0160i'lnikov homoclinic theorem and its applications are still discussed. We establish another extended version of the \u0160i'lnikov homoclinic theorem. In addition, we construct a new three-dimensional chaotic system which meets all the conditions in this extended \u0160i'lnikov homoclinic theorem. Finally, we list all well-known three-dimensional autonomous quadratic chaotic systems and classify them in the light of the \u0160i'lnikov theorems.<\/jats:p>","DOI":"10.1155\/2013\/287123","type":"journal-article","created":{"date-parts":[[2013,3,18]],"date-time":"2013-03-18T17:03:37Z","timestamp":1363626217000},"page":"1-5","source":"Crossref","is-referenced-by-count":1,"title":["The Related Extension and Application of the \u0160i'lnikov Theorem"],"prefix":"10.1155","volume":"2013","author":[{"given":"Baoying","family":"Chen","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Guangdong University of Technology, Guangzhou 510006, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"98","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1177\/074873099129000948"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1063\/1.1345727"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(78)90115-4"},{"key":"4","first-page":"163","volume":"6","year":"1965","journal-title":"Soviet Mathematics, Doklady"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1070\/SM1970v010n01ABEH001588"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1109\/81.246142"},{"issue":"4","key":"7","first-page":"403","volume":"30","year":"1994","journal-title":"Kybernetika"},{"key":"8","year":"1996"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127402005467"},{"key":"10","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1142\/S0218127403006698","volume":"13","year":"2003","journal-title":"International Journal of Bifurcation and Chaos"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127408021063"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1109\/TCSI.2004.838151"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127409023779"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127406016203"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1142\/S0217979211101120"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.50.R647"},{"key":"17","series-title":"Texts in Applied Mathematics","volume":"2","year":"1991"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2013\/287123.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2013\/287123.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2013\/287123.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,21]],"date-time":"2017-06-21T05:35:36Z","timestamp":1498023336000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/jam\/2013\/287123\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013]]},"references-count":17,"alternative-id":["287123","287123"],"URL":"https:\/\/doi.org\/10.1155\/2013\/287123","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"value":"1110-757X","type":"print"},{"value":"1687-0042","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013]]}}}