{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,2]],"date-time":"2026-04-02T04:07:54Z","timestamp":1775102874744,"version":"3.50.1"},"reference-count":17,"publisher":"World Scientific Pub Co Pte Lt","issue":"05","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2009,5]]},"abstract":"<jats:p> The classical \u0160il'nikov homoclinic theorem provides an analytic criterion for proving the existence of chaos in three-dimensional autonomous systems, but it can only be applied to systems with fixed points of the saddle-focus type. This paper extends this powerful theorem to a degenerate case where one of the eigenvalues of the Jacobian evaluated at an equilibrium point is zero and the other two are a pair of conjugate complex numbers, and consequently establishes a set of criteria for proving the existence of chaos in the sense of having Smale horseshoes. Based on this new extended \u0160il'nikov homoclinic theorem, a new chaotic system is constructed, whose corresponding bounded chaotic attractor is first verified numerically through phase trajectories, Lyapunov exponents, bifurcation routes and Poincar\u00e9 mappings, followed by theoretical analysis on the existence of one homoclinic orbit, the key component of the extended \u0160il'nikov homoclinic theorem. <\/jats:p>","DOI":"10.1142\/s0218127409023779","type":"journal-article","created":{"date-parts":[[2009,8,17]],"date-time":"2009-08-17T09:02:26Z","timestamp":1250499746000},"page":"1679-1693","source":"Crossref","is-referenced-by-count":25,"title":["AN EXTENDED \u0160IL'NIKOV HOMOCLINIC THEOREM AND ITS APPLICATIONS"],"prefix":"10.1142","volume":"19","author":[{"given":"BAOYING","family":"CHEN","sequence":"first","affiliation":[{"name":"Department of Mathematics, Zhongshan University, Guangzhou 510275, P. R. China"},{"name":"Department of Applied Mathematics, Guangdong University of Technology, Guangzhou 510006, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"TIANSHOU","family":"ZHOU","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Zhongshan University, Guangzhou 510275, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"GUANRONG","family":"CHEN","sequence":"additional","affiliation":[{"name":"Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1142\/3033"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01221359"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(78)90115-4"},{"key":"rf4","first-page":"163","volume":"6","author":"\u0160il'nikov L. 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