{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,4]],"date-time":"2026-05-04T16:48:43Z","timestamp":1777913323880,"version":"3.51.4"},"reference-count":15,"publisher":"World Scientific Pub Co Pte Lt","issue":"08","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2012,8]]},"abstract":"<jats:p>Homoclinic bifurcation is a difficult and important topic of bifurcation theory. As we know, a general theory for a homoclinic loop passing through a hyperbolic saddle was established by [Roussarie, 1986]. Then the method of stability-changing to find limit cycles near a double homoclinic loop passing through a hyperbolic saddle was given in [Han &amp; Chen, 2000], and further developed by [Han et al., 2003; Han &amp; Zhu, 2007]. For a homoclinic loop passing through a nilpotent saddle there are essentially two different cases, which we distinguish by cuspidal type and smooth type, respectively. For the cuspidal type a general theory was recently established in [Zang et al., 2008]. In this paper, we consider limit cycle bifurcation near a double homoclinic loop passing through a nilpotent saddle by studying the analytical property of the first order Melnikov functions for general near-Hamiltonian systems and obtain the conditions for the perturbed system to have 8, 10 or 12 limit cycles in a neighborhood of the loop with seven different distributions. In particular, for the homoclinic loop of smooth type, a general theory is obtained as a consequence. We finally consider some polynomial systems and find a lower bound of the maximal number of limit cycles as an application of our main results.<\/jats:p>","DOI":"10.1142\/s0218127412501891","type":"journal-article","created":{"date-parts":[[2012,9,17]],"date-time":"2012-09-17T08:52:34Z","timestamp":1347871954000},"page":"1250189","source":"Crossref","is-referenced-by-count":34,"title":["LIMIT CYCLE BIFURCATIONS NEAR A DOUBLE HOMOCLINIC LOOP WITH A NILPOTENT SADDLE"],"prefix":"10.1142","volume":"22","author":[{"given":"MAOAN","family":"HAN","sequence":"first","affiliation":[{"name":"Department of Mathematics, Shanghai Normal University, Shanghai 200234, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"JUNMIN","family":"YANG","sequence":"additional","affiliation":[{"name":"College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050016, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"DONGMEI","family":"XIAO","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Shanghai Jiaotong University, Shanghai 200030, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2012,9,17]]},"reference":[{"key":"rf1","first-page":"401","volume":"30","author":"Han M.","journal-title":"Sci. China Ser. A"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1016\/S0362-546X(02)00301-2"},{"key":"rf3","series-title":"Ordinary Differential Equations","volume-title":"Handbook of Differential Equations","volume":"3","author":"Han M.","year":"2006"},{"key":"rf4","doi-asserted-by":"crossref","first-page":"67","DOI":"10.3934\/mbe.2006.3.67","volume":"3","author":"Han M.","journal-title":"Math. Biosci. 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