{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T05:29:28Z","timestamp":1649136568886},"reference-count":26,"publisher":"World Scientific Pub Co Pte Lt","issue":"09","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2011,9]]},"abstract":"<jats:p> An explicit upper bound Z(2, n) \u2264 n + m - 1 is derived for the number of zeros of Abelian integrals M<jats:sub>1<\/jats:sub>(h) = \u222e<jats:sub>\u03b3(h)<\/jats:sub> P(x, y) dy - Q(x, y) dx on the open interval (0, 1\/6), where \u03b3(h) is an oval lying on the algebraic curve H<jats:sub>\u03bb<\/jats:sub> = (1\/2)x<jats:sup>2<\/jats:sup> + (1\/2)y<jats:sup>2<\/jats:sup> - (1\/3)x<jats:sup>3<\/jats:sup> - \u03bby<jats:sup>3<\/jats:sup> = h, P(x, y), Q(x, y) are polynomials of x and y, and max { deg P(x, y), deg Q(x, y)} = n. The proof exploits the expansion of the first order Melnikov function M<jats:sub>1<\/jats:sub>(h, \u03bb) near \u03bb = 0 and assume (\u2202<jats:sup>m<\/jats:sup>\/\u2202\u03bb<jats:sup>m<\/jats:sup>)M<jats:sub>1<\/jats:sub>(h, \u03bb)|<jats:sub>\u03bb = 0<\/jats:sub> not vanish identically. <\/jats:p>","DOI":"10.1142\/s0218127411030052","type":"journal-article","created":{"date-parts":[[2011,7,12]],"date-time":"2011-07-12T09:19:23Z","timestamp":1310462363000},"page":"2723-2727","source":"Crossref","is-referenced-by-count":2,"title":["ABELIAN INTEGRALS FOR THE ONE-PARAMETER BOGDANOV\u2013TAKENS SYSTEM"],"prefix":"10.1142","volume":"21","author":[{"given":"YONGKANG","family":"ZHANG","sequence":"first","affiliation":[{"name":"LMIB and School of Mathematics and Systems Science, Beihang University, Beijing 100191, P. R. 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