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	Perturbation from an elliptic Hamiltonian of degree four - IV figure eight-loop
	Dumortier, F ; Li, CZ
	2003
关键词	NUMBER
英文摘要	The paper deals with Lienard equations of the form <(x)over dot>= y, <(y) over dot>= P(x) + yQ(x) with P and Q polynomials of degree, respectively, 3 and 2. Attention goes to perturbations of the Hamiltonian vector fields with an elliptic Hamiltonian of degree four, exhibiting a figure eight-loop. It is proved that the least upper bound of the number of zeros of the related elliptic integral is five, and this is a sharp bound, multiplicity taken into account. Moreover, if restricting to the level curves "inside" a saddle loop or "outside" the figure eight-loop the sharp upper bound is respectively two or four; also the multiplicity of the zeros is at most four. This is the last one in a series of papers on this subject. The results of this paper, together with (J. Differential Equations 176 (2001) 114; J. Differential Equations 175 (2001) 209; J. Differential Equations, to be published), largely finish the study of the cubic perturbations of the elliptic Hamiltonians of degree four and presumably provide a complete description of the number and the possible configurations of limit cycles for cubic Lienard equations with small quadratic damping. As a special case, we obtain a configuration of four limit cycles surrounding three singularities together with a "small" limit cycle which surrounds one of the singularities. (C) 2002 Elsevier Science (USA). All rights reserved.; http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000180988300007&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=8e1609b174ce4e31116a60747a720701 ; Mathematics; SCI(E); 54; ARTICLE; 2; 512-554; 188
语种	英语
出处	SCI
出版者	微分方程杂志
内容类型	其他
源URL	[http://hdl.handle.net/20.500.11897/256062]
专题	数学科学学院
推荐引用方式 GB/T 7714	Dumortier, F,Li, CZ. Perturbation from an elliptic Hamiltonian of degree four - IV figure eight-loop. 2003-01-01.

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