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Please use this identifier to cite or link to this item: http://hdl.handle.net/1942/2453

Title: Perturbation from an elliptic Hamiltonian of degree four - IV figure eight-loop
Authors: DUMORTIER, Freddy
Li, CZ
Issue Date: 2003
Publisher: ACADEMIC PRESS INC ELSEVIER SCIENCE
Citation: JOURNAL OF DIFFERENTIAL EQUATIONS, 188(2). p. 512-554
Abstract: 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.
Notes: Limburgs Univ Ctr, B-3590 Diepenbeek, Belgium. Peking Univ, Dept Math, Beijing 100871, Peoples R China.Dumortier, F, Limburgs Univ Ctr, Univ Campus, B-3590 Diepenbeek, Belgium.
URI: http://hdl.handle.net/1942/2453
DOI: 10.1016/S0022-0396(02)00111-0
ISI #: 000180988300007
ISSN: 0022-0396
Category: A1
Type: Journal Contribution
Validation: ecoom, 2004
Appears in Collections: Research publications

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