Perturbation from an elliptic Hamiltonian of degree four—IV figure eight-loop

• Published in: Journal of Differential Equations Vol. 188; no. 2; pp. 512 - 554
• Main Authors: Dumortier, Freddy, Li, Chengzhi
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.03.2003
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/S0022-0396(02)00111-0

The paper deals with Liénard equations of the form x ̇ =y , y ̇ =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 Liénard 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.