Bifurcation of limit cycles in the neighborhood of generic hyperbolic polycycles with three vertices
The author, a brilliant student of R. Roussarie (Dijon) works on the 16th Hilbert problem, with the aim of obtaining explicit bounds on the number of limit cycles bifurcating from some type of an elementary polycycle [the author, Lect. Notes Math. 1455, 272-314 (1990; Zbl 0719.58031) and Ann. Inst. Fourier 41, No. 3, 719-753 (1991; Zbl 0725.58032)]. In the present paper a very important result is obtained. It is shown that generic hyperbolic polycycles with three vertices are of cyclicity at most three in \(C^ \infty\) families of vector fields on the plane. A complete topological classification is given. As a counterexample, a class of degenerate hyperbolic polycycles of cyclicity at least four is presented. The paper is virtually self-contained and well written. The source of inspiration was certainly the work of \textit{N. N. Bautin} [Mat. Sb., n. Ser. 30(72), 181-196 (1952; Zbl 0046.094)] and that of \textit{R. Roussarie} [Bol. Soc. Bras. Mat. 17, No. 1, 67-101 (1986; Zbl 0628.34032)], but the proofs go far beyond the sources, with new ideas and new methods, which are not all easy or obvious. This work is worth reading and learning from.
scientific article; zbMATH DE number 635811