Abstract
This paper is a part of the proof of the existential part of Hilbert's 16th problem for quadratic vector fields initiated in [2]. Its principal aim is the proof of the finite cyclicity of four elementary graphics among quadratic systems with two semi-hyperbolic points and surrounding a center, namely, the graphics (I 25 , (I 19 ), (H 25 ) and (H 111 ) (using the names introduced in [2]). The technique used is a refinement of the technique of Khovanskii as adapted to the finite cyclicity of graphics by Il'yashenko and Yakovenko, together with an equivalent of the Bautin trick to treat the center case. We show that the cyclicity of each of the first three graphics is equal to 2 and that the cyclicity of the fourth one is equal to three. We improve the known results about finite cyclicity of the graphics (H 18 ), (H 110 ), (I 24 ) by showing that their cyclicities are equal to 2.
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N. N. Bautin, On the number of limit cycles appearing with variations of coeficients from an equilibrium state of the type of a focus or a center. (Russian) Mat. Sb. (N.S.) 30 (1952), 181–196; English translation: Am. Math. Soc. Transl. (1954). Reprinted in: Stability and Dynamical Systems. Am. Math. Soc. Transl. Ser. 1, 5 (1962), 396–413.
F. Dumortier, R. Roussarie, and C. Rousseau, Hilbert's 16th problem for quadratic vector fields. J. Differ. Equ. 110 (1994), 86–133.
_____, Elementary graphics of cyclicity one and two. Nonlinearity 7 (1994), 1001–1043.
F. Dumortier, M. El Morsalani, and C. Rousseau, Hilbert's 16th problem for quadratic systems and cyclicity of elementary graphics. Nonlinearity 9 (1996), 1209–1261.
M. El Morsalani, Bifurcations de polycycles infinis pour les champs de vecteurs polynomiaux. Ann. Faculté Sci. Toulouse 3 (1994), 387–410.
_____, Perturbations of grahics with semi-hyperbolic singularities. Bull. Sci. Math. 120 (1996), 337–366.
A. Guzmán and C. Rousseau, Genericity conditions for finite cyclicity of elementary graphics. Preprint, Univ. Montréal. Sumbitted to J. Differ. Equ. (1997).
D. Hilbert, Mathematische Probleme (lecture). The second Int. Cong. Math., Paris, 1990, Nachr. Ges. Wiss. Gottingen Math.-Phys. Kl. (1990), 253–297; Mathematical developments arising from Hilbert's problems. In: Proc. Symp. Pure Math., F. Brower, Ed., Am. Math. Soc. 28 (1976), 50–51.
Y. Il'yashenko and S. Yakovenko, Finitely smooth normal forms of local families of diffeomorphisms and vector fields. Russ. Math. Surv. 46 (1991), 1–43.
_____, Finite cyclicity of elementary polycycles in generic families. Am. Math. Soc. Transl. 165 (1995), 21–95.
A. G. Khovanskii, Fewnomials. Am. Math. Soc. Transl. Math. Monographs 88 (1991).
A. Mourtada, Cyclicité finie des polycycles hyperboliques des champs de vecteurs du plan: mise sous forme normale. Springer Lect. Notes Math. 1455 (1990), 272–314.
R. Moussu and C. Roche, Théorie de Khovanskii et problème de Dulac. Inv. Math. 105 (1991), 431–441.
M. El Morsalani and A. Mourtada, Degenerate and nontrivial hyperbolic 2-polycycles: appearance of two independant Ecalle—Roussarie compensators and Khovanskii's theory. Nonlinearity 7 (1994), 68–83.
R. Roussarie, A note on finite cyclicity and Hilbert's 16th problem. Springer Lect. Notes Math. 1331 (1988), 161–168.
_____, Cyclicité finie des lacets et des points cuspidaux. Nonlinearity 2 (1989), 73–117.
H. Żoł{ie189-1}dek, Asymptotic properties of abelian integrals arising in quadratic systems. In: Proc. Conf. “Bifurcations in Differentiable Dynamics” Diepenbeek, 1992 (to appear).
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Rousseau, C., Świrszcz, G. & Żoładek, H. Cyclicity of Graphics with Semi-Hyperbolic Points Inside Quadratic Systems. Journal of Dynamical and Control Systems 4, 149–189 (1998). https://doi.org/10.1023/A:1022887001627
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DOI: https://doi.org/10.1023/A:1022887001627