Cyclicity of graphics with semi-hyperbolic points inside quadratic systems

Journal of Dynamical and Control Systems

Volumn 4, Issue 2, 1998, Pages 149-189

  Rousseau, C ^a   Swirszcz G ^b   Zoladek H ^b  

^a UNIVERSITÉ DE MONTRÉAL   (Canada)

^b UNIVERSITY OF WARSAW   (Poland)

Author keywords

Finite cyclicity of graphics; Finite number of limit cycles; Khovanskii procedure; Polynomial vector fields; Semi hyperbolic points

Indexed keywords

GRAPH THEORY; POLYNOMIALS; VECTORS; CONFORMAL MAPPING; FUNCTIONS; INTEGRAL EQUATIONS; MATHEMATICAL TRANSFORMATIONS; ORDINARY DIFFERENTIAL EQUATIONS; PERTURBATION TECHNIQUES; SET THEORY; THEOREM PROVING;

CYCLICITY OF GRAPHICS; KHOVANSKII TECHNIQUES; POLYNOMIAL VECTOR FIELDS; QUADRATIC SYSTEMS; FINITE CYCLICITY OF GRAPHICS; FINITE NUMBER; KHOVANSKII PROCEDURE; LIMIT CYCLES; QUADRATIC VECTOR FIELDS; SEMIHYPERBOLIC POINTS;

SYSTEM THEORY; GRAPH THEORY;

EID: 0032048419     PISSN: 10792724     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1022887001627     Document Type: Article

References (20)

1. 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.
2. 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.
3. 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.
4. F. Dumortier, R. Roussarie, and C. Rousseau, Hilbert's 16th problem for quadratic vector fields. J. Differ. Equ. 110 (1994), 86-133.
5. _, Elementary graphics of cyclicity one and two. Nonlinearity 7 (1994), 1001-1043.
6. 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.
7. M. El Morsalani, Bifurcations de polycycles infinis pour les champs de vecteurs polynomiaux. Ann. Faculté Sci. Toulouse 3 (1994), 387-410.
8. _, Perturbations of grahics with semi-hyperbolic singularities. Bull. Sci. Math. 120 (1996), 337-366.
9. A. Guzmán and C. Rousseau, Genericity conditions for finite cyclicity of elementary graphics. Preprint, Univ. Montréal. Sumbitted to J. Differ. Equ. (1997).
10. D. Hilbert, Mathematische Probleme (lecture). The second Int. Cong. Math., Paris, 1900, Nachr. Ges. Wiss. Gottingen Math.-Phys. Kl. (1900), 253-297; Mathematical developments arising from Hilbert's problems. In: Proc. Symp. Pure Math., F. Brower, Ed., Am. Math. Soc. 28 (1976), 50-51.
11. D. Hilbert, Mathematische Probleme (lecture). The second Int. Cong. Math., Paris, 1900, Nachr. Ges. Wiss. Gottingen Math.-Phys. Kl. (1900), 253-297; Mathematical developments arising from Hilbert's problems. In: Proc. Symp. Pure Math., F. Brower, Ed., Am. Math. Soc. 28 (1976), 50-51.
12. 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.
13. _, Finite cyclicity of elementary polycycles in generic families. Am. Math. Soc. Transl. 165 (1995), 21-95.
14. A. G. Khovanskii, Fewnomials. Am. Math. Soc. Transl. Math. Monographs 88 (1991).
15. 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.
16. R. Moussu and C. Roche, Théorie de Khovanskii et problème de Dulac. Inv. Math. 105 (1991), 431-441.
17. 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.
18. R. Roussarie, A note on finite cyclicity and Hubert's 16th problem. Springer Lect. Notes Math. 1331 (1988), 161-168.
19. _, Cyclicité finie des lacets et des points cuspidaux. Nonlinearity 2 (1989), 73-117.
20. H. Żoła̧dek, Asymptotic properties of abelian integrals arising in quadratic systems. In: Proc. Conf. "Bifurcations in Differentiable Dynamics" Diepenbeek, 1992 (to appear).