The existential Hilbert 16-th problem and an estimate for cyclicity of elementary polycycles

Inventiones Mathematicae

Volumn 151, Issue 3, 2003, Pages 451-512

  Kaloshin, V ^a  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0037934272     PISSN: 00209910     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00222-002-0244-9     Document Type: Article

References (52)

1. [AAIS] Aframovich, V., Arnold, V., Ilyashenko, Yu., Shilnikov, L.: Encyclopedia Math. Sci., Vol. 5, Bifurcation Theory. Dynamical Systems VINITI (Russian) & Springer-Verlag (English) 1986 & 1994
2. [A] Arnold, V.: Problems for Arnold's seminar, 1989
3. [AGV] Arnold, V., Gusein-Zade, S., Varchenko, A.: Singularities of differentiable maps, Vol. I. Monographs in Mathematics 82. Birkhäuser, Boston 1985
4. [BCR] Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Springer-Verlag, Berlin 1998
5. [DeR] Denkowska, Z., Roussarie, R.: A Method of Desingularization for Analytic twodimensional vector field families. Bol. Soc. Bras. Mat. 22,93-126 (1991)
6. [Du] Dulac, H.: Sur les cycles limite. Bull. Soc. Math. Fr. 51,45-188 (1923)
7. [D] Dumortier, F.: Singularities of vector fields on the plane. J. Differ. Equations 23, 53-106(1977)
8. [DIR] Dumortier, F., Ilyashenko, Yu., Rousseau, C.: Normal forms near a saddle-node and applications to finite cyclicity of graphics. Preprint 2000
9. [DMR] Dumortier, F., El Morsalani, M., Rousseau, C.: Hilbert's 16th problem for quadratic systems and cyclicity of elementary graphics. Nonlinearity 9, 1209- 1251 (1996)
10. [DRR] Dumortier, F., Roussarie, R., Rousseau, C.: Hilbert's 16th problem for quadratic vector fields. Differ. Equations 110, 86-133 (1994)
11. [E] Écalle, J.: Introduction aux fonctions analysables et preuve consrustive de la conjecture de Dulac. Herman, Paris 1992
12. [FP] Francoise, J.-P., Pugh, C.: Keeping track of limit cycles. J. Differ. Equations 65, 139-157(1986)
13. [GK] Gabrielov, A., Khovanski, A.: Multiplicity of a Noetherian intersection. Geometry of differential equations, 119-130. AMS Transl. Ser. 2, 186. AMS, Providence, RI 1998
14. [GWPL] Gibson, C., Wirthmuller, K., du Plessis, A., Loojenga, E.: Topological Stability of Smooth Mappings. Lectures Notes in Mathematics 552. Springer 1976
15. [GG] Golubitsky, M., Guillemin, V.: Stable Mappings and Their Singularities. Graduate Texts in Mathematics 14. Springer-Verlag 1973
16. [GM] Goresky, M., MacPherson, R.: Stratified Morse Theory. Springer-Verlag 1987
17. [G] Grozovskii, T.: Bifurcations of polycycles an "apple" and a "half-apple" in generic two-parameter families (in Russian). Differ. Equations 32, 458-469 (1996)
18. [GR] Guzman, A., Rousseau, C.: Genericity condition for cyclicity of elementary graphics. Differ. Equations 155, 44-72 (1999)
19. [Hir1] Hironaka, H.: Stratification and flatness Real and Complex singularities. Nordic Summer School (Oslo, 1976). Sijthoff-Noordhoff, Growhgen 1977
20. [Hir2] Hironaka, H.: Introduction to real-algebraic sets and real-analytic maps. Instituto Mathematico L. Tonelli, Dell' Universita'di Piza 1973
21. [I1] Ilyashenko, Yu.: Finiteness Theorem for limit cycles. Am. Math. Soc., Providence, RI 1991
22. [I2] Ilyashenko, Yu.: Normal forms for local families and nonlocal bifurcations. Astérisque 222, 233-258 (1994)
23. [I3] Ilyashenko, Yu.: Centennial history of the Hilbert 16th problem. Bull. Am. Math. Soc. 39, 301-354(2002)
24. [IK] Ilyashenko, Yu., Kaloshin, V.: Bifurcations of planar and spatial polycycles: Arnold's program and its development. Fields Inst. Commun. 24,241-271 (1999)
25. [IY1] Ilyashenko, Yu., Yakovenko, S.: Finitely smooth normal forms of local families of diffeomorphisms and vector fields. Russ. Math. Surv. 46, 1-43 (1991)
26. [IY2] Ilyashenko, Yu., Yakovenko, S.: Concerning Hilbert sixteenth problem. Am. Math. Soc. Transl., Ser. 2 165, 1-20 (1995)
27. [IY3] Ilyashenko, Yu., Yakovenko, S.: Finite Cyclicity of Elementary Polycycles in Generic Families. Am. Math. Soc. Transl. 165, 21-95 (1995)
28. [J] Jacobson, N.: Basic Algebra, Vol. 1, 1974
29. [Ka1] Kaloshin, V.: A Geometric Proof of Existence of Whitney's stratifications. Preprint, http://www.aimath.org/preprints.html, 2000-11
30. [Ka2] Kaloshin, V.: Around Hilbert-Arnold Problem. CRM, Montreal, to appear
31. [Ka3] Kaloshin, V. : Hilbert-Arnold Problem and an estimate for cyclicity of polycycles on the plane and in the space. Funct. Anal. Appl. 35, 78-81 (2001)
32. [Ka4| Kaloshin, V.: Some prevalent properties of smooth dynamical systems. Proc. Steklov Inst. Math. 213, 115-140 (1996)
33. [Kh] Khovanski, A.: Fewnomials. Am. Math. Soc., Providence, RI 1991
34. [Kl] Kleban, O.: Order of the topologically sufficient jet of a smooth vector field on the real plane at a singular point of finite multiplicity. Am. Math. Soc. Transl., Ser. 2 165, 131-153(1995)
35. [KS] Kotova, A., Stanzo, V.: Few-Parameter Generic Families on the Sphere. Am. Math. Soc. Transl., Ser. 2, Providence, RI 213, 155-202 (1996)
36. [Ma] Mather, J.: Notes on topological stability. Harvard Uninersity 1970
37. [Mi] Milnor, J.: Singular points of complex hypersurfaces. Princeton University Press, Princeton, NJ 61, 1968
38. [MR] Moussu, R., Roche, C.: Khovanski's theory and the Dulac Problem. Invent. Math. 105,431-441 (1991)
39. [Mu] Mumford, D.: Algebraic Geometry I, Complex Projective Algebraic Varieties. Springer-Verlag 1976
40. [P] Perko, L.: Differential Equations and Dynamical Systems. Appl. Math. 7. Springer-Verlag 1993
41. [PW] du Plessis, A., Wall, T.: The Geometry of Topological Stability. Oxford 1995
42. [R1] Roussarie, R.: Cyclicite finie et le 16 problem d'Hilbert, Dynamical systems (Volparaiso 1986), R. Bauon, R. Lavarca, J. Palis (eds.). Lecture Notes in Mathematics 1331. Springer-Verlag, Berlin, New York 1988, pp. 161-188
43. [R2] Roussarie, R.: Bifurcation of planar vector fields and Hilbert's sixteenth problem. Progr. Math. 164. Birkhäuser Verlag, Basel 1998
44. [RSZ] Rousseau. C., Świrszcz, G., Zoladek. H.: Cyclicity of graphics with semihyperbolic points inside quadratic systems. Dyn. Control Syst. 4,149-189 (1988)
45. [S] Schlomiuk, D. (ed.): Bifurcations and periodic orbits of vector fields. NATO AS 1, Series C (Math, and Phys. Sciences), Vol. 408. Kluwer, Dordrecht, Boston, London 1993
46. [Ta] Takens, F.: Unfoldings of certain singularities of vector fields: generalized Hopf bifurcations. J. Differ. Equations 14, 476-493 (1973)
47. [Th] Thorn, R.: Ensembles etmorphismes stratifiés. Bull. Am. Math. Soc. 75, 240-284 (1969)
48. [Ti] Tikhomirov, V.: Fundamental principles of the theory of extremal problems. Transl. by Bernd Luderer. John Wiley & Sons 1986
49. [Tr1] Trifonov, S.: Desingularization in Families of Analytic Differential Equations. Am. Math. Soc. Transl., Ser. 2 165, 97-129 (1995)
50. [Tr2] Trifonov, S.: Cyclicity of elementary polycycles in generic smooth vector fields, Differential equations with real and complex time, Yu. Ilyashenko (ed.). Proc. Steklov Inst. Math. 213, 141-199 (1996)
51. [Wa] Wall, T.: Regular Stratifications. Lecture Notes in Mathematics 468, 332-344 (1975)
52. [W] Whitney, H.: Elementary structure of real algebraic varieties. Ann. Math. 66, 545-556 (1957)