Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 

 

Centennial History of Hilbert's 16th Problem


Author: Yu. Ilyashenko
Journal: Bull. Amer. Math. Soc. 39 (2002), 301-354
MSC (2000): Primary 34Cxx, 34Mxx, 37F75
Published electronically: April 9, 2002
MathSciNet review: 1898209
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The second part of Hilbert's 16th problem deals with polynomial differential equations in the plane. It remains unsolved even for quadratic polynomials. There were several attempts to solve it that failed. Yet the problem inspired significant progress in the geometric theory of planar differential equations, as well as bifurcation theory, normal forms, foliations and some topics in algebraic geometry. The dramatic history of the problem, as well as related developments, are presented below.


References [Enhancements On Off] (What's this?)

  • [ALGM] A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maĭer, Theory of bifurcations of dynamic systems on a plane, Halsted Press [A division of John Wiley & Sons], New York-Toronto, Ont.; Israel Program for Scientific Translations, Jerusalem-London, 1973. Translated from the Russian. MR 0344606
  • [A] V. I. Arnol′d, Geometrical methods in the theory of ordinary differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 250, Springer-Verlag, New York-Berlin, 1983. Translated from the Russian by Joseph Szücs; Translation edited by Mark Levi. MR 695786
  • [AAIS] V. Arnold, V. Afrajmovich, Yu. Ilyashenko, L. Shil'nikov, Bifurcation theory and catastrophe theory, Translated from the 1986 Russian original Enciclopaedia Math. Sci., Dynamical systems. V, Springer-Verlag, Berlin, 1999. CMP 2000:07
  • [AGV] V. I. Arnol′d, A. N. Varchenko, and S. M. Guseĭn-Zade, Osobennosti differentsiruemykh otobrazhenii. II, “Nauka”, Moscow, 1984 (Russian). Monodromiya i asimptotiki integralov [Monodromy and the asymptotic behavior of integrals]. MR 755329
  • [AI] V. I. Arnol′d and Yu. S. Il′yashenko, Ordinary differential equations, Current problems in mathematics. Fundamental directions, Vol. 1, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985, pp. 7–149, 244 (Russian). MR 823489
  • [AKL] Joan C. Artés, Robert E. Kooij, and Jaume Llibre, Structurally stable quadratic vector fields, Mem. Amer. Math. Soc. 134 (1998), no. 639, viii+108. MR 1432139, 10.1090/memo/0639
  • [B] I. Bendixson, Sur les courbes définies par des équations différentielles, Acta Math. 24 (1901), 1-88.
  • [Ba] N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, American Math. Soc. Translation 1954 (1954), no. 100, 19. MR 0059426
  • [Ber] Lipman Bers, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960), 94–97. MR 0111834, 10.1090/S0002-9904-1960-10413-2
  • [BFY] M. Briskin, J.-P. Francoise, and Y. Yomdin, Center conditions, compositions of polynomials and moments on algebraic curves, Ergodic Theory Dynam. Systems 19 (1999), no. 5, 1201–1220. MR 1721616, 10.1017/S0143385799141737
  • [BM] F. S. Berezovskaya and N. B. Medvedeva, The asymptotics of the return map of a singular point with fixed Newton diagram, J. Soviet Math. 60 (1992), no. 6, 1765–1781. MR 1181103, 10.1007/BF01102588
  • [Bo76] R. I. Bogdanov, The versal deformation of a singular point of a vector field on the plane in the case of zero eigenvalues, Trudy Sem. Petrovsk. Vyp. 2 (1976), 37–65 (Russian). MR 0442996
  • [Bo77] R. I. Bogdanov, Bifurcations of a limit cycle of a certain family of vector fields on the plane, Trudy Sem. Petrovsk. Vyp. 2 (1976), 23–35 (Russian). MR 0442988
  • [Bo79] R. I. Bogdanov, Local orbital normal forms of vector fields on the plane, Trudy Sem. Petrovsk. 5 (1979), 51–84 (Russian). MR 549622
  • [Bo85] R. I. Bogdanov, Invariants of elementary singular points on the plane, Uspekhi Mat. Nauk 40 (1985), no. 3(243), 199–200 (Russian). MR 795192
  • [B64] A. D. Brjuno, The normal form of differential equations, Dokl. Akad. Nauk SSSR 157 (1964), 1276–1279 (Russian). MR 0166457
  • [B71] A. D. Brjuno, Analytic form of differential equations. I, II, Trudy Moskov. Mat. Obšč. 25 (1971), 119–262; ibid. 26 (1972), 199–239 (Russian). MR 0377192
  • [B72] A. D. Brjuno, Analytic form of differential equations. I, II, Trudy Moskov. Mat. Obšč. 25 (1971), 119–262; ibid. 26 (1972), 199–239 (Russian). MR 0377192
  • [C] Kuo-Tsai Chen, Equivalence and decomposition of vector fields about an elementary critical point, Amer. J. Math. 85 (1963), 693–722. MR 0160010
  • [Ca] César Camacho, Problems on limit sets of foliations on complex projective spaces, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 1235–1239. MR 1159308
  • [CGM] A. Candel and X. Gómez-Mont, Uniformization of the leaves of a rational vector field, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 4, 1123–1133 (English, with English and French summaries). MR 1359843
  • [Ch] Marc Chaperon, On the local classification of holomorphic vector fields, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 96–103. MR 730265, 10.1007/BFb0061412
  • [ChL] C. J. Christopher and N. G. Lloyd, Polynomial systems: a lower bound for the Hilbert numbers, Proc. Roy. Soc. London Ser. A 450 (1995), no. 1938, 219–224. MR 1349062, 10.1098/rspa.1995.0081
  • [CKP] César Camacho, Nicolaas H. Kuiper, and Jacob Palis, La topologie du feuilletage d’un champ de vecteurs holomorphes près d’une singularité, C. R. Acad. Sci. Paris Sér. A-B 282 (1976), no. 17, Ai, A959–A961 (French, with English summary). MR 0413182
  • [CLW] Shui-Nee Chow, Cheng Zhi Li, and Duo Wang, Normal forms and bifurcation of planar vector fields, Cambridge University Press, Cambridge, 1994. MR 1290117
  • [CM] D. Cerveau and R. Moussu, Groupes d’automorphismes de (𝐶,0) et équations différentielles 𝑦𝑑𝑦+\cdots=0, Bull. Soc. Math. France 116 (1988), no. 4, 459–488 (1989) (French, with English summary). MR 1005391
  • [Co] W. A. Coppel, A survey of quadratic systems, J. Differential Equations 2 (1966), 293–304. MR 0196182
  • [CS] César Camacho and Paulo Sad, Invariant varieties through singularities of holomorphic vector fields, Ann. of Math. (2) 115 (1982), no. 3, 579–595. MR 657239, 10.2307/2007013
  • [CSc] C. Camacho, A. Scardua, Holomorphic foliations with Liouvillian first integrals, Ergodic Theory Dynam. Systems 21 (2001), no. 3, 717-756.
  • [CW] Lan Sun Chen and Ming Shu Wang, The relative position, and the number, of limit cycles of a quadratic differential system, Acta Math. Sinica 22 (1979), no. 6, 751–758 (Chinese, with English summary). MR 559742
  • [D] H. Dulac, Sur les cycles limite, Bull. Soc. Math France 51 (1923), 45-188.
  • [De] Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970 (French). MR 0417174
  • [Du] Freddy Dumortier, Singularities of vector fields on the plane, J. Differential Equations 23 (1977), no. 1, 53–106. MR 0650816
  • [DHP] Freddy Dumortier, Chris Herssens, and Lawrence Perko, Local bifurcations and a survey of bounded quadratic systems, J. Differential Equations 165 (2000), no. 2, 430–467. MR 1772568, 10.1006/jdeq.2000.3777
  • [DIR] F. Dumortier, Yu. Ilyashenko, C. Rousseau, Normal forms near a saddle-node and applications to finite cyclicity of graphics, Preprint (2000), 29 pp. CRM-2697.
  • [DMR] F. Dumortier, M. El Morsalani, and C. Rousseau, Hilbert’s 16th problem for quadratic systems and cyclicity of elementary graphics, Nonlinearity 9 (1996), no. 5, 1209–1261. MR 1416474, 10.1088/0951-7715/9/5/008
  • [DRR] F. Dumortier, R. Roussarie, and C. Rousseau, Hilbert’s 16th problem for quadratic vector fields, J. Differential Equations 110 (1994), no. 1, 86–133. MR 1275749, 10.1006/jdeq.1994.1061
  • [DRRa] F. Dumortier, R. Roussarie, and C. Rousseau, Elementary graphics of cyclicity 1 and 2, Nonlinearity 7 (1994), no. 3, 1001–1043. MR 1275538
  • [DRS] Freddy Dumortier, Robert Roussarie, and Jorge Sotomayor, Bifurcations of cuspidal loops, Nonlinearity 10 (1997), no. 6, 1369–1408. MR 1483548, 10.1088/0951-7715/10/6/001
  • [DRSZ] F. Dumortier, R. Roussarie, J. Sotomayor, and H. Żołpolhk adek, Bifurcations of planar vector fields, Lecture Notes in Mathematics, vol. 1480, Springer-Verlag, Berlin, 1991. Nilpotent singularities and Abelian integrals. MR 1166189
  • [E81a] Jean Écalle, Les fonctions résurgentes. Tome I, Publications Mathématiques d’Orsay 81 [Mathematical Publications of Orsay 81], vol. 5, Université de Paris-Sud, Département de Mathématique, Orsay, 1981 (French). Les algèbres de fonctions résurgentes. [The algebras of resurgent functions]; With an English foreword. MR 670417
  • [E81b] Jean Écalle, Les fonctions résurgentes. Tome II, Publications Mathématiques d’Orsay 81 [Mathematical Publications of Orsay 81], vol. 6, Université de Paris-Sud, Département de Mathématique, Orsay, 1981 (French). Les fonctions résurgentes appliquées à l’itération. [Resurgent functions applied to iteration]. MR 670418
  • [E85] Jean Écalle, Les fonctions résurgentes. Tome III, Publications Mathématiques d’Orsay [Mathematical Publications of Orsay], vol. 85, Université de Paris-Sud, Département de Mathématiques, Orsay, 1985 (French). L’équation du pont et la classification analytique des objects locaux. [The bridge equation and analytic classification of local objects]. MR 852210
  • [E92] Jean Écalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1992 (French, with French summary). MR 1399559
  • [EISV] P. Elizarov, Yu. Ilyashenko, A. Shcherbakov, S. Voronin, Finitely generated groups of germs of one-dimensional conformal mappings, and invariants for complex singular points of analytic foliations of the complex plane, Nonlinear Stokes Phenomena, Amer. Math. Soc., Providence, RI, 1993, pp. 57-105. MR 94e:3205
  • [EMMRa] Jean Écalle, Jean Martinet, Robert Moussu, and Jean-Pierre Ramis, Non-accumulation des cycles-limites. I, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), no. 13, 375–377 (French, with English summary). MR 889742
  • [EMMRb] Jean Écalle, Jean Martinet, Robert Moussu, and Jean-Pierre Ramis, Non-accumulation des cycles-limites. II, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), no. 14, 431–434 (French, with English summary). MR 888240
  • [F] R. Fedorov, Growth of the number of orbital topological types of planar polynomial vector fields modulo limit cycles, Moscow Math. J., to appear.
  • [GLMM] Armengol Gasull, Jaume Llibre, Víctor Mañosa, and Francesc Mañosas, The focus-centre problem for a type of degenerate system, Nonlinearity 13 (2000), no. 3, 699–729. MR 1758996, 10.1088/0951-7715/13/3/311
  • [Ga98] Lubomir Gavrilov, Petrov modules and zeros of Abelian integrals, Bull. Sci. Math. 122 (1998), no. 8, 571–584. MR 1668534, 10.1016/S0007-4497(99)80004-9
  • [Ga01] L. Gavrilov, The infinitesimal 16th Hilbert problem in the quadratic case, Invent. Math. 143, no. 3 (2001), 449-497. CMP 2001:09
  • [Gb] A. M. Gabrièlov, Projections of semianalytic sets, Funkcional. Anal. i Priložen. 2 (1968), no. 4, 18–30 (Russian). MR 0245831
  • [G94] A. A. Glutsyuk, Hyperbolicity of phase curves of a general polynomial vector field in 𝐶ⁿ, Funktsional. Anal. i Prilozhen. 28 (1994), no. 2, 1–11, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 28 (1994), no. 2, 77–84. MR 1283247, 10.1007/BF01076493
  • [G96] A. A. Glutsyuk, Hyperbolicity of the leaves of a generic one-dimensional holomorphic foliation on a nonsingular projective algebraic variety, Tr. Mat. Inst. Steklova 213 (1997), no. Differ. Uravn. s Veshchestv. i Kompleks. Vrem., 90–111 (Russian); English transl., Proc. Steklov Inst. Math. 2 (213) (1996), 83–103. MR 1632233
  • [G01] Alexey Glutsyuk, Nonuniformizable skew cylinders. A counterexample to the simultaneous uniformization problem, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 3, 209–214 (English, with English and French summaries). MR 1817363, 10.1016/S0764-4442(00)01843-7
  • [GI*] A. Glutsuk, Yu. Ilyashenko, Restricted version of the Infinitesimal Hilbert 16th problem, to appear.
  • [GLW] É. Ghys, R. Langevin, and P. Walczak, Entropie géométrique des feuilletages, Acta Math. 160 (1988), no. 1-2, 105–142 (French). MR 926526, 10.1007/BF02392274
  • [GM] Xavier Gómez-Mont, The transverse dynamics of a holomorphic flow, Ann. of Math. (2) 127 (1988), no. 1, 49–92. MR 924673, 10.2307/1971416
  • [GR] Ana Guzmán and Christiane Rousseau, Genericity conditions for finite cyclicity of elementary graphics, J. Differential Equations 155 (1999), no. 1, 44–72. MR 1693218, 10.1006/jdeq.1998.3570
  • [Gu] John Guckenheimer, Hartman’s theorem for complex flows in the Poincaré domain, Compositio Math. 24 (1972), 75–82. MR 0301765
  • [H] D. Hilbert, Mathematical problems, Reprinted from Bull. Amer. Math. Soc. 8 (1902), 437-479, in Bull. Amer. Math. Soc. 37 (2000), 407-436. CMP 2000:17
  • [Hi-a] Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326. MR 0199184
  • [Hi-b] Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326. MR 0199184
  • [Ho] E. I. Horozov, Versal deformations of equivariant vector fields for cases of symmetry of order 2 and 3, Trudy Sem. Petrovsk. 5 (1979), 163–192 (Russian). MR 549627
  • [I69a] Ju. S. Il′jašenko, The appearance of limit cycles under a perturbation of the equation 𝑑𝑤/𝑑𝑧=-𝑅_{𝑧}/𝑅_{𝑤}, where 𝑅(𝑧,𝑤) is a polynomial, Mat. Sb. (N.S.) 78 (120) (1969), 360–373 (Russian). MR 0243155
  • [I69b] Ju. S. Il′jašenko, An example of equations 𝑑𝑤/𝑑𝑧=𝑃_{𝑛}(𝑧,𝑤)/𝑄_{𝑛}(𝑧,𝑤) having a countable number of limit cycles and arbitrarily high Petrovskiĭ-Landis genus, Mat. Sb. (N.S.) 80 (122) (1969), 388–404 (Russian). MR 0259239
  • [I72] Ju. S. Iljašenko, Algebraic unsolvability and almost algebraic solvability of the problem for the center-focus, Funkcional. Anal. i Priložen. 6 (1972), no. 3, 30–37 (Russian). MR 0315200
  • [I77] Ju. S. Il′jašenko, Remarks on the topology of the singular points of analytic differential equations in a complex domain, and Ladis’ theorem, Funkcional. Anal. i Priložen. 11 (1977), no. 2, 28–38, 95 (Russian). MR 0442372
  • [I78a] Yu. S. Il′yashenko, Topology of phase portraits of analytic differential equations on a complex projective plane, Trudy Sem. Petrovsk. 4 (1978), 83–136 (Russian). MR 524528
  • [I78b] Yu. Ilyashenko, Global and local aspects of the theory of complex differential equations, Proceedings of International Congress of Mathematicians, Helsinki, 1978, v. II, Springer-Verlag, Berlin, 1979, pp. 821-826.
  • [I82] Yu. Ilyashenko, Singular points and limit cycles of differential equations in the real and complex plane, Preprint NIVTS AN SSSR, Pushchino (1982), 38.
  • [I84] Yu. S. Il′yashenko, Limit cycles of polynomial vector fields with nondegenerate singular points on the real plane, Funktsional. Anal. i Prilozhen. 18 (1984), no. 3, 32–42 (Russian). MR 757247
  • [I85] Yu. S. Il′yashenko, Dulac’s memoir “On limit cycles” and related questions of the local theory of differential equations, Uspekhi Mat. Nauk 40 (1985), no. 6(246), 41–78, 199 (Russian). MR 815489
  • [I90] Yu. S. Il′yashenko, Finiteness theorems for limit cycles, Uspekhi Mat. Nauk 45 (1990), no. 2(272), 143–200, 240 (Russian); English transl., Russian Math. Surveys 45 (1990), no. 2, 129–203. MR 1069351, 10.1070/RM1990v045n02ABEH002335
  • [I91] Yu. S. Il′yashenko, Finiteness theorems for limit cycles, Translations of Mathematical Monographs, vol. 94, American Mathematical Society, Providence, RI, 1991. Translated from the Russian by H. H. McFaden. MR 1133882
  • [I93] Yu. S. Il′yashenko (ed.), Nonlinear Stokes phenomena, Advances in Soviet Mathematics, vol. 14, American Mathematical Society, Providence, RI, 1993. MR 1206039
  • [I00] Yu. Ilyashenko, Hilbert-type numbers for Abel equations, growth and zeros of holomorphic functions, Nonlinearity 13 (2000), no. 4, 1337–1342. MR 1767962, 10.1088/0951-7715/13/4/319
  • [IK] Yu. S. Ilyashenko and V. Yu. Kaloshin, Bifurcation of planar and spatial polycycles: Arnold’s program and its development, The Arnoldfest (Toronto, ON, 1997) Fields Inst. Commun., vol. 24, Amer. Math. Soc., Providence, RI, 1999, pp. 241–271. MR 1733580
  • [IKh] Yu. S. Il′yashenko and A. G. Khovanskiĭ, Galois groups, Stokes operators and a theorem of Ramis, Funktsional. Anal. i Prilozhen. 24 (1990), no. 4, 31–42, 96 (Russian); English transl., Funct. Anal. Appl. 24 (1990), no. 4, 286–296 (1991). MR 1092801, 10.1007/BF01077333
  • [IL] Yu. Ilyashenko and Weigu Li, Nonlocal bifurcations, Mathematical Surveys and Monographs, vol. 66, American Mathematical Society, Providence, RI, 1999. MR 1650842
  • [IP*] Yu. Ilyashenko, A. Panov, Some upper estimates of the number of limit cycles of planar vector fields with applications to the Lienard equation, to appear.
  • [IPy95] Yulij S. Ilyashenko and A. S. Pyartli, The monodromy group at infinity of a generic polynomial vector field on the complex projective plane, Russian J. Math. Phys. 2 (1994), no. 3, 275–315. MR 1330871
  • [IPy97] Yu. S. Il′yashenko and A. S. Pyartli, Rational differential equations with a nonfree monodromy group at infinity, Tr. Mat. Inst. Steklova 213 (1997), no. Differ. Uravn. s Veshchestv. i Kompleks. Vrem., 56–73 (Russian); English transl., Proc. Steklov Inst. Math. 2 (213) (1996), 50–67. MR 1632225
  • [IS] Ilia Itenberg and Eugeniĭ Shustin, Singular points and limit cycles of planar polynomial vector fields, Duke Math. J. 102 (2000), no. 1, 1–37. MR 1741776, 10.1215/S0012-7094-00-10211-6
  • [IYa91] Yu. S. Il′yashenko and S. Yu. Yakovenko, Finitely smooth normal forms of local families of diffeomorphisms and vector fields, Uspekhi Mat. Nauk 46 (1991), no. 1(277), 3–39, 240 (Russian); English transl., Russian Math. Surveys 46 (1991), no. 1, 1–43. MR 1109035, 10.1070/RM1991v046n01ABEH002733
  • [IYa95a] Zhi Yong Chen, Asymptotic behavior of the solutions to a class of second-order homogeneous linear differential equations, Math. Practice Theory 3 (1994), 57–60 (Chinese, with Chinese summary). MR 1329933
  • [IYa95b] Yuliĭ Il′yashenko and Sergeĭ Yakovenko, Double exponential estimate for the number of zeros of complete abelian integrals and rational envelopes of linear ordinary differential equations with an irreducible monodromy group, Invent. Math. 121 (1995), no. 3, 613–650. MR 1353310, 10.1007/BF01884313
  • [IYa95c] Yu. Il′yashenko and S. Yakovenko, Concerning the Hilbert sixteenth problem, Concerning the Hilbert 16th problem, Amer. Math. Soc. Transl. Ser. 2, vol. 165, Amer. Math. Soc., Providence, RI, 1995, pp. 1–19. MR 1334339
  • [IYa96] Yuliĭ Il′yashenko and Sergeĭ Yakovenko, Counting real zeros of analytic functions satisfying linear ordinary differential equations, J. Differential Equations 126 (1996), no. 1, 87–105. MR 1382058, 10.1006/jdeq.1996.0045
  • [JKM] A. Jacquemard, F. Z. Khechichine-Mourtada, and A. Mourtada, Algorithmes formels appliqués à l’étude de la cyclicité d’un polycycle algébrique générique à quatre sommets hyperboliques, Nonlinearity 10 (1997), no. 1, 19–53 (French, with English and French summaries). MR 1430738, 10.1088/0951-7715/10/1/003
  • [K*] V. Kaloshin, Hilbert 16th problem and an estimate for cyclicity of an elementary polycycle, to appear.
  • [Kh84] A. G. Khovanskiĭ, Real analytic manifolds with the property of finiteness, and complex abelian integrals, Funktsional. Anal. i Prilozhen. 18 (1984), no. 2, 40–50 (Russian). MR 745698
  • [Kh91] A. G. Khovanskiĭ, Fewnomials, Translations of Mathematical Monographs, vol. 88, American Mathematical Society, Providence, RI, 1991. Translated from the Russian by Smilka Zdravkovska. MR 1108621
  • [KS] A. Kotova and V. Stanzo, On few-parameter generic families of vector fields on the two-dimensional sphere, Concerning the Hilbert 16th problem, Amer. Math. Soc. Transl. Ser. 2, vol. 165, Amer. Math. Soc., Providence, RI, 1995, pp. 155–201. MR 1334343
  • [K-V] M.G. Khudai-Verenov, A property of the solutions of a differential equation (Russian), Mat. Sb. 56 (1962), 301-308.
  • [La] N. N. Ladis, Topological equivalence of hyperbolic linear systems, Differencial′nye Uravnenija 13 (1977), no. 2, 255–264, 379–380 (Russian). MR 0481271
  • [Le] Leau, Étude sur les équation fonctionelles à une ou plusieère variables, Ann. Fac. Sci. Toulouse 11 (1897), E1-E110.
  • [Lef] Solomon Lefschetz, On a theorem of Bendixson, J. Differential Equations 4 (1968), 66–101. MR 0219800
  • [L] E. Leontovič, On the generation of limit cycles from separatrices, Doklady Akad. Nauk SSSR (N.S.) 78 (1951), 641–644 (Russian). MR 0042576
  • [LMP] A. Lins, W. de Melo, and C. C. Pugh, On Liénard’s equation, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976) Springer, Berlin, 1977, pp. 335–357. Lecture Notes in Math., Vol. 597. MR 0448423
  • [LN80] Alcides Lins Neto, On the number of solutions of the equation 𝑑𝑥/𝑑𝑡=∑ⁿⱼ₌₀𝑎ⱼ(𝑡)𝑥^{𝑗}, 0≤𝑡≤1, for which 𝑥(0)=𝑥(1), Invent. Math. 59 (1980), no. 1, 67–76. MR 575082, 10.1007/BF01390315
  • [LN94] Alcides Lins Neto, Simultaneous uniformization for the leaves of projective foliations by curves, Bol. Soc. Brasil. Mat. (N.S.) 25 (1994), no. 2, 181–206. MR 1306560, 10.1007/BF01321307
  • [LN00] A. Lins Neto, Uniformization and the Poincaré metric on the leaves of a foliation by curves, Bol. Soc. Brasil. Mat. (N.S.) 31 (2000), no. 3, 351–366. MR 1817093, 10.1007/BF01241634
  • [LR] F. Loray, J.C. Rebello, Stably chaotic rational vector fields on $ \mathbb{CP}^{n} $, Stony Brook IMS preprint no. 2000/5 (2000).
  • [LSSc] A. Lins Neto, P. Sad, and B. Scárdua, On topological rigidity of projective foliations, Bull. Soc. Math. France 126 (1998), no. 3, 381–406 (English, with English and French summaries). MR 1682801
  • [Ma] Bernard Malgrange, Travaux d’Écalle et de Martinet-Ramis sur les systèmes dynamiques, Bourbaki Seminar, Vol. 1981/1982, Astérisque, vol. 92, Soc. Math. France, Paris, 1982, pp. 59–73 (French). MR 689526
  • [Mar] Pavao Mardešić, An explicit bound for the multiplicity of zeros of generic Abelian integrals, Nonlinearity 4 (1991), no. 3, 845–852. MR 1124336
  • [MM] J.-F. Mattei and R. Moussu, Holonomie et intégrales premières, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 4, 469–523 (French). MR 608290
  • [MR82] Jean Martinet and Jean-Pierre Ramis, Problèmes de modules pour des équations différentielles non linéaires du premier ordre, Inst. Hautes Études Sci. Publ. Math. 55 (1982), 63–164 (French). MR 672182
  • [MR83] Jean Martinet and Jean-Pierre Ramis, Classification analytique des équations différentielles non linéaires résonnantes du premier ordre, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 4, 571–621 (1984) (French). MR 740592
  • [M92] N. B. Medvedeva, The principal term of the monodromy transformation of a monodromic singular point is linear, Sibirsk. Mat. Zh. 33 (1992), no. 2, 116–124, 221 (Russian); English transl., Siberian Math. J. 33 (1992), no. 2, 280–288. MR 1174066, 10.1007/BF00971099
  • [M96] N. B. Medvedeva, The principal term of the asymptotics of the monodromy transformation: computation by the Newton diagram, Tr. Mat. Inst. Steklova 213 (1997), no. Differ. Uravn. s Veshchestv. i Kompleks. Vrem., 226–238 (Russian); English transl., Proc. Steklov Inst. Math. 2 (213) (1996), 212–223. MR 1632253
  • [MMa*] N. Medvedeva, E. Mazaeva, Sufficient condition for a monodromic singular point to be a focus, Transactions of Moscow Math. Soc. (to appear).
  • [Mo] Abderaouf Mourtada, Cyclicité finie des polycycles hyperboliques de champs de vecteurs du plan. Algorithme de finitude, Ann. Inst. Fourier (Grenoble) 41 (1991), no. 3, 719–753 (French, with English summary). MR 1136601
  • [MRo] R. Moussu and C. Roche, Théorie de Hovanskiĭ et problème de Dulac, Invent. Math. 105 (1991), no. 2, 431–441 (French). MR 1115550, 10.1007/BF01232274
  • [MS] Jean-François Mattei and Eliane Salem, Complete systems of topological and analytical invariants for a generic foliation of (𝐶²,0), Math. Res. Lett. 4 (1997), no. 1, 131–141. MR 1432816, 10.4310/MRL.1997.v4.n1.a12
  • [Mu] B. Muller, On the density of solutions of an equation in $ \mathbb{C}P^{2}$, Mat. Sbornik 98, no. 3 (1975), 325-338.
  • [Muc] Jesús Muciño-Raymundo, Deformations of holomorphic foliations having a meromorphic first integral, J. Reine Angew. Math. 461 (1995), 189–219. MR 1324214, 10.1515/crll.1995.461.189
  • [N] Isao Nakai, Separatrices for nonsolvable dynamics on 𝐶,0, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 2, 569–599 (English, with English and French summaries). MR 1296744
  • [Na] V. A. Naĭshul′, Topological invariants of analytic and area-preserving mappings and their application to analytic differential equations in 𝐶² and 𝐶𝑃², Trudy Moskov. Mat. Obshch. 44 (1982), 235–245 (Russian). MR 656288
  • [NYa95] D. Novikov and S. Yakovenko, Simple exponential estimate for the number of real zeros of complete Abelian integrals, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 4, 897–927 (English, with English and French summaries). MR 1359833
  • [NYa99a] D. Novikov and S. Yakovenko, Tangential Hilbert problem for perturbations of hyperelliptic Hamiltonian systems, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 55–65 (electronic). MR 1679454, 10.1090/S1079-6762-99-00061-X
  • [NYa99b] D. Novikov and S. Yakovenko, Trajectories of polynomial vector fields and ascending chains of polynomial ideals, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 2, 563–609 (English, with English and French summaries). MR 1697373
  • [NYa*] D. Novikov, S. Yakovenko, Redundant Picard-Fuchs system for Abelian integrals, to appear.
  • [O-B] Laura Ortíz-Bobadilla, Quadratic vector fields in 𝐶𝑃² with two saddle-node type singularities at infinity, J. Dynam. Control Systems 1 (1995), no. 3, 295–317. MR 1354537, 10.1007/BF02269371
  • [P] A. A. Panov, Variety of Poincaré mappings for cubic equations with variable coefficients, Funktsional. Anal. i Prilozhen. 33 (1999), no. 4, 84–88 (Russian); English transl., Funct. Anal. Appl. 33 (1999), no. 4, 310–312 (2000). MR 1746436, 10.1007/BF02467118
  • [Pe] G. Petrov, Elliptic integrals and their nonoscillation (Russian), Funktsional. Anal. i Prilozhen. 20, no. 1 (1986), 46-49. MR 87f:5803
  • [PL1] I. G. Petrovskiĭ and E. M. Landis, On the number of limit cycles of the equation 𝑑𝑦/𝑑𝑥=𝑃(𝑥,𝑦)/𝑄(𝑥,𝑦), where 𝑃 and 𝑄 are polynomials of 2nd degree, Mat. Sb. N.S. 37(79) (1955), 209–250 (Russian). MR 0073004
  • [PL2] E. M. Landis and I. G. Petrovskiĭ, On the number of limit cycles of the equation 𝑑𝑦/𝑑𝑥=𝑃(𝑥,𝑦)/𝑄(𝑥,𝑦), where 𝑃 and 𝑄 are polynomials, Mat. Sb. N.S. 43(85) (1957), 149–168 (Russian). MR 0089968
  • [P-M] Ricardo Pérez-Marco, Fixed points and circle maps, Acta Math. 179 (1997), no. 2, 243–294. MR 1607557, 10.1007/BF02392745
  • [Pu] I. A. Pushkar′, A multidimensional generalization of Il′yashenko’s theorem on abelian integrals, Funktsional. Anal. i Prilozhen. 31 (1997), no. 2, 34–44, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 31 (1997), no. 2, 100–108. MR 1475322, 10.1007/BF02466015
  • [Py] A. Pyartli, Rational differential equations with a commutative monodromy group at infinity, Trans. Moscow Math. Soc. 61 (2000), 67-95.
  • [R86] R. Roussarie, On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields, Bol. Soc. Brasil. Mat. 17 (1986), no. 2, 67–101. MR 901596, 10.1007/BF02584827
  • [R88] R. Roussarie, A note on finite cyclicity property and Hilbert’s 16th problem, Dynamical systems, Valparaiso 1986, Lecture Notes in Math., vol. 1331, Springer, Berlin, 1988, pp. 161–168. MR 961099, 10.1007/BFb0083072
  • [R89] Robert Roussarie, Cyclicité finie des lacets et des points cuspidaux, Nonlinearity 2 (1989), no. 1, 73–117 (French, with English summary). MR 980858
  • [R98] Robert Roussarie, Bifurcation of planar vector fields and Hilbert’s sixteenth problem, Progress in Mathematics, vol. 164, Birkhäuser Verlag, Basel, 1998. MR 1628014
  • [RSZ] C. Rousseau, G. Świrszcz, and H. Żołpolhk adek, Cyclicity of graphics with semi-hyperbolic points inside quadratic systems, J. Dynam. Control Systems 4 (1998), no. 2, 149–189. MR 1626537, 10.1023/A:1022887001627
  • [S] Steve Smale, Mathematical problems for the next century, Math. Intelligencer 20 (1998), no. 2, 7–15. MR 1631413, 10.1007/BF03025291
  • [Sa] A. P. Sadovskiĭ, A problem of distinguishing the center and focus for a case of a complex singular point, Differentsial′nye Uravneniya 22 (1986), no. 5, 789–794, 916 (Russian). MR 846508
  • [Se] A. Seidenberg, Reduction of singularities of the differential equation 𝐴𝑑𝑦=𝐵𝑑𝑥, Amer. J. Math. 90 (1968), 248–269. MR 0220710
  • [Sh] S. Shahshahani, Periodic solutions of polynomial first order differential equations, Nonlinear Anal. 5 (1981), no. 2, 157–165. MR 606697, 10.1016/0362-546X(81)90041-9
  • [Shch82] A. A. Shcherbakov, Density of the orbit of a pseudogroup of conformal mappings and generalization of the Khudaĭ-Verenov theorem, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 4 (1982), 10–15, 84 (Russian, with English summary). MR 671879
  • [Shch84] A. A. Shcherbakov, Topological and analytic conjugation of noncommutative groups of germs of conformal mappings, Trudy Sem. Petrovsk. 10 (1984), 170–196, 238–239 (Russian, with English summary). MR 778885
  • [Shi] Song Ling Shi, A concrete example of the existence of four limit cycles for plane quadratic systems, Sci. Sinica 23 (1980), no. 2, 153–158. MR 574405
  • [Si] Michael F. Singer, Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc. 333 (1992), no. 2, 673–688. MR 1062869, 10.1090/S0002-9947-1992-1062869-X
  • [SRO] A. A. Shcherbakov, E. Rosales-González, and L. Ortiz-Bobadilla, Countable set of limit cycles for the equation 𝑑𝑤/𝑑𝑧=𝑃_{𝑛}(𝑧,𝑤)/𝑄_{𝑛}(𝑧,𝑤), J. Dynam. Control Systems 4 (1998), no. 4, 539–581. MR 1662926, 10.1023/A:1021819201777
  • [St] Shlomo Sternberg, On the structure of local homeomorphisms of euclidean 𝑛-space. II., Amer. J. Math. 80 (1958), 623–631. MR 0096854
  • [SV] J. Seade, A. Verjovsky, Actions of discrete groups on complex projective spaces, Laminations and foliations in dynamics, geometry and topology (Stony Brook, NY, 1998), Contemp. Math. 269, Amer. Math. Soc., Providence, RI, 2001, pp. 155-178.
  • [T] Floris Takens, Forced oscillations and bifurcations, Applications of global analysis, I (Sympos., Utrecht State Univ., Utrecht, 1973) Math. Inst. Rijksuniv. Utrecht, Utrecht, 1974, pp. 1–59. Comm. Math. Inst. Rijksuniv. Utrecht, No. 3-1974. MR 0478235
  • [Ti] E. Titčmarš, Teoriya funktsii, 2nd ed., “Nauka”, Moscow, 1980 (Russian). Translated from the English and with a preface by V. A. Rohlin. MR 593142
  • [Tr] S. I. Trifonov, Cyclicity of elementary polycycles of generic smooth vector fields, Tr. Mat. Inst. Steklova 213 (1997), no. Differ. Uravn. s Veshchestv. i Kompleks. Vrem., 152–212 (Russian); English transl., Proc. Steklov Inst. Math. 2 (213) (1996), 141–199. MR 1632245
  • [V] A. N. Varchenko, Estimation of the number of zeros of an abelian integral depending on a parameter, and limit cycles, Funktsional. Anal. i Prilozhen. 18 (1984), no. 2, 14–25 (Russian). MR 745696
  • [Ve] A. Verjovsky, Private communication.
  • [Vo] S. M. Voronin, Analytic classification of germs of conformal mappings (𝐶,0)→(𝐶,0), Funktsional. Anal. i Prilozhen. 15 (1981), no. 1, 1–17, 96 (Russian). MR 609790
  • [Y] Jean-Christophe Yoccoz, Théorème de Siegel, nombres de Bruno et polynômes quadratiques, Astérisque 231 (1995), 3–88 (French). Petits diviseurs en dimension 1. MR 1367353
  • [Ya95] S. Yakovenko, A geometric proof of the Bautin theorem, Concerning the Hilbert 16th problem, Amer. Math. Soc. Transl. Ser. 2, vol. 165, Amer. Math. Soc., Providence, RI, 1995, pp. 203–219. MR 1334344
  • [Ya99] Sergei Yakovenko, On functions and curves defined by ordinary differential equations, The Arnoldfest (Toronto, ON, 1997) Fields Inst. Commun., vol. 24, Amer. Math. Soc., Providence, RI, 1999, pp. 497–525. MR 1733590
  • [Z83] Khenrik Zholondek, Versality of a family of symmetric vector fields on the plane, Mat. Sb. (N.S.) 120(162) (1983), no. 4, 473–499 (Russian). MR 695955
  • [Z87] Henryk Żołpolhk adek, Bifurcations of certain family of planar vector fields tangent to axes, J. Differential Equations 67 (1987), no. 1, 1–55. MR 878251, 10.1016/0022-0396(87)90138-0

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 34Cxx, 34Mxx, 37F75

Retrieve articles in all journals with MSC (2000): 34Cxx, 34Mxx, 37F75


Additional Information

Yu. Ilyashenko
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
Address at time of publication: Moscow State and Independent Universities, Steklov Mathematical Institute, Moscow (MIAN) Gubkina st. 8, Moscow, Russia, 117966
Email: yulij@math.cornell.edu, yulijs@mccme.ru

DOI: http://dx.doi.org/10.1090/S0273-0979-02-00946-1
Keywords: Limit cycles, polynomial vector fields, normal forms, bifurcations, foliations, Abelian integrals
Received by editor(s): December 1, 2001
Published electronically: April 9, 2002
Additional Notes: The author was supported in part by grants NSF DMS 997-0372, NSF 0010404, and CRDF RM1-2086. The main results of the paper were presented at colloquium talks at Cornell University, December 1999, and Northeastern University (Harvard - MIT - Brandeis - Northeastern Colloquium), November 2000. The author thanks Dr. S. Gelfand, who assisted with the latter talk and suggested the idea of writing a survey on the subject
Article copyright: © Copyright 2002 American Mathematical Society