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Bulletin of the American Mathematical Society

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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
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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.

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  • [ALGM] A. Andronov; E. Leontovic; I. Gordon; A. Mauier, Theory of bifurcations of dynamic systems on a plane, Translated from the Russian, Halsted Press [A division of John Wiley & Sons], New York-Toronto, Ont., 1973. MR 49:9345
  • [A] V. I. Arnold, Geometrical methods in the theory of ordinary differential equations, Springer-Verlag, New York, 1983. MR 84d:58023.
  • [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. Arnold, S. Gusein-Zade, A. Varchenko, Osobennosti differentsiruemykh otobrazhenii. II [Singularities of differentiable maps], Monodromiya i asimptotiki integralov [Monodromy and the asymptotic behavior of integrals], ``Nauka'', Moscow, 1984. MR 86m:58026
  • [AI] V. Arnold, Yu. Ilyashenko, Ordinary differential equations [Translated from the 1985 Russian original], Enciclopaedia Math. Sci., Dynamical systems. 1, Springer, Berlin, 1988, pp. 1-148. MR 87e:34049
  • [AKL] J. Artes, R. Kooij, J. Llibre, Structurally stable quadratic vector fields, Mem. Amer. Math. Soc. 134 (1998), viii+100. MR 98m:34053
  • [B] I. Bendixson, Sur les courbes définies par des équations différentielles, Acta Math. 24 (1901), 1-88.
  • [Ba] 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), 19. MR 15:527h
  • [Ber] L. Bers, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960), 94-97. MR 22:2694
  • [BFY] M. Briskin, J.-P. Francoise, Y. Yomdin, Center conditions, compositions of polynomials and moments on algebraic curves, Ergodic Theory and Dynamical Systems 19 (1999), 1201-1220. MR 2000k:34051
  • [BM] F. Berezovskaya, N. Medvedeva, The asymptotics of the return map of a singular point with fixed Newton diagram, Trudy Sem. Petrovsk. 15 (1991), 156-177. MR 93f:58198
  • [Bo76] R. Bogdanov, The versal deformation of a singular point of a vector field on the plane in the case of zero eigenvalues, Trudy Sem. Petrovsk 2 (1976), 37-65. MR 56:1371
  • [Bo77] R. Bogdanov, Bifurcations of a limit cycle of a certain family of vector fields on the plane, Trudy Sem. Petrovsk 2 (1976), 23-35. MR 56:1363
  • [Bo79] R. Bogdanov, Local orbital normal forms of vector fields on the plane, Trudy Sem. Petrovsk 5 (1979), 51-84. MR 80j:58056
  • [Bo85] R. Bogdanov, Invariants of elementary singular points on the plane, Uspekhi Mat. Nauk 40 (1985), no. 3(243), 199-200. MR 86k:58087
  • [B64] A. Brjuno, The normal form of differential equations, Dokl. Akad. Nauk SSSR 157 (1964), 1276-1279. MR 29:3733
  • [B71] A. Brjuno, Analytic form of differential equations I, Trans. Moscow Math. Soc. 25 (1971), 119-262. MR 51:13365
  • [B72] A. Brjuno, Analytic form of differential equations II, Trudy Moskov. Mat. Obsc. 26 (1972), 199-239. MR 51:13365
  • [C] K.T. Chen, Equivalence and decomposition of vector fields about an elementary critical point, Amer. J. Math. 85 (1963), 693-722. MR 28:3224
  • [Ca] C. 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 93j:32039
  • [CGM] A. Candel, X. Gómez-Mont, Uniformization of the leaves of a rational vector field, Ann. Inst. Fourier (Grenoble) 45, no. 4 (1995), 1123-1133. MR 96k:32068
  • [Ch] M. Chaperon, On the local classification of holomorphic vector fields, Geometric dynamics (Rio de Janeiro, 1981), Springer, Berlin, 1983, pp. 96-103. MR 85d:58049
  • [ChL] C.J. Christopher, N.G. Lloyd, Polynomial systems: a lower bound for the Hilbert numbers, Proc. Roy. Soc. London Ser. A 450, no. 1938 (1995), 219-224. MR 96f:34041
  • [CKP] C. Camacho, N. Kuiper, J. Palis, La topologie du feuilletage d'un champ de vecteurs holomorphes près d'une singularité, C. R. Acad. Sci. Paris Sér. A-B 282, no. 17 (1976), Ai, A959-A961. MR 54:1301
  • [CLW] S-N. Chow, Ch.Z. Li, D. Wang, Normal forms and bifurcation of planar vector fields, Cambridge University Press, Cambridge, 1994. MR 95i:58161
  • [CM] D. Cerveau, R. Moussu, Groupes d'automorphismes de $({C},0)$ et équations différentielles $ydy+\cdots =0$, Bull. Soc. Math. France 116, no. 4 (1988), 459-488. MR 90m:58192
  • [Co] W. A. Coppel, A survey of quadratic systems, J. Differential Equations 2 (1966), 293-304. MR 33:4374
  • [CS] C. Camacho, P. Sad, Invariant varieties through singularities of holomorphic vector fields, Ann. of Math. 115, no. 3 (1982), 579-595. MR 83m:58062
  • [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, Ming Shu Wang, The relative position, and the number, of limit cycles of a quadratic differential system, Acta Math. Sinica 22 (1979), 751-758. MR 81g:34031
  • [D] H. Dulac, Sur les cycles limite, Bull. Soc. Math France 51 (1923), 45-188.
  • [De] P. Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, 163, Springer-Verlag, Berlin, 1970, pp. iii+133. MR 54:5232
  • [Du] F. Dumortier, Singularities of vector fields on the plane, Equations 23, no. 1 (1977), 53-106. MR 58:31276
  • [DHP] F. Dumortier, Ch. Herssens, L. Perko, Local bifurcations and a survey of bounded quadratic systems, Differential Equations 165, no. 2 (2000), 430-467. MR 2001i:37074
  • [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 Morsalami, C. Rousseau, Hilbert's 16th problem for quadratic systems and cyclicity of elementary graphics, Nonlinearity 9 (1996), 1209-1261. MR 97g:58136
  • [DRR] F. Dumortier, R. Roussarie, C. Rousseau, Hilbert's 16th problem for quadratic vector fields, Differential Equations 110, no. 1 (1994), 86-133. MR 95g:58179
  • [DRRa] F. Dumortier, R. Roussarie, C. Rousseau, Elementary graphics of cyclicity $1$ and $2$, Nonlinearity 7 (1994), no. 3, 1001-1043. MR 95d:58095
  • [DRS] F. Dumortier, R. Roussarie, J. Sotomayor, Bifurcations of cuspidal loops, Nonlinearity 10, no. 6 (1997), 1369-1408. MR 98k:58167
  • [DRSZ] F. Dumortier, R. Roussarie, J. Sotomayor, H. Zoladek, Bifurcations of planar vector fields. Nilpotent singularities and Abelian integrals, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1991. MR 93f:58165
  • [E81a] J. Écalle, Les fonctions résurgentes. Tome I Les algèbres de fonctions résurgentes, Université de Paris-Sud, Département de Mathématique, Orsay, 1981. MR 84h:30077a
  • [E81b] J. Écalle, Les fonctions résurgentes. Tome II, Les fonctions résurgentes appliquées à l'itération, Université de Paris-Sud, Département de Mathématique, Orsay, 1981. MR 84h:30077b
  • [E85] J. Écalle, Les fonctions résurgentes. Tome III, L'équation du pont et la classification analytique des objects locaux, Université de Paris-Sud, Département de Mathématiques, Orsay, 1985. MR 87k:32009
  • [E92] J. Écalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac (French), Hermann, Paris, 1992. MR 97f:58104
  • [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] J. Écalle, J. Martinet, R. Moussu, J.-P. Ramis, Non-accumulation des cycles-limites. I, C. R. Acad. Sci. Paris Sér. I Math. 304, no. 13 (1987), 375-377. MR 89i:58121a
  • [EMMRb] J. Écalle, J. Martinet, R. Moussu, J.-P. Ramis, Non-accumulation des cycles-limites. II, C. R. Acad. Sci. Paris Sér. I Math. 304, no. 14 (1987), 431-434. MR 89i:58121b
  • [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] A. Gasull, J. Llibre, V. Mañosa, F. Mañosas, The focus-centre problem for a type of degenerate system, Nonlinearity 13, no. 3 (2000), 699-729. MR 2001b:34061
  • [Ga98] L. Gavrilov, Petrov modules and zeros of Abelian integrals, Bull. Sci. Math 122, no. 8 (1998), 571-584. MR 99m:32043
  • [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. Gabrielov, Projections of semianalytic sets (Russian), Funktsional. Anal. i Prilozen. 2, no. 4 (1968), 18-30. MR 39:7137
  • [G94] A. Glutsyuk, Hyperbolicity of phase curves of a general polynomial vector field in ${C}\sp{n}$ (Russian), Funktsional. Anal. i Prilozhen 28, no. 2 (1994), 1-11. MR 95i:32045
  • [G96] A. Glutsyuk, Hyperbolicity of the leaves of a generic one-dimensional holomorphic foliation on a nonsingular projective algebraic variety (Russian), Tr. Mat. Inst. Steklova 213 (1997), 90-111. MR 99m:32041
  • [G01] A. Glutsyuk, Nonuniformizable skew cylinders. A counterexample to the simultaneous uniformization problem, C. R. Acad. Sci. Paris Ser. I Math. 332, no. 3 (2001), 209-214. MR 2002c:32045
  • [GI*] A. Glutsuk, Yu. Ilyashenko, Restricted version of the Infinitesimal Hilbert 16th problem, to appear.
  • [GLW] É. Ghys, R. Langevin, P. Walczak, Entropie géométrique des feuilletages, Acta Mathematica 160, nos. 1-2 (1988), 105-142. MR 89a:57034
  • [GM] X. Gómez-Mont, The transverse dynamics of a holomorphic flow, Annals of Mathematics. Second Series 127, no. 1 (1988), 49-92. MR 89d:32049
  • [GR] A. Guzman, C. Rousseau, Genericity conditions for cyclicity of elementary graphics, Differential Equations 155, no. 1 (1999), 44-72. MR 2000d:34066
  • [Gu] J. Guckenheimer, Hartman's theorem for complex flows in the Poincaré domain, Compositio Math. 24 (1972), 75-82. MR 46:920
  • [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] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, Ann. of Math. 79 (1964), 109-203. MR 33:7333
  • [Hi-b] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. II, Ann. of Math. 79 (1964), 205-326. MR 33:7333
  • [Ho] E. Horozov, Versal deformations of equivariant vector fields for cases of symmetry of order $2$ and $3$ (Russian), Trudy Sem. Petrovsk. 5 (1979), 163-192. MR 80k:58079
  • [I69a] Yu. Ilyashenko, The appearance of limit cycles under a perturbation of the equation $dw/dz=-R\sb {z}/R\sb {w}$, where $R(z,\,w)$ is a polynomial (Russian), Mat. Sb. (N.S.) 78 (120) no. 3 (1969), 360-373. MR 39:4479
  • [I69b] Yu. Ilyashenko, An example of equations $dw/dz=P\sb {n}\,(z,\,w)/Q\sb {n}\,(z,\,w)$ having a countable number of limit cycles and arbitrarily high Petrovskii- Landis genus (Russian), Mat. Sb. (N.S.) 80 (122), no. 3 (1969), 388-404. MR 41:3881
  • [I72] Yu. Ilyashenko, Algebraic unsolvability and almost algebraic solvability of the problem for the center-focus (Russian), Funktsional. Anal. i Prilozen 6, no. 3 (1972), 30-37. MR 47:3749
  • [I77] Yu. Ilyashenko, Remarks on the topology of the singular points of analytic differential equations in a complex domain, and Ladis' theorem, Funktsional. Anal. i Prilozen. 11, no. 2 (1977), 28-38, 95. MR 56:755
  • [I78a] Yu. Ilyashenko, Topology of phase portraits of analytic differential equations on a complex projective plane (Russian), Trudy Sem. Petrovsk. no. 4 (1978), 83-136. MR 84k:58164
  • [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. Ilyashenko, Limit cycles of polynomial vector fields with nondegenerate singular points on the real plane (Russian), Funktsional. Anal. i Prilozhen. 18, no. 3 (1984), 32-42. MR 86a:34054
  • [I85] Yu. Ilyashenko, Dulac's memoir ``On limit cycles" and related questions of the local theory of differential equations (Russian), Uspekhi Mat. Nauk 40, no. 6 (1985), 41-78. MR 87j:34052
  • [I90] Yu. Ilyashenko, Finiteness theorems for limit cycles, Uspekhi Mat. Nauk 45, no. 2 (1990), 143-200. MR 92a: 58110
  • [I91] Yu. Ilyashenko, Finiteness theorems for limit cycles, American Mathematical Society, Providence, RI, 1991. MR 92k:58221
  • [I93] Yu. Ilyashenko (editor), Nonlinear Stokes phenomena, American Mathematical Society, Providence, RI, 1993. MR 93i:32002
  • [I00] Yu. Ilyashenko, Hilbert-type numbers for Abel equations, growth and zeros of holomorphic functions, Nonlinearity 13, no. 4 (2000), 1337-1342. MR 2001h:34047
  • [IK] Yu. Ilyashenko, V. Kaloshin, Bifurcation of planar and spatial polycycles: Arnold's program and its development, The Arnoldfest (Toronto, ON, 1997), Amer. Math. Soc., Providence, RI, 1999, pp. 241-271. MR 2001b:34064
  • [IKh] Yu. Ilyashenko, A. Khovanskii, Galois groups, Stokes operators and a theorem of Ramis (Russian), Funktsional. Anal. i Prilozhen. 24, no. 4 (1990), 31-42. MR 92f:32038
  • [IL] Yu. Ilyashenko, W. Li, Nonlocal bifurcations, Mathematical Surveys and Monographs, 66, American Mathematical Society, Providence, RI, 1999. MR 99m:58139
  • [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] Yu. Ilyashenko, A. Pyartli, The monodromy group at infinity of a generic polynomial vector field on the complex projective plane, Russian J. Math. Phys. 2, no. 3 (1994), 275-315. MR 96e:34009
  • [IPy97] Yu. Ilyashenko, A. Pyartli, Rational differential equations with a nonfree monodromy group at infinity (Russian), Tr. Mat. Inst. Steklova 213 (1997), 50-67. MR 2001f:34174
  • [IS] I. Itenberg, E. Shustin, Singular points and limit cycles of planar polynomial vector fields, Duke Math. J. 102 (2000), 1-37. MR 2001d:34049
  • [IYa91] Yu. Ilyashenko, S. Yakovenko, Finitely smooth normal forms of local families of diffeomorphisms and vector fields (Russian), Uspekhi Mat. Nauk 46, no. 1 (277) (1991), 3-39. MR 92i:58165
  • [IYa95a] Yu. Ilyashenko, S. Yakovenko, Finite cyclicity of elementary polycycles in generic families, Concerning the Hilbert 16th problem, vol. 165, Amer. Math. Soc. Transl. Ser. 2, Amer. Math. Soc., Providence, RI, 1995, pp. 21-95. MR 96b:34042
  • [IYa95b] Yu. Ilyashenko, S. 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, no. 3 (1995), 613-650. MR 96g:58157
  • [IYa95c] Yu. Ilyashenko, S. Yakovenko, editors, Concerning the Hilbert 16th problem, American Mathematical Society, Providence, RI, 1995. MR 96j:34057
  • [IYa96] Yu. Ilyashenko, S. Yakovenko, Counting real zeros of analytic functions satisfying linear ordinary differential equations, Differential Equations 126, no. 1 (1996), 87-105. MR 97a:34010
  • [JKM] A. Jacquemard, F. Z. Khechichine-Mourtada, A. Mourtada, Algorithmes formels appliqués à l'étude de la cyclicité d'un polycycle algébrique générique à quatre sommets hyperboliques, Nonlinearity 10, no. 1 (1997), 19-53. MR 98b:58142
  • [K*] V. Kaloshin, Hilbert 16th problem and an estimate for cyclicity of an elementary polycycle, to appear.
  • [Kh84] A. Khovanskii, Real analytic manifolds with the property of finiteness, and complex abelian integrals (Russian), Funktsional. Anal. i Prilozhen. 18, no. 2 (1984), 40-50. MR 86a:32024
  • [Kh91] A. Khovanskii, Fewnomials, Translations of Mathematical Monographs, 88, American Mathematical Society, Providence, RI, 1991. MR 92h:14039
  • [KS] A. Kotova, V. Stanzo, On few-parameter generic families of vector fields on the two-dimensional sphere, Concerning the Hilbert 16th problem, vol. 165, Amer. Math. Soc. Transl. Ser. 2, Amer. Math. Soc., Providence, RI, 1995, pp. 155-201. MR 96i:34055
  • [K-V] M.G. Khudai-Verenov, A property of the solutions of a differential equation (Russian), Mat. Sb. 56 (1962), 301-308.
  • [La] N. Ladis, Topological equivalence of hyperbolic linear systems, Differencial'nye Uravnenija 13, no. 2 (1977), 255-264, 379-380. MR 58:1396
  • [Le] Leau, Étude sur les équation fonctionelles à une ou plusieère variables, Ann. Fac. Sci. Toulouse 11 (1897), E1-E110.
  • [Lef] S. Lefchetz, On a theorem of Bendixson, J. Diff. Equat. 4 (1968), 66-101. MR 36:2879
  • [L] E. Leontovich, On the generation of limit cycles from separatrices (Russian), Doklady Akad. Nauk SSSR (N.S.) 78 (1951), 641-644. MR 13:132b
  • [LMP] A. Lins, W. de Melo, C. C. Pugh, On Liénard's equation, Lecture Notes in Math., 597, Springer, Berlin, 1977, pp. 335-357. MR 56:6730
  • [LN80] A. Lins Neto, On the number of solutions of the equation $dx/dt=\sum \sp{n}\sb {j=0}\,a\sb {j}(t)x\sp{j}$, $0\leq t\leq 1$, for which $x(0)=x(1)$, Invent. Math. 59, no. 1 (1980), 67-76. MR 81i:34009
  • [LN94] A. Lins Neto, Simultaneous uniformization for the leaves of projective foliations by curves, Bol. Soc. Brasil. Mat. (N.S.) 25, no. 2 (1994), 181-206. MR 95k:32034
  • [LN00] A. Lins Neto, Uniformization and the Poincaré metric on the leaves of a foliation by curves, Bol. Soc. Brasil. Mat. (N.S.) 31, no.3 (2000), 351-366. MR 2002c:37069
  • [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, B. Scardua, On topological rigidity of projective foliations, Bull. Soc. Math. France 126, no. 3 (1998), 381-406. MR 2000b:32027
  • [Ma] B. Malgrange, Travaux d'Écalle et de Martinet-Ramis sur les systémes dynamiques (French), Bourbaki Seminar, Astérisque 92-93 (1982), 59-73. MR 84m:58023
  • [Mar] P. Mardesic, An explicit bound for the multiplicity of zeros of generic Abelian integrals, Nonlinearity 4, no. 3 (1991), 845-852. MR 92h:58163
  • [MM] J.-F. Mattei, R. Moussu, Holonomie et integrales premiéres (French), Ann. Sci. École Norm. Sup. (4) 13, no. 4 (1980), 469-523. MR 83b:58005
  • [MR82] J. Martinet, J-P. Ramis, Problémes de modules pour des equations differentielles non lineaires du premier ordre (French), Inst. Hautes Études Sci. Publ. Math. 55 (1982), 63-164. MR 84k:34011
  • [MR83] J. Martinet, J-P. Ramis, Classification analytique des equations differentielles non lineaires resonnantes du premier ordre (French), Ann. Sci. Ecole Norm. Sup. (4) 16, no. 4 (1983), 571-621. MR 86k:34034
  • [M92] N. Medvedeva, The principal term of the monodromy transformation of a monodromic singular point is linear (Russian), Sibirsk. Mat. Zh. 33, no. 2 (1992), 116-124. MR 93f:58200
  • [M96] N. Medvedeva, The principal term of the asymptotics of the monodromy transformation: computation by the Newton diagram (Russian), Proc. Steklov Inst. Math 213 (1996), 212-223. MR 2000g:34048
  • [MMa*] N. Medvedeva, E. Mazaeva, Sufficient condition for a monodromic singular point to be a focus, Transactions of Moscow Math. Soc. (to appear).
  • [Mo] A. Mourtada, Cyclicite finie des polycycles hyperboliques de champs de vecteurs du plan. Algorithme de finitude (French), Ann. Inst. Fourier 41, no. 3 (1991), 719-753. MR 93e:58155
  • [MRo] R. Moussu, C. Roche, Théorie de Hovanskii et problème de Dulac, Inventiones Mathematicae 105, no. 2 (1991), 431-441. MR 92e:58169
  • [MS] J.F. Mattei, E. Salem, Complete systems of topological and analytical invariants for a generic foliation of $({C}\sp{2},0)$, Math. Res. Lett. 4, no. 1 (1997), 131-141. MR 98a:32044
  • [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] J. Muciño-Raymundo, Deformations of holomorphic foliations having a meromorphic first integral, Journal für die Reine und Angewandte Mathematik 461 (1995), 189-219. MR 96e:32026
  • [N] I. Nakai, Separatrices for nonsolvable dynamics on ${C},0$, Ann. Inst. Fourier 44 (1994), no. 2, 569-599. MR 95j:58124
  • [Na] V. Naishul, Topological invariants of analytic and area-preserving mappings and their application to analytic differential equations in ${C}\sp{2}$ and ${C}P\sp{2}$ (Russian), Trudy Moskov. Mat. Obshch. 44 (1982), 235-245. MR 84f:58092
  • [NYa95] D. Novikov, S. Yakovenko, Simple exponential estimate for the number of real zeros of complete Abelian integrals, Ann. Inst. Fourier 45, no. 4 (1995), 897-927. MR 97b:14053
  • [NYa99a] D. Novikov, S. Yakovenko, Tangential Hilbert problem for perturbations of hyperelliptic Hamiltonian systems, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 55-65. MR 2000a:34065
  • [NYa99b] D. Novikov, S. Yakovenko, Trajectories of polynomial vector fields and ascending chains of polynomial ideals, Ann. Inst. Fourier 49, no. 2 (1999), 563-609. MR 2001h:32054
  • [NYa*] D. Novikov, S. Yakovenko, Redundant Picard-Fuchs system for Abelian integrals, to appear.
  • [O-B] L. Ortiz-Bobadilla, Quadratic vector fields in $CP^{2}$with two saddle-node type singularities at infinity, Dynam. Control Systems 1, no. 3 (1995), 295-317. MR 97a:58152
  • [P] A. Panov, Variety of Poincaré mappings for cubic equations with variable coefficients (Russian), Funktsional. Anal. i Prilozhen 33, no. 4 (1999), 84-88. MR 2001f:34076
  • [Pe] G. Petrov, Elliptic integrals and their nonoscillation (Russian), Funktsional. Anal. i Prilozhen. 20, no. 1 (1986), 46-49. MR 87f:5803
  • [PL1] I. Petrovskii, E. Landis, On the number of limit cycles of the equation $dy/dx=P(x,y)/$$Q(x,y)$, where $P$ and $Q$ are polynomials of 2nd degree (Russian), Mat. Sb. N.S. 37(79) (1955), 209-250. MR 17:364d
  • [PL2] I. Petrovskii, E. Landis, On the number of limit cycles of the equation $dy/dx=P(x,y)/$$Q(x,y)$, where $P$ and $Q$ are polynomials (Russian), Mat. Sb. N.S. 85 (1957), 149-168. MR 19:746c
  • [P-M] R. Perez-Marco, Fixed points and circle maps, Acta Math. 179, no. 2 (1997), 243-294. MR 99a:58130
  • [Pu] I. Pushkar', A multidimensional generalization of Ilyashenko's theorem on abelian integrals (Russian), Funktsional. Anal. i Prilozhen. 31, no. 2 (1997), 34-44, 95. MR 98k:58183
  • [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, no. 2 (1986), 67-101. MR 88i:34061
  • [R88] R. Roussarie, A note on finite cyclicity property and Hilbert's 16th problem, Lecture Notes in Math., 1331, Springer, Berlin, 1986, pp. 161-168. MR 90b:58227
  • [R89] R. Roussarie, Cyclicité finie des lacets et des points cuspidaux, Nonlinearity 2, no. 1 (1989), 73-117. MR 90m:58169
  • [R98] R. Roussarie, Bifurcation of planar vector fields and Hilbert's sixteenth problem, Birkhauser Verlag, Basel, 1998. MR 99k:58129
  • [RSZ] C. Rousseau, G. Swirszcz, H. Zoladek, Cyclicity of graphics with semi-hyperbolic points inside quadratic systems, Dynam. Control Systems 4, no. 2 (1988), 149-189. MR 99h:58149
  • [S] S. Smale, Mathematical problems for the next century, Math. Intelligencer 20, no. 2 (1998), 7-15. MR 99h:01033
  • [Sa] A. Sadovskii, A problem of distinguishing the center and focus for a case of a complex singular point (Russian), Differentsial'nye Uravneniya 22, no. 5 (1986), 789-794. MR 87h:34042
  • [Se] A. Seidenberg, Reduction of singularities of the differential equation $ Adx = Bdy $, Amer. J. Math. 90 (1968), 248-269. MR 36:3762
  • [Sh] S. Shahshahani, Periodic solutions of polynomial first order differential equations, Nonlinear Anal. 5, no. 2 (1981), 157-165. MR 82d:34052
  • [Shch82] A. Shcherbakov, Density of the orbit of a pseudogroup of conformal mappings and generalization of the Khudai-Verenov theorem (Russian), Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1982), no. 4, 10-15. MR 84m:30015
  • [Shch84] A. Shcherbakov, Topological and analytic conjugation of noncommutative groups of germs of conformal mappings (Russian), Trudy Sem. Petrovsk. no. 10 (1984), 170-196, 238-239. MR 86g:58083
  • [Shi] Song Ling Shi, A concrete example of the existence of four limit cycles for plane quadratic systems, Sci. Sinica 23 (1980), 153-158. MR 81f:34037
  • [Si] M. Singer, Liouvillian first integrals of differential equations., Trans. Amer. Math. Soc. 333 (1992), 673-688. MR 92m:12014
  • [SRO] A. Shcherbakov, E. Rosales-Gonzalez, L. Ortiz-Bobadilla, Countable set of limit cycles for the equation $dw/dz=P\sb n(z,w)/Q\sb n(z,w)$, Dynam. Control Systems 4, no. 4 (1998), 539-581. MR 99m:58150
  • [St] S. Sternberg, On the structure of local homeomorphisms of Euclidean $n$-space, II. Amer. J. Math. 80 (1958), 623-631. MR 20:3336
  • [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] F. Takens, Forced oscillations and bifurcations, Comm. Math. Inst. Rijksuniv. Utrecht, Math. Inst. Rijksuniv. Utrecht 3 (1974), 1-59. MR 57:17720
  • [Ti] E. Titcmars, Teoriya funktsii (Translated from the English), Nauka, Moscow, 1980. MR 82b:30001
  • [Tr] S. Trifonov, Cyclicity of elementary polycycles of generic smooth vector fields, Proc. Steklov Inst. Math. 213, no. 2 (1996), 141-199. MR 99i:58115
  • [V] A. Varchenko, Estimation of the number of zeros of an abelian integral depending on a parameter, and limit cycles (Russian), Funktsional. Anal. i Prilozhen. 18, no. 2 (1984), 14-25. MR 85g:32033
  • [Ve] A. Verjovsky, Private communication.
  • [Vo] S. Voronin, Analytic classification of germs of conformal mappings $({C},\,0)\rightarrow ({C},\,0)$ (Russian), Funktsional. Anal. i Prilozhen. 15, no. 1, (1981), 1-17. MR 82h:58008
  • [Y] J.-Ch. Yoccoz, Théorème de Siegel, nombres de Bruno et polynômes quadratiques. Petits diviseurs en dimension $1$, Astérisque no. 231 (1995), 3-88. MR 96m:58214
  • [Ya95] S. Yakovenko, A geometric proof of the Bautin theorem, Concerning the Hilbert 16th problem, Amer. Math. Soc, Providence, RI, 1995, pp. 203-219. MR 96j:34056
  • [Ya99] S. Yakovenko, On functions and curves defined by ordinary differential equations, The Arnoldfest (Toronto, ON, 1997), Amer. Math. Soc. , Providence, RI, 1999, pp. 497-525. MR 2001k:34065
  • [Z83] H. Zoladek, Versality of a family of symmetric vector fields on the plane (Russian), Mat. Sb. (N.S.) 120(162), no. 4 (1983), 473-499. MR 84i:58094
  • [Z87] H. Zoladek, Bifurcations of certain family of planar vector fields tangent to axes, Differential Equations 67, no. 1 (1987), 1-55. MR 89e:58086

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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

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

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