Skip to main content
SpringerLink
Log in
SpringerLink
Log in

The Qualitative Theory of Differential Equations in the Plane

  • Chapter
Mathematical Events of the Twentieth Century

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. V. I. Arnold. Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields. Funct. Anal. Appl., 1977, 11(2), 85–92.

    Article  MATH  Google Scholar 

  2. V. I. Arnold, V. S. Afraimovich, Yu. S. Il’yashenko, L. P. Shil’nikov. Bifurcation Theory and Catastrophe Theory. Berlin: Springer, 1994 (Dynamical Systems V; Encyclopædia Math. Sci., 5).

    MATH  Google Scholar 

  3. V. I. Arnold, Yu. S. Il’yashenko. Ordinary differential equations. In: Ordinary Differential Equations and Smooth Dynamical Systems. Berlin: Springer, 1988, 1–148 (Dynamical Systems I; Encyclopædia Math. Sci., 1).

    Google Scholar 

  4. A. F. Andreev. Singular Points of Differential Equations. Minsk: Vysheishaya Shkola, 1979 (Russian).

    Google Scholar 

  5. A.A. Andronov, E.A. Leontovich, I. I. Gordon, A.G. Maier. Qualitative Theory of Second-Order Dynamic Systems. New York: Halsted Press, 1973.

    MATH  Google Scholar 

  6. A.A. Andronov, E.A. Leontovich, I. I. Gordon, A.G. Maier. Theory of Bifurcations of Dynamic Systems on a Plane. New York: Halsted Press, 1973.

    Google Scholar 

  7. A.A. Andronov, L. S. Pontryagin. Structurally stable systems. Dokl. Akad Nauk SSSR, 1937, 14, 247–250 (Russian).

    MATH  Google Scholar 

  8. A.A. Andronov, A.A. Vitt, S. E. Khaikin. Theory of Oscillators. New York: Dover, 1987.

    MATH  Google Scholar 

  9. I. Bendixson. Sur les courbes définies par des équations différentielles. Acta Math., 1901, 24, 1–88.

    Article  MathSciNet  Google Scholar 

  10. A.D. Bryuno. Analytic forms of differential equations, I. Trudy Mosk Matem. Obshch., 1971, 25, 119–262 (Russian).

    MATH  Google Scholar 

  11. A.D. Bryuno. Analytic forms of differential equations, II. Trudy Mosk Matem. Obshch., 1972, 26, 199–239 (Russian).

    MATH  Google Scholar 

  12. 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. Matem Sb., 1952, 30, 181–196 (Russian); AMS Transl., 1954, 100, 19 pp.

    Google Scholar 

  13. R. I. Bogdanov. Local orbital normal forms of vector fields on the plane. Trudy Semin. im. I.G. Petrovskogo, 1979, 5, 51–84 (Russian).

    MATH  MathSciNet  Google Scholar 

  14. C. Camacho. Problems on limit sets of foliations on complex projective spaces. In: Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990). Tokyo: Math. Soc. Japan, 1991, 1235–1239.

    Google Scholar 

  15. S.-N. Chow, Ch. Zh. Li, D. Wang. Normal Forms and Bifurcation of Planar Vector Fields. Cambridge: Cambridge University Press, 1994.

    MATH  Google Scholar 

  16. S.-N. Chow, Ch. Zh. Li, Y. Yi. The cyclicity of period annuli of degenerate quadratic Hamiltonian systems with elliptic segment. Ergodic Theory Dynam. Systems, 2002, 22(2), 349–374.

    Article  MATH  MathSciNet  Google Scholar 

  17. H. Dulac. Détermination et intégration d’une certaine classe d’équations différentielles ayant pour point singulier un centre. Bull. Soc. Math. France, 1908, 32, 230–252.

    Google Scholar 

  18. H. Dulac. Sur les cycles limites. Bull. Soc. Math. France, 1923, 51, 45–188.

    MATH  MathSciNet  Google Scholar 

  19. F. Dumortier. Singularities of vector fields on the plane. Equations, 1977, 23(1), 53–106.

    Article  MATH  MathSciNet  Google Scholar 

  20. F. Dumortier, R. Roussarie, C. Rousseau. Hilbert’s 16th problem for quadratic vector fields. J. Diff. Equations, 1994, 110(1), 86–133.

    Article  MATH  MathSciNet  Google Scholar 

  21. F. Dumortier, R. Roussarie, J. Sotomayor. Bifurcations of cuspidal loops. Nonlinearity, 1997, 10(6), 1369–1408.

    Article  MATH  MathSciNet  Google Scholar 

  22. F. Dumortier, R. Roussarie, J. Sotomayor, H. Żołądek. Bifurcations of Planar Vector Fields. Nilpotent Singularities and Abelian Integrals. Berlin: Springer, 1991 (Lecture Notes in Math., 1480).

    MATH  Google Scholar 

  23. J. Écalle. Les fonctions résurgentes. Tome I. Les algèbres de fonctions résurgentes. Orsay: Université Paris-Sud, 1981.

    Google Scholar 

  24. J. Écalle. Les fonction résurgentes. Tome II. Les fonctions résurgentes appliquées à l’itération. Orsay: Université Paris-Sud, 1981.

    Google Scholar 

  25. J. Écalle. Les fonctions résurgentes. Tome III. L’équation du pont et la classification analytique des objets locaux. Orsay: Université Paris-Sud, 1985.

    Google Scholar 

  26. J. Écalle. Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac. Paris: Hermann, 1992.

    Google Scholar 

  27. L. Gavrilov, The infinitesimal 16th Hilbert problem in the quadratic case. Invent. Math., 2001, 143(3), 449–497.

    Article  MATH  MathSciNet  Google Scholar 

  28. Yu. S. Il’yashenko. The origin of limit cycles under perturbation of the equation dw/dz = −R z/R w, where R(z, w) is a polynomial. Math. USSR, Sb., 1969, 7, 353–364.

    Article  MATH  Google Scholar 

  29. Yu. S. Il’yashenko. An example of equations dw/dz = P n(z, w)/Q n(z, w) having a countable number of limit cycles and arbitrarily large Petrovskii-Landis genus. Math. USSR, Sb., 1969, 9, 365–378.

    Article  MATH  Google Scholar 

  30. Yu. S. Il’yashenko. Algebraic nonsolvability and almost algebraic solvability of the center-focus problem. Funct. Anal. Appl., 1972, 6(3), 197–202.

    Article  MATH  Google Scholar 

  31. Yu. S. Il’yashenko. Topology of phase portraits of analytic differential equations on the complex projective plane. Trudy Semin. im. I.G. Petrovskogo, 1978, 4, 83–136 (Russian).

    MathSciNet  MATH  Google Scholar 

  32. Yu. S. Il’yashenko. Singular points and limit cycles of differential equations in the real and complex plane. Preprint No. 38, Research Computing Center of the USSR Academy of Sciences. Pushchino, 1982 (Russian).

    Google Scholar 

  33. Yu. S. Il’yashenko. Dulac’s memoir ‘On limit cycles’ and related questions of the local theory of differential equations. Uspekhi Mat. Nauk, 1985, 40(6), 41–78 (Russian).

    MathSciNet  Google Scholar 

  34. Yu. S. Il’yashenko. Finiteness Theorems for Limit Cycles. Providence, RI: Amer. Math. Soc., 1991.

    MATH  Google Scholar 

  35. Yu. S. Il’yashenko, ed. Nonlinear Stokes Phenomena. Providence, RI: Amer. Math. Soc., 1993.

    MATH  Google Scholar 

  36. Yu. S. Il’yashenko. Local dynamics and nonlocal bifurcations. In: Bifurcations and Periodic Orbits of Vector Fields (Montreal, 1992). Dordrecht: Kluwer Acad. Publ., 1993, 279–319.

    Google Scholar 

  37. Yu. S. Il’yashenko. Hilbert-type numbers for Abel equations, growth and zeros of holomorphic functions. Nonlinearity, 2000, 13(4), 1337–1342.

    Article  MathSciNet  Google Scholar 

  38. Yu. S. Il’yashenko. Centennial history of Hilbert’s 16th problem. Bull. Amer Math. Soc., 2002, 39(3), 301–354.

    Article  MathSciNet  Google Scholar 

  39. Yu. S. Il’yashenko, A.G. Khovanskii. Galois groups, Stokes operators, and a theorem of Ramis. Funct. Anal. Appl., 1990, 24(4), 286–296.

    Article  MathSciNet  MATH  Google Scholar 

  40. Yu. S. Il’yashenko, J. Llibre. Restricted Hilbert problem for quadratic vector fields, to appear.

    Google Scholar 

  41. Yu. S. Il’yashenko, A. Panov. Some upper estimates of the number of limit cycles of planar vector fields with applications to the Liénard equation. Moscow Math. J., 2001, 1(4), 583–599.

    MathSciNet  Google Scholar 

  42. Yu. S. Il’yashenko, S.Yu. Yakovenko. Finitely-smooth normal forms of local families of diffeomorphisms and vector fields. Russ. Math. Surveys, 1991, 46(1), 1–43.

    Article  MathSciNet  MATH  Google Scholar 

  43. Yu. S. Il’yashenko, S.Yu. Yakovenko. Finite cyclicity of elementary polycycles in generic families. In: Concerning the Hilbert 16th Problem. Providence, RI: Amer. Math. Soc., 1995, 21–95 (AMS Transl., Ser. 2, 165).

    Google Scholar 

  44. V.Yu. Kaloshin. The existential Hilbert 16th problem and an estimate for cyclicity of elementary polycycles. Invent. Math., 2003, 151(3), 451–512.

    Article  MATH  MathSciNet  Google Scholar 

  45. A.G. Khovanskii. Real analytic varieties with the finiteness property and complex Abelian integrals. Funct. Anal. Appl., 1984, 18(2), 119–127.

    Article  MATH  MathSciNet  Google Scholar 

  46. A.G. Khovanskii. Fewnomials. Providence, RI: Amer. Math. Soc., 1991 (Transl. Math. Monographs, 88).

    MATH  Google Scholar 

  47. M.G. Khudai-Verenov. On a property of the solutions of a differential equation. Matem. Sb., 1962, 56, 301–308 (Russian).

    Google Scholar 

  48. E. I. Khorozov. Versal deformations of equivariant vector fields in the case of symmetries of order 2 and 3. Trudy Semin. im. I.G. Petrovskogo, 1979, 5, 163–192 (Russian).

    MATH  MathSciNet  Google Scholar 

  49. O. Kleban. On the order of topologically sufficient jets of the germ of a characteristic vector field in the plane. In: Concerning the Hilbert 16th Problem. Providence, RI: Amer. Math. Soc., 1995, 125–153 (AMS Transl., Ser. 2, 165).

    Google Scholar 

  50. A.Yu. Kotova, V.V. Stanzo. On few-parameter generic families of vector fields on the two-dimensional sphere. In: Concerning the Hilbert 16th Problem. Providence, RI: Amer. Math. Soc., 1995, 155–201 (AMS Transl., Ser. 2, 165).

    Google Scholar 

  51. E.A. Leontovich. On the appearance of limit cycles of the loop of a separatrix. Dokl. Akad. Nauk SSSR, 1951, 78, 641–644 (Russian).

    Google Scholar 

  52. L. Leau. Étude sur les équations fonctionelles à une ou plusieurs variables. Ann. Fac. Sci. Toulouse, 1897, 11, E1–E110.

    MathSciNet  Google Scholar 

  53. S. Lefschetz. On a theorem of Bendixson. J. Diff. Equations, 1968, 4, 66–101.

    Article  MATH  MathSciNet  Google Scholar 

  54. B. Malgrange. Travaux d’ Écalle et de Martinet-Ramis sur les systèmes dynamiques. In: Séminaire Bourbaki. Astérisque, 1982, 92–93, 59–73.

    MathSciNet  Google Scholar 

  55. P. Mardešić. Chebyshev Systems and the Versal Unfoldings of the Cusp of Order n. Paris: Hermann, 1998.

    Google Scholar 

  56. J.-F. Mattei, R. Moussu. Holonomie et intégrales premières. Ann. Sci. École Norm. Supér. (4), 1980, 13(4), 469–523.

    MATH  MathSciNet  Google Scholar 

  57. N.B. Medvedeva, E.V. Mazaeva. A sufficient focus condition for a monodromic singular point. Trudy Mosk. Matem. Obshch., 2002, 77–103 (Russian).

    Google Scholar 

  58. J. Martinet, J.-P. Ramis. Problèmes de modules pour des équations différentielles non linéaires du premier ordre. Publ. Math. IHES, 1982, 55, 63–164.

    MATH  MathSciNet  Google Scholar 

  59. J. Martinet, J.-P. Ramis. Classification analytique des équations différentielles non-linéaires résonantes du premier ordre. Ann. Sci. École Norm. Supér. (4), 1983, 16(4), 571–621.

    MATH  MathSciNet  Google Scholar 

  60. I. Nakai. Separatrices for nonsolvable dynamics on (ℂ,0). Ann. Inst. Fourier (Grenoble), 1994, 44(2), 569–599.

    MATH  MathSciNet  Google Scholar 

  61. D. I. Novikov, S.Yu. Yakovenko. Trajectories of polynomial vector fields and ascending chains of polynomial ideals. Ann. Inst. Fourier (Grenoble), 1999, 49(2), 563–609.

    MATH  MathSciNet  Google Scholar 

  62. D. I. Novikov, S.Yu. Yakovenko. Redundant Picard-Fuchs systems for Abelian integrals. J. Diff. Equations, 2001, 177(2), 267–306.

    Article  MATH  MathSciNet  Google Scholar 

  63. H. Poincaré. Sur les propriétés des fonctions définies par les équations aux différences partielles. Thèse, 1879.

    Google Scholar 

  64. H. Poincaré. Memoire sur les courbes définies par une équation différentielle. J. Math. Pures et Appl., 1881, 7, 251–296.

    Google Scholar 

  65. R. Perez-Marco. Fixed points and circle maps. Acta Math., 1997, 179(2), 243–294.

    Article  MATH  MathSciNet  Google Scholar 

  66. G. S. Petrov. Elliptic integrals and their nonoscillation. Funct. Anal. Appl., 1986, 20(1), 37–40.

    Article  MATH  Google Scholar 

  67. I.G. Petrovskii, 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 degree 2. Matem. Sb., 1955, 37, 209–250 (Russian).

    MATH  MathSciNet  Google Scholar 

  68. I.G. Petrovskii, 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 degree 2. Matem. Sb., 1957, 85, 149–168 (Russian).

    MATH  MathSciNet  Google Scholar 

  69. L. S. Pontryagin. On dynamical systems that are close to Hamiltonian. Zh. Eksp. Teoret. Fiz., 1934, 4(8), 234–238 (Russian).

    Google Scholar 

  70. I.A. Pushkar’. Multidimensional generalization of the Il’yashenko theorem on Abelian integrals. Funct. Anal. Appl., 1997, 31(2), 100–108.

    Article  MathSciNet  Google Scholar 

  71. A. S. Pyartli. Birth of complex invariant manifolds close to a singular point of a parametrically dependent vector field. Funct. Anal. Appl., 1972, 6(4), 339–340.

    Article  MATH  Google Scholar 

  72. A. S. Pyartli. Cycles of a system of two complex differential equations in a neighborhood of a fixed point. Trudy Mosk. Matem. Obshch., 1978, 37, 95–106 (Russian).

    MATH  MathSciNet  Google Scholar 

  73. R. Roussarie, On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields. Bol. Soc. Brasil Mat., 1986, 17(2), 67–101.

    MATH  MathSciNet  Google Scholar 

  74. J.-P. Ramis, Y. Sibuya. Hukuhara domains and fundamental existence and uniqueness theorems for asymptotic solutions of G every type. Asymptotic Anal., 1989, 2(1), 39–94.

    MATH  MathSciNet  Google Scholar 

  75. S. Smale. Mathematical problems for the next century. Math. Intelligencer, 1998, 20(2), 7–15.

    Article  MATH  MathSciNet  Google Scholar 

  76. A. Seidenberg. Reduction of singularities of the differential equation Adx = Bdy. Amer. J. Math., 1968, 90, 248–269.

    Article  MATH  MathSciNet  Google Scholar 

  77. A.A. Shcherbakov. Topological and analytic conjugacy of noncommutative groups of germs of conformal mappings. Trudy Semin. im. I.G. Petrovskogo, 1984, 10, 170–196 (Russian).

    MATH  Google Scholar 

  78. C. L. Siegel. Iteration of analytic functions. Ann. Math., 1942, 43(2), 607–612.

    MATH  Google Scholar 

  79. A.A. Shcherbakov, E. Rosales-González, L. Ortiz-Bobadilla. Countable set of limit cycles for the equation dw/dz= Pn(z,w)/Qn(z,w). J. Dynam. Control Systems, 1998, 4(4), 539–581.

    Article  MATH  MathSciNet  Google Scholar 

  80. A.N. Varchenko. Estimate of the number of zeros of an Abelian integral depending on a parameter and limit cycles. Funct. Anal. Appl., 1984, 18(2), 98–108.

    Article  MATH  MathSciNet  Google Scholar 

  81. S.M. Voronin. Analytic classification of germs of conformal mappings (ℂ,0)→(ℂ,0). Funct. Anal. Appl., 1981, 15(1), 1–13.

    Article  MATH  MathSciNet  Google Scholar 

  82. J.-C. Yoccoz. Théorème de Siegel, nombres de Bruno et polynômes quadratiques. Petits diviseurs en dimension 1. Astérisque, 1995, 231, 3–88.

    MathSciNet  Google Scholar 

  83. H. Żołądek. Versality of a family of symmetric vector fields on the plane. Matem. Sb., 1983, 120(4), 473–499 (Russian).

    Google Scholar 

  84. H. Żołądek. Bifurcations of certain family of planar vector fields tangent to axes. Diff. Equations, 1987, 67(1), 1–55.

    Article  MATH  Google Scholar 

Download references

Authors
  1. Yu. S. Il’yashenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Academy of Sciences, Leninsky Pr. 14, 117901, Moscow, Russia

    Yu. S. Osipov

  2. Department of Mathematics, University of Princeton, Washington Road, 08544-1000, Princeton, USA

    Ya. G. Sinai

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Il’yashenko, Y.S. (2006). The Qualitative Theory of Differential Equations in the Plane. In: Bolibruch, †.A.A., et al. Mathematical Events of the Twentieth Century. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-29462-7_6

Download citation

Publish with us

Policies and ethics

Search

Navigation