Some applications of the Euler-Jacobi formula to differential equations
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- by Anna Cima, Armengol Gasull and Francesc Mañosas
- Proc. Amer. Math. Soc. 118 (1993), 151-163
- DOI: https://doi.org/10.1090/S0002-9939-1993-1150647-9
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Abstract:
The Euler-Jacobi formula gives an algebraic relation between the critical points of a vector field and their indices. Using this formula we obtain an upper bound for the number of centers that a planar polynomial differential equation can have and study the distribution of the critical points for planar quadratic and cubic differential equations.References
- V. Arnold, A. Varchenko, and S. Goussein-Zude, Singularités des applications différentiables, Mir, Moscow, 1982.
- A. N. Berlinskiĭ, On the behavior of the integral curves of a differential equation, Izv. Vysš. Učebn. Zaved. Matematika 1960 (1960), no. 2 (15), 3–18 (Russian). MR 0132249 E. Brieskorn and H. Knörrer, Plane algebraic curves, Birkhäuser, Basel, Boston, and Stuttgart, 1986.
- W. A. Coppel, A survey of quadratic systems, J. Differential Equations 2 (1966), 293–304. MR 196182, DOI 10.1016/0022-0396(66)90070-2
- Carmen Chicone and Jing Huang Tian, On general properties of quadratic systems, Amer. Math. Monthly 89 (1982), no. 3, 167–178. MR 645790, DOI 10.2307/2320199
- Anna Cima and Jaume Llibre, Configurations of fans and nests of limit cycles for polynomial vector fields in the plane, J. Differential Equations 82 (1989), no. 1, 71–97. MR 1023302, DOI 10.1016/0022-0396(89)90168-X A. Cima, A. Gasull, and F. Mañosas, On Hamiltonian planar vector fields, J. Differential Equations (to appear).
- William Fulton, Algebraic curves. An introduction to algebraic geometry, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes written with the collaboration of Richard Weiss. MR 0313252
- A. G. Hovanskiĭ, The index of a polynomial vector field, Funktsional. Anal. i Prilozhen. 13 (1979), no. 1, 49–58, 96 (Russian). MR 527521
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725 R. E. Kooij, A cubic system with $(4, - 3,2)$ configuration, preprint, Univ. Tech. of Delft, 1991.
- Andrés Sestier, Note on a theorem of Berlinskiĭ, Proc. Amer. Math. Soc. 78 (1980), no. 3, 358–360. MR 553376, DOI 10.1090/S0002-9939-1980-0553376-6
- Yan Qian Ye and Wei Yin Ye, A generalization of Berlinskiĭ’s theorem to cubic and quartic differential system, Ann. Differential Equations 4 (1988), no. 4, 503–509. MR 977808
Bibliographic Information
- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 118 (1993), 151-163
- MSC: Primary 58F21; Secondary 34C05
- DOI: https://doi.org/10.1090/S0002-9939-1993-1150647-9
- MathSciNet review: 1150647