## Some applications of the Euler-Jacobi formula to differential equations

HTML articles powered by AMS MathViewer

- 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
- PDF | Request permission

## 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,
- 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, - 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, - 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, - 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**

*Singularités des applications différentiables*, Mir, Moscow, 1982.

*Plane algebraic curves*, Birkhäuser, Basel, Boston, and Stuttgart, 1986.

*On Hamiltonian planar vector fields*, J. Differential Equations (to appear).

*A cubic system with*$(4, - 3,2)$

*configuration*, preprint, Univ. Tech. of Delft, 1991.

## 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