Kodai Mathematical Journal

A note on the Poincaré-Bendixson index theorem

Marek Izydorek, Sławomir Rybicki, and Zbigniew Szafraniec

Full-text: Open access

Article information

Source
Kodai Math. J., Volume 19, Number 2 (1996), 145-156.

Dates
First available in Project Euclid: 23 January 2006

Permanent link to this document
https://projecteuclid.org/euclid.kmj/1138043594

Digital Object Identifier
doi:10.2996/kmj/1138043594

Mathematical Reviews number (MathSciNet)
MR1397416

Zentralblatt MATH identifier
0863.34045

Subjects
Primary: 58F25
Secondary: 34C05: Location of integral curves, singular points, limit cycles

Citation

Izydorek, Marek; Rybicki, Sławomir; Szafraniec, Zbigniew. A note on the Poincaré-Bendixson index theorem. Kodai Math. J. 19 (1996), no. 2, 145--156. doi:10.2996/kmj/1138043594. https://projecteuclid.org/euclid.kmj/1138043594


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References

  • [A. F. N] AOKI, K., FUKUDA, T. AND NISHIMURA, T., On the number of branches of the zero locus of a map germ (Rn, 0) -+ (Rn~\ 0), Topology and Computer Science, ed. by S. Suzuki, Kinokumya, 1987, 347-363.
  • [ARD] ARNOLD, V. I., Index of a singular point of a vector field, the Petrovski-Oleini inequality, and mixed Hodge structures, Funct. Anal. Appl., 2 (1978), 1-11.
  • [BER1] BERLINSKII, A. N., On the number of elliptic domains adherent to a singu larity, Soviet Math. Dokl., 9 (1968), 169-173.
  • [BER2] BERLINSKII, A. N., On the structure of the neighbourhood of a singular poin of a two-dimensional autonomous system, Soviet Math. Dokl., 10 (1969), 882-885.
  • [BDX] BENDIXSON, L., Sur les courbes definies par des equations differentielles, Act Math., 24 (1913), 14-22.
  • [B. M. ] BRUNELLA, M. AND MIARI, M., Topological equivalence of a plane vecto field with its principal part defined through Newton polyhedra, J. Differential Equations, 85 (1990), 338-366.
  • [CON] CONLEY, C., Isolated invariant sets and the Morse index, CBMS-NSF Regiona Conf. Ser. in Math., 38, Amer. Math. Soc, Providence, 1978.
  • [E. L. ] EISENBUD, D. AND LEVINE, H. I., An algebraic formula for the degree of C??-map germ, Ann. of Math., 106 (1977), 19-44.
  • [HRT] HARTMAN, P., Ordinary Differntial Equations, Wiley, New York, 1964
  • [HLE] HALE, J. K., Ordinary Differential Equations, Interscience, New York, 1969
  • [I. R. ] IZYDOREK, M. AND RYBICKI, S., On the computation of the set of bifurcatio points for ordinary differential equations, J. Differential Equations, 107 (1994), 418-427.
  • [P] POINCARE, H., Memoire sur les courbes definies par une equation differentiells, " J. Math. Pura Appl., 7(1881), 375-422, 8(1882), 252-296, 1 (1885), 167-244, (1886), 151-217.
  • [PCR2] POINCARE, H., Les Methodes Nouvelles de la Mecanique Celeste, 3 vols., Gauthiers-Villars, 1892-1899
  • [PCR3] POINCARE, H., Surun theoreme de Geometrie, Rend. Circ. Mat. Palermo, 3 (1912), 375-407.
  • [SAG1] SAGALOVICH, M. E., Topological structure of the nieghbourhood of a critica point of a differential equation, Differential Equations, 11 (1975), 1498-1503.
  • [SAG2] SAGALOVICH, M. E., Classes of local topological structures of an equilibriu state, Differential Equations, 15 (1979), 253-255.
  • [S. S. I] SCHECTER, S. AND SINGER, M. F., Separatrices at singular points of plana vector fields, Acta Math., 145 (1980), 47-78.
  • [S. S. 2] SCHECTER, S. AND SINGER, M. F., Correction to Separatrices atsingular point of planar vector fields, Acta Math., 151 (1983), 297-298.
  • [SFR1] SZAFRANIEC, Z., On the number of branches of an 1-dimensional semianalytic set, Kodai Math. J., 11 (1988), 78-85.
  • [SFR2] SZAFRANIEC, Z., On the number of singular points of real projective hyper surfaces, Math. Ann., 291 (1991), 487-496.
  • [SMR] SMOLLER, J., Shock Waves and Reaction-Diffusion Equations, Springer Verlag, New York-Berlin, 1983
  • [WLL] WALL, C. T. C., Topological invariance of the Milnor number mod 2, Topology, 22 (1983), 345-350