## Geometric properties of homogeneous vector fields of degree two in $\textbf {R}^{3}$

HTML articles powered by AMS MathViewer

- by M. Izabel T. Camacho PDF
- Trans. Amer. Math. Soc.
**268**(1981), 79-101 Request permission

## Abstract:

In the space of homogeneous polynomial vector fields of degree two, those that project on Morse-Smale vector fields on ${S^2}$ by the PoincarĂ© central projection form a generic subset. The classification of those vector fields on ${S^2}$ without periodic orbits is given and applications to the study of local actions of the affine group of the line are derived.## References

- A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. MaÄer,
*Theory of bifurcations of dynamic systems on a plane*, Halsted Press [John Wiley & Sons], New York-Toronto; Israel Program for Scientific Translations, Jerusalem-London, 1973. Translated from the Russian. MR**0344606** - JosĂ© ArgĂ©mi,
*Sur les points singuliers multiples de systĂ¨mes dynamiques dans $R^{2}$*, Ann. Mat. Pura Appl. (4)**79**(1968), 35â€“69 (French). MR**235199**, DOI 10.1007/BF02415178 - Ivar Bendixson,
*Sur les courbes dĂ©finies par des Ă©quations diffĂ©rentielles*, Acta Math.**24**(1901), no.Â 1, 1â€“88 (French). MR**1554923**, DOI 10.1007/BF02403068 - Courtney Coleman,
*A certain class of integral curves in 3-space*, Ann. of Math. (2)**69**(1959), 678â€“685. MR**104885**, DOI 10.2307/1970031 - David Hilbert,
*Mathematical problems*, Bull. Amer. Math. Soc.**8**(1902), no.Â 10, 437â€“479. MR**1557926**, DOI 10.1090/S0002-9904-1902-00923-3 - Lawrence Markus,
*Quadratic differential equations and non-associative algebras*, Contributions to the theory of nonlinear oscillations, Vol. V, Princeton Univ. Press, Princeton, N.J., 1960, pp.Â 185â€“213. MR**0132743** - J. Palis and S. Smale,
*Structural stability theorems*, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp.Â 223â€“231. MR**0267603**
C. Pugh, - Geovan Tavares dos Santos,
*Classification of generic quadratic vector fields with no limit cycles*, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976) Lecture Notes in Math., Vol. 597, Springer, Berlin, 1977, pp.Â 605â€“640. MR**0455046**

*Hilbertâ€™s 16th problem: Limit cycles of polynomial vector fields in the plane*, Dynamical Systems, Lecture Notes in Math., vol. 468, Springer-Verlag, Berlin and New York, 1975, pp. 55-57. Sh. R. Sharipov,

*Classification of integral manifolds of a homogeneous three-dimensional system according to the structure of limit sets*, Differencialâ€™nye Uravnenija

**7**(1971), 355-363.

## Additional Information

- © Copyright 1981 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**268**(1981), 79-101 - MSC: Primary 58F09; Secondary 34D30
- DOI: https://doi.org/10.1090/S0002-9947-1981-0628447-1
- MathSciNet review: 628447