Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Geometric properties of homogeneous vector fields of degree two in $ {\bf R}\sp{3}$

Author: M. Izabel T. Camacho
Journal: Trans. Amer. Math. Soc. 268 (1981), 79-101
MSC: Primary 58F09; Secondary 34D30
MathSciNet review: 628447
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 Poincaré 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 [Enhancements On Off] (What's this?)

  • [1] A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maier, Theory of bifurcations of dynamic systems on a plane, Program for Scientific Translations, Jerusalem, 1971, pp. 377-401. MR 0344606 (49:9345)
  • [2] J. Argemi, Sur les points singuliers multiples de systèmes dynamiques dans $ {R^2}$, Ann. Mat. Pura Appl. (4) 79 (1968), 35-69. MR 0235199 (38:3510)
  • [3] I. Bendixson, Sur les courbes définies par des équations différentielles, Acta Math. Scand. 24 (1901), 1-88. MR 1554923
  • [4] C. Coleman, A certain class of integral curves in $ 3$-space, Ann. of Math. (2) 69 (1959), 678-685. MR 0104885 (21:3636)
  • [5] D. Hilbert, Mathematical problems, Bull. Amer. Math. Soc. 8 (1902), 437-479. MR 1557926
  • [6] L. Markus, Quadratic differential equations and non-associative algebras, Contributions to the Theory of Non-linear Oscillations, vol. 5, (S. Lefschetz, editor), Princeton, Univ. Press, Princeton, N. J., 1960, pp. 185-213. MR 0132743 (24:A2580)
  • [7] J. Palis and S. Smale, Structural stability theorems, Global Analysis, Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, R. I., 1970, pp. 223-231. MR 0267603 (42:2505)
  • [8] C. Pugh, Hilbert'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.
  • [9] Sh. R. Sharipov, Classification of integral manifolds of a homogeneous three-dimensional system according to the structure of limit sets, Differencial'nye Uravnenija 7 (1971), 355-363.
  • [10] G. Tavares, Classification of generic quadratic vector fields with no limit cycles, Geometry and Topology, III, Lecture Notes in Math., Springer-Verlag, Berlin and New York, 1977, pp. 605-640. MR 0455046 (56:13287)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F09, 34D30

Retrieve articles in all journals with MSC: 58F09, 34D30

Additional Information

Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society