Geometric properties of homogeneous vector fields of degree two in 𝑅³

Camacho, M. Izabel T.

doi:10.1090/S0002-9947-1981-0628447-1

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 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

A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maĭer, 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
José Argémi, Sur les points singuliers multiples de systèmes dynamiques dans $R^{2}$, Ann. Mat. Pura Appl. (4) 79 (1968), 35–69 (French). MR 235199, DOI 10.1007/BF02415178
Ivar Bendixson, Sur les courbes définies par des équations différentielles, Acta Math. 24 (1901), no. 1, 1–88 (French). MR 1554923, DOI 10.1007/BF02403068
Courtney Coleman, A certain class of integral curves in 3-space, Ann. of Math. (2) 69 (1959), 678–685. MR 104885, DOI 10.2307/1970031
David Hilbert, Mathematical problems, Bull. Amer. Math. Soc. 8 (1902), no. 10, 437–479. 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. 185–213. 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. 223–231. MR 0267603

Hilbert’s 16th problem: Limit cycles of polynomial vector fields in the plane

Classification of integral manifolds of a homogeneous three-dimensional system according to the structure of limit sets

7

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. 605–640. MR 0455046