Positive Quadratic Differential Forms: Topological Equivalence Through Newton Polyhedra
Tóm tắt
The purpose of this article is to establish conditions under which a positive quadratic differential form is topologically equivalent to its principal part defined by Newton polyhedra. The problem is to study the simultaneous behavior of two foliations in the plane having a common point as a singularity.
Tài liệu tham khảo
1. S. Kh. Aranson, V. Z. Grines, and V. A. Kaimanovich, Classification of supertransitive 2-webs on surfaces. J. Dynam. Control Systems 9 (2003), No. 4, 455–468.
2. S. Kh. Aranson, Flows on closed surfaces and behavior of trajectories lifted to the universal covering plane. J. Dynam. Control Systems 1 (1995) No. 1, 125–138.
3. J. W. Bruce and D. L. Fidal, On binary differential equation and umbilics. Proc. Roy. Soc. Edinburgh Sec. A 111 (1989), 147–168.
4. J. W. Bruce and F. Tari, On binary equations. Nonlinearity 8 (1995), 255–271.
5. M. Brunella, Complete polynomial vector fields on the complex plane. Topology 43 (2004), No. 2, 247–496.
6. M. Brunella and M. Miari, Topological equivalence of a plane vector field with its principal part defined through Newton polyhedra. J. Differential Equations 85 (1990), 338–366.
7. C. Christopher, P. Marsedic, and C. Rousseau, Normalizable, integrable and linearizable saddle points for complex quadratic systems in C 2. J. Dynam. Control Systems 9 (2003), No. 3 311–363.
8. A. A. Davydov, Qualitative theory of control systems. Transl. Math. Monogr. 141 (1994).
9. F. Dumortier, Singularities of vector fields on the plane. J. Differential Equations 23 (1977), No. 1, 53–106.
10. R. Garcia and J. Sotomayor, Lines of curvature near singular points of implicit surfaces. Bull. Sci. Math. 117 (1993), No. 3, 313–331.
11. I. Guadalupe, V. Guíñez, C. Gutierrez, and R. Tribuzy, Lines of curvature on surfaces immersed in R4. Bol. Soc. Brasil. Mat. 28 (1997) No. 2, 233–251.
12. V. Guíñez, Positive quadratic differential forms and foliations with singularities on surfaces. Trans. Amer. Math. Soc. 309 (1988), No. 2, 477–502.
13. _______, Locally stable singularities for PQD forms. J. Differential Equations 110 (1994), 1–37.
14. _______, Rank 2 codimension 1 singularities of PQD forms. Nonlinearity 10 (1997), 631–654.
15. P. Hartman, Ordinary differential equations. Birkhäuser, Boston (1982).
16. P. Hartman and A. Wintner, On the singularities in nets of curves defined by differential equations. Amer. J. Math. 75 (1953), 277–297.
17. A. G. Kuzmin, Nonclassical equations of mixed type and their applications in gas dynamics. Int. Ser. Numer. Math. 109 (1992).
18. M.-F. Michel, Formes normales de certaines #x00E9;quations différentielles quadratiques. C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), No. 3, 283–286.
19. R. D. S. Oliveira, Families of pairs of Hamiltonian vector fields in the plane. Contemp. Math. 354 (2004).
20. R. D. S. Oliveira and F. Tari, On pairs of foliations in the plane. Discrete Contin. Dynam. Systems 6 (2000), No. 3, 519–536.
21. J. Sotomayor and C. Gutierrez, Structural stable configurations of lines of principal curvature. Astérisque (1982), 98–99.
22. _______, An approximation theorem for immersions with structural stable configurations of lines of principal curvature. Lect. Notes Math. 1007, Springer-Verlag (1983).
23. _______, Configurations of lines of principal curvature and their bifurcations. Aportaciones Mat. Notas Investigación 1 (1985), 115–126.
24. _______, Principal lines on surfaces immersed with constant mean curvature. Trans. Amer. Math. Soc. 293 (1986), No. 2, 751–766.
25. _______, Lines of principal curvature on surfaces with Whitney umbrella singularities. Tohoku Math. J. 38 (1986), No. 4, 551–559.
26. _______, Closed lines of curvature and bifurcation. Bol. Soc. Brasil. Mat. 17 (1986), No. 1, 1–19.
27. _______, Periodics lines of curvature bifurcating from Darbouxian umbilical connections. Lect. Notes Math. 1455, Springer-Verlag (1990).
28. J. Sotomayor and M. Zhitomirskii, On pairs of foliations defined by vector fields in the plane. Discrete Contin. Dynam. Systems 6 (2000), No. 3, 741–749.