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Bifurcations of umbilic points and related principal cycles. (English) Zbl 1075.53004

The paper is motivated by the theory of first order structural stability of vector fields that was initiated by A. A. Andronov and E. A. Leontovich [Theory of bifurcations of dynamical systems on a plane, ITPS, Jerusalem (1971); translation from Moscow: Nauka (1967; Zbl 0257.34001)] and further developed by the second author on manifolds in [Publ. Math., Inst. Hautes Étud. Sci. 43, 5–46 (1973; Zbl 0279.58008)], where transversality of Banach submanifolds is used.
A study of quantitative changes of bifurcations on principal configurations, under small perturbations on the immersion, which violates in a minimal fashion the Darboux structural stability condition on umbilic points, is performed.
Precise definitions and bifurcation analysis of umbilic patterns are given. Global implications on bifurcations of principal cycles, such as the appearance and the annihilation of periodic principal lines are also studied.

MSC:

53A05 Surfaces in Euclidean and related spaces
37G10 Bifurcations of singular points in dynamical systems
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
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[1] Andronov, A., and Leontovich, E. et al. (1971). Theory of bifurcations of dynamical systems on a plane, ITPS, Jerusalem.
[2] Bruce, J.W., and Fidal, D.L. (1987). On binary differential equations and umbilics. Proc. Royal Soc. Edinburgh 111A, 147-168. · Zbl 0685.34004
[3] Bruce, J.W., and Tari, F. (1995). On binary differential equations. Nonlinearity 8, 255-271. · Zbl 0830.34021
[4] Cayley, A. (1863). On differential equations and umbilici. Philos. Mag. 26, Collected Works, Vol. vi.
[5] Chicone, C. (1999). Ordinary differential equations and applications, Texts in Applied Mathematics, Vol. 34, Springer-Verlag, Berlin. · Zbl 0937.34001
[6] Chow, S., and Hale, J. (1982). Methods of bifurcation theory, Springer-Verlag, Berlin. · Zbl 0487.47039
[7] Darboux G. (1896). Sur la forme des lignes de courbure dans la voisingage d ’un ombilic. Lec ?ons sur la th?orie des surfaces, IV, Note 7, Gauthier Villars.
[8] Garcia, R. (1993). Lines of curvature near partially umbilic points of hypersurfaces of ?4, Pr?publication do Laboratoire de Topologie, Universit? de Bourgogne, Vol. 27, pp. 1-36.
[9] Garcia, R. (2001). Principal curvature lines near Darbouxian partially umbilic points of hypersurfaces immersed in ?4. Comput. Appl. Math. SBMAC 20, 121-148. · Zbl 1169.53302
[10] Garcia, R., Gutierrez, C., and Sotomayor, J. (1999). Structural stability of asymptotic lines on surfaces immersed in ?3. Bull. Sci. Math. 123, 599-622.
[11] Garcia, R., Mello, L.F., and Sotomayor, J. (2003). Principal Mean Curvature Foliations on Surfaces immersed in ?4, to appear in Proceedings Equadiff-2003, math. DS/0311215, www.arxiv.org.
[12] Garcia, R., and Sotomayor, J. (2000). Lines of axial curvature on surfaces immersed in ?4. Diff. Geom. Appl. 12, 253-269. · Zbl 0992.53010
[13] Garcia, R., Sotomayor, J. (2001). Structurally stable con gurations of lines of mean curvature and umbilic points on surfaces immersed in ?3. Publ. Mat. 45, 431-466. · Zbl 1005.53003
[14] Garcia, R., Sotomayor, J. (2002). Geometric mean curvature lines on surfaces immersed in ?3. Annales de la Facult? de Sciences de Toulouse 11, 377-401. · Zbl 1052.53008
[15] Garcia, R., Sotomayor, J. (2003). Harmonic mean curvature lines on surfaces immersed in ?3. Bull. Braz. Math. Soc. 34, 303-331. · Zbl 1079.53007
[16] Garcia, R., Sotomayor, J. (1992). Principal lines near principal cycles. Annals Global Anal. Geometry 10, 275-289. · Zbl 0768.53002
[17] Garcia, R., and Sotomayor, J. (1996). Lines of curvature on algebraic surfaces. Bull. des Sci. Math. 120, 367-395. · Zbl 0871.53007
[18] Guckenheimer, J., and Holmes, P. (1983). Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Applied Math. Sciences, Vol. 42, Springer-Verlag, Berlin. · Zbl 0515.34001
[19] Gui?ez, V., and Gutierrez, C. (1996). Positive Quadratic Differential Forms: Linearization, Finite Determinacy and Versal Unfolding. Annales de Toulouse S?rie 6, V, Fasc. 4, 661-690.
[20] Gui?ez, V., and Gutierrez, C. (2003). Rank-1 codimension one singularities of positive quadratic differential forms, Cadernos de Matem?tica, ICMC-USP 4, 87-114.
[21] Gui?ez, V. (1997). Rank two codimension 1 singularities of positive quadratic differential forms. Nonlinearity 10, 631-654. · Zbl 0908.58056
[22] Gui?ez, V. (1988). Positive quadratic differential forms and foliations with singularities on surfaces. Trans. Amer. Math. Soc. 309, 477-502. · Zbl 0707.57014
[23] Gullstrand, A. (1905). Zur Kenntiss der Kreispunkte. Acta Math. 29, 59-100. · JFM 35.0614.01
[24] Gutierrez, C., and Sotomayor, J. (1982). Stable con gurations of lines of principal curvature. Asterisque 98-99, 195-215. · Zbl 0521.53003
[25] Gutierrez, C., Sotomayor, J. (1983). An approximation theorem for immersions with stable configurations of lines of principal curvature. Springer Lect. Notes Math. 1007, 332-368. · Zbl 0528.53002
[26] Gutierrez, C., and Sotomayor, J. (1986). Closed principal lines and bifurcations. Bol. Soc. Bras. Mat. 17, 1-19. · Zbl 0668.58027
[27] Gutierrez, C., Sotomayor, J. (1991). Lines of curvature and umbilic points on surfaces, Lecture Notes, 18th Brazilian Math. Colloq, IMPA, Reprinted and updated as Structurally Stable Configurations of Lines of Curvature and Umbilic Points on Surfaces, Lima, Monogra as del IMCA, 1998.
[28] Gutierrez, C., and Sotomayor, J. (1990). Periodic lines of curvature bifurcating from Darbouxian umbilical connections. Springer Lect. Notes Math. 1455, 196-229. · Zbl 0714.34057
[29] Gutierrez, C., and Sotomayor, J. (1985). Bifurcations of lines of curvature and umbilic points. Aportaciones Math. Soc. Mat. Mex. 1, 115-126.
[30] Ilyashenko, Yu., and Li Weigu (1999). Nonlocal Bifurcations, Surveys and Monographs, Vol. 66, American Math. Soc. · Zbl 1120.37308
[31] Lang, S. (2002). Introduction to differentiable manifolds, Springer-Verlag, Berlin. · Zbl 1008.57001
[32] Mello, L.F. (2003). Mean Directionally Curved Lines on Surfaces Immersed in ?4. Publicacions Matematiques 47, 415-440. · Zbl 1069.53006
[33] Monge, G. (1796). Sur les lignes de courbure de la surface de l ’ellipsoide, J. Ecole Polytech. II cah.
[34] Palmeira, C.F. (1989). Line elds de ned by eigenspaces o derivatives of maps from the plane into itself, Curs. y Congr., Vol. 61, Univ. Santiago de Compostela, Spain. · Zbl 0697.58009
[35] Paterlini, R., and Sotomayor, J. (1987). Bifurcations of polynomial vector fields, Oscillations, Bifurcation and Chaos (Toronto, Ont. 1986), CMS Conf. Proc., Vol. 8, Amer. Math. Soc., Providence, RI, pp. 665-685.
[36] Porteous, I.R. (1994). Geometric Differentiation, Cambridge University Press. · Zbl 0806.53001
[37] Peixoto, M. (1962). Structural stability on two-dimensional manifolds. Topology 1, 101-120. · Zbl 0107.07103
[38] Roussarie, R. (1998). Bifurcations of Planar Vector Fields and Hilbert ’s Sixteen Problem, Progress in Mathematics, Vol. 164, Birkha ?user Verlag, Basel. · Zbl 0898.58039
[39] S?nchez-Bringas, F., and Ram?rez-Galarza, A. (1995). Lines of curvature near umbilical points on surfaces immersed in ?4. Ann. Global Anal. Geom. 13, 129-140. · Zbl 0836.53003
[40] Sotomayor, J. (1974). Generic one paramater families of vector fields on two dimensional manifolds. Publ. Math. IHES, 43, 5-46. · Zbl 0279.58008
[41] Spivak, M. (1980). Introduction to Comprehensive Differential Geometry, Vol. III, IV Berkeley, Publish or Perish. · Zbl 0202.52001
[42] Struik, D. (1950). Lectures on Classical Differential Geometry, Addison Wesley, Reprinted by Dover Publications Inc., 1988. · Zbl 0041.48603
[43] Thom, R. (1972). Stabilit? Structurelle et Morphogenese, Benjamin.
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