Abstract
In this paper we give a description of the sets of accumulation of secants for orbits of real analytic vector fields in dimension three with the origin as only ω-limit point. It is an infinitesimal version of the Poincaré-Bendixson problem in dimension three. These sets have structure of cyclic graph when the singularities are isolated under one blow-up. If the reduction of singularities is hyperbolic, under conditions of Morse-Smale type, we prove that the accumulation set is a single point or homeomorphic to \({\mathbb{S}}^{1}\).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Alonso-González, C., Cano, F., Rosas, R.: Infinitesimal Poincaré-Bendixson problem in dimension three. ArXiv:1212.2134 [math.DS]
Alonso-González, C., Camacho, M.I., Cano, F.: Topological classification of multiple saddle connections. Discret. Contin. Dyn. Syst. 15, 395–414 (2006)
Alonso-González, C., Camacho, M.I., Cano, F.: Topological invariants for singularities of real vector fields in dimension three. Discret. Contin. Dyn. Syst. 20, 275–291 (2008)
Belotto, A.: Analytic varieties as limit periodic sets. ArXiv: 1109.0877v1 [math.DS]
Camacho, M.I.: A contribution to the topological classification of homogeneus vector fields. J. Differ. Equ. 57, 159–171 (1983)
Camacho, C., Cano, F., Sad, P.: Desingularization of absolutely isolated singularities of vector fields. Invent. Math. 98, 351–369 (1989)
Cano, F., Moussu, R., Sanz, F.: Oscillation, spiralement, tourbillonnement. Comment. Math. Helv. 75, 284–318 (2000)
Kurdyka, K., Mostowski, T., Parusinsky, A.: Proof of the gradient conjecture of R. Thom. Ann. Math. 152(2), 763–792 (2000)
Palis, J., de Melo, W.: Geometric Theory of Dynamical Systems. Springer, New York (1982)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Alonso-González, C., Cano, F., Rosas, R. (2013). Secants of Trajectories in Dimension Three. In: Ibáñez, S., Pérez del Río, J., Pumariño, A., Rodríguez, J. (eds) Progress and Challenges in Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol 54. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38830-9_1
Download citation
DOI: https://doi.org/10.1007/978-3-642-38830-9_1
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38829-3
Online ISBN: 978-3-642-38830-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)