Secants of Trajectories in Dimension Three

  • C. Alonso-GonzálezEmail author
  • F. Cano
  • R. Rosas
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 54)


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 \({\mathbb{S}}^{1}\).


Vector Field Periodic Orbit Singular Point Exceptional Divisor Weight Transition 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Universidad de AlicanteAlicanteSpain
  2. 2.University of ValladolidValladolidSpain
  3. 3.Pontificia Universidad Católica del PerúLimaPeru

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