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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)

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

Keywords

Vector Field Periodic Orbit Singular Point Exceptional Divisor Weight Transition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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