Journal of Dynamical and Control Systems

, Volume 22, Issue 1, pp 45–54 | Cite as

Examples of Infinitesimal Nontrivial Accumulation of Secants in Dimension Three

  • André Belotto


We present new examples of accumulation of secants for orbits (of a real analytic three-dimensional vector fields) having the origin as only ω-limit point. These new examples have the structure of a proper algebraic variety of \(\mathbb {S}^{2}\) intersected with a cone. In particular, we present explicit examples of accumulation of secants sets which are not in the list of possibilities of the classical Poincaréé-Bendixson Theorem.


Accumulation of secants Ordinary differential equations Real analytic geometry 

Mathematics Subject Classification (2010)

34Cxx 14Pxx 



I would like to thank my thesis advisor, Professor Daniel Panazzolo, for the useful discussions, suggestions and revision of the manuscript. I would also like to thank Professor Felipe Cano for bringing the problem to my attention and Professor Patrick Speisseger for a very useful discussion. Finally, I would like to express my gratitude to the anonymous reviewer for all the suggestions and for indicating a shorter way of proving Proposition 3.1. This work was supported by the Universit´e de Haute-Alsace.


  1. 1.
    Belotto A. Analytic varieties as limit periodic sets. Qualitative Theory of Dynamical Systems 2012;11 (2): 449–465.CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Bierstone E, Milman PD. Semianalytic and subanalytic sets. Inst Hautes Études Sci Publ Math No. 1988;67:5–42.Google Scholar
  3. 3.
    Cano F, Moussu R, Rolin JP. Non-oscillating integral curves and valuations. J Reine Angew Math 2005;582:107–141.CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Cano F, Moussu R, Sanz F. Nonoscillating projections for trajectories of vector fields. J Dyn Control Syst 2007;13(2):173–176.CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Cano F, Moussu R, Sanz F. Oscillation, spiralement, tourbillonnement. Comment Math Helv 2000;75:284–318.CrossRefMathSciNetGoogle Scholar
  6. 6.
    Cano F, Moussu R, Sanz F. Pinceaux de courbes intégrales d’un champ de vecteurs analytique. Astérisque 2004;297:1–34.MathSciNetGoogle Scholar
  7. 7.
    Coste M. In: Colliot-Thélène JE, Coste ME, Mahé LE, Roy M-F, editors. Ensembles semi-algebriques, Géomtrie Algébrique Réelle et Formes Quadratiques. Berlin Heidelberg: Springer; 1982, pp. 109–138.Google Scholar
  8. 8.
    Alonso-Gonzlez C, Cano F, Rosas R. Infinitesimal Poincaré-Bendixson Problem in dimension 3. arXiv:1212.2134[math.DS].
  9. 9.
    Lefschetz S. Differential equations: geometric theory. New York: Dover Publ.; 1977.Google Scholar
  10. 10.
    Lyapunov AM. Stability of motion.Google Scholar
  11. 11.
    Poincaré H. Mémoire sur les courbes définies par une équation différentielle. (I) Journal de mathématiques pures et appliquées 3e série, tome 7 (1881), p. 375-422.Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.University of TorontoTorontoCanada

Personalised recommendations