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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
Article
  • 128 Downloads

Abstract

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.

Keywords

Accumulation of secants Ordinary differential equations Real analytic geometry 

Mathematics Subject Classification (2010)

34Cxx 14Pxx 

Notes

Acknowledgments

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.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.University of TorontoTorontoCanada

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