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Examples of Infinitesimal Nontrivial Accumulation of Secants in Dimension Three

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.

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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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Correspondence to André Belotto.

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Belotto, A. Examples of Infinitesimal Nontrivial Accumulation of Secants in Dimension Three. J Dyn Control Syst 22, 45–54 (2016). https://doi.org/10.1007/s10883-014-9248-1

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Keywords

  • Accumulation of secants
  • Ordinary differential equations
  • Real analytic geometry

Mathematics Subject Classification (2010)

  • 34Cxx
  • 14Pxx