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Global Saddles for Planar Maps

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Abstract

We study the dynamics of planar diffeomorphisms having a unique fixed point that is a hyperbolic local saddle. We obtain sufficient conditions under which the fixed point is a global saddle. We also address the special case of \(D_2\)-symmetric maps, for which we obtain a similar result for \(C^1\) homeomorphisms. Some applications to differential equations are also given.

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Acknowledgements

B. Alarcón thanks Prof. Pedro Torres for his hospitality and fruitful conversations during her stay at the University of Granada, Spain. She also thanks Prof. Christian Bonatti for giving her an example which improved her understanding of global saddles. Centro de Matemática da Universidade do Porto (CMUP — UID/MAT/00144/2013) is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020. B. Alarcón was also supported in part by CAPES from Brazil and Grants MINECO-15-MTM2014-56953-P from Spain and CNPq 474406/2013-0 from Brazil.

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Authors and Affiliations

  1. Departamento de Matemática Aplicada, Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rua Professor Marcos Waldemar de Freitas Reis, S/N, Bloco H, Campus do Gragoatá, São Domingos — Niterói, RJ, CEP 24.210-201, Brazil

    B. Alarcón

  2. Centro de Matemática da Universidade do Porto, Rua do Campo Alegre 687, 4169-007, Porto, Portugal

    S. B. S. D. Castro & I. S. Labouriau

  3. Faculdade de Economia do Porto, Rua Dr. Roberto Frias, 4200-464, Porto, Portugal

    S. B. S. D. Castro

  4. Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre 687, 4169-007, Porto, Portugal

    I. S. Labouriau

Authors
  1. B. Alarcón
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  2. S. B. S. D. Castro
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  3. I. S. Labouriau
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Corresponding author

Correspondence to I. S. Labouriau.

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Handling editor: Yingfei Yi.

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Alarcón, B., Castro, S.B.S.D. & Labouriau, I.S. Global Saddles for Planar Maps. J Dyn Diff Equat 30, 601–612 (2018). https://doi.org/10.1007/s10884-016-9561-3

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  • DOI: https://doi.org/10.1007/s10884-016-9561-3

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