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Different Approaches to the Global Periodicity Problem

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Book cover Difference Equations, Discrete Dynamical Systems and Applications (ICDEA 2012)

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Abstract

Let F be a real or complex n-dimensional map. It is said that F is globally periodic if there exists some \(p\in \mathbb {N}^+\) such that \(F^p(x)=x\) for all x,  where \(F^k=F\circ F^{k-1}\), \(k\ge 2.\) The minimal p satisfying this property is called the period of F. Given a m-dimensional parametric family of maps, say \(F_\lambda \), a problem of current interest is to determine all the values of \(\lambda \) such that \(F_\lambda \) is globally periodic, together with their corresponding periods. The aim of this paper is to show some techniques that we use to face this question, as well as some recent results that we have obtained. We will focus on proving the equivalence of the problem with the complete integrability of the dynamical system induced by the map F, and related issues; on the use of the local linearization given by the Bochner Theorem; and on the use the Normal Form theory. We also present some open questions in this setting.

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Acknowledgments

The authors are partially supported by Spanish Ministry of Economy and Competitiveness through grants MTM2008-03437 (first and second authors); DPI2011-25822 and DPI2016-77407-P (third author); and MTM2011-26995-C02-01 (fourth author). Both CoDALab and GSD-UAB groups are supported by Generalitat de Catalunya through the SGR program.

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Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Barcelona, Spain

    Anna Cima, Armengol Gasull & Francesc Mañosas

  2. Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain

    Víctor Mañosa

Authors
  1. Anna Cima
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  2. Armengol Gasull
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  3. Víctor Mañosa
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  4. Francesc Mañosas
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Corresponding author

Correspondence to Armengol Gasull .

Editor information

Editors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Cerdanyola del Vallès, Barcelona, Spain

    Lluís Alsedà i Soler

  2. Department of Mathematics, University of Arizona, Tucson, Arizona, USA

    Jim M. Cushing

  3. Department of Mathematics, Trinity University, San Antonio, Texas, USA

    Saber Elaydi

  4. Faculty of Sciences, Department of Mathematics, University of Porto, Porto, Portugal

    Alberto A. Pinto

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Cima, A., Gasull, A., Mañosa, V., Mañosas, F. (2016). Different Approaches to the Global Periodicity Problem. In: Alsedà i Soler, L., Cushing, J., Elaydi, S., Pinto, A. (eds) Difference Equations, Discrete Dynamical Systems and Applications. ICDEA 2012. Springer Proceedings in Mathematics & Statistics, vol 180. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-52927-0_7

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