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Results and Conjectures about Order Lyness' Difference Equation in , with a Particular Study of the Case

Abstract

We study order Lyness' difference equation in , with and the associated dynamical system in . We study its solutions (divergence, permanency, local stability of the equilibrium). We prove some results, about the first three invariant functions and the topological nature of the corresponding invariant sets, about the differential at the equilibrium, about the role of 2-periodic points when is odd, about the nonexistence of some minimal periods, and so forth and discuss some problems, related to the search of common period to all solutions, or to the second and third invariants. We look at the case with new methods using new invariants for the map and state some conjectures on the associated dynamical system in in more general cases.

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Correspondence to G. Bastien.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Bastien, G., Rogalski, M. Results and Conjectures about Order Lyness' Difference Equation in , with a Particular Study of the Case . Adv Differ Equ 2009, 134749 (2009). https://doi.org/10.1155/2009/134749

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Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis
  • Functional Equation