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Periodic solutions of arbitrary length in a simple integer iteration
Advances in Difference Equations volume 2006, Article number: 035847 (2006)
Abstract
We prove that all solutions to the nonlinear second-order difference equation in integers yn+1 = ⌈ay n ⌉-yn-1, {a ∈ ℝ:|a|<2, a≠0,±1}, y0, y1 ∈ ℤ, are periodic. The first-order system representation of this equation is shown to have self-similar and chaotic solutions in the integer plane.
References
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Niven I: Irrational Numbers, The Carus Mathematical Monographs, no. 11. The Mathematical Association of America. Distributed by John Wiley & Sons, New York; 1956.
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Clark, D. Periodic solutions of arbitrary length in a simple integer iteration. Adv Differ Equ 2006, 035847 (2006). https://doi.org/10.1155/ADE/2006/35847
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DOI: https://doi.org/10.1155/ADE/2006/35847