Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: PERIODIC SOLUTIONS OF ARBITRARY LENGTH IN A SIMPLE INTEGER ITERATION | PERIODIC SOLUTIONS OF ARBITRARY LENGTH IN A SIMPLE INTEGER ITERATION DEAN CLARK Received 28 May 2005 Accepted 19 July 2005 We prove that all solutions to the nonlinear second-order difference equation in integers yn 1 fayn - yn-1 a e R a 2 a 0 1 y0 y1 e Z are periodic. The first-order system representation of this equation is shown to have self-similar and chaotic solutions in the integer plane. Copyright 2006 Dean Clark. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction We study the nonlinear second-order difference equation in integers yn 1 fayn - yn-1 a e R a 2 a 0 1 y0 y1 e Z where x denotes the smallest integer not smaller than x the ceiling function . The reader is already familiar with the linear cases a 0 1 therefore we do not consider these values in this paper. Besides the natural generalization to discrete space there are at least three reasons why is interesting. First when a 3 2 becomes 3 yn 1 2 - yn-1 - yn-1 if yn is odd if yn is even a second-order variant of the notorious 3x 1 iteration. So far as we know the ultimate convergence to 1 of the 3x 1 iterates remains an unproven conjecture. In contrast we will prove an ultimate recurrence property for for all initial states y0 y1 e Z and parameter values -2 a 2. It is the initial state that is always recurrent. Moreover solutions to can exhibit periods of arbitrary length Theorems below . Hindawi Publishing Corporation Advances in Difference Equations Volume 2006 Article ID 35847 Pages 1-9 DOI ADE 2006 35847 2 Periodic solutions of arbitrary length Second the method used to establish the periodicity of all solutions to seems novel elegant and of potentially wider applicability. Without this method we could not prove that solutions of the special case above were even bounded 1 . .
