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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Sharkovskii's theorem ： ウィキペディア英語版 Sharkovskii's theorem In mathematics, Sharkovskii's theorem, named after Oleksandr Mikolaiovich Sharkovsky who published it in 1964, is a result about discrete dynamical systems.〔 *A.N. Sharkovskii, ''Co-existence of cycles of a continuous mapping of the line into itself'', Ukrainian Math. J., 16:61-71 (1964).〕 One of the implications of the theorem is that if a discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period. ==The theorem== For some interval $I\backslash subset\; \backslash mathbb$, suppose :$f\; :\; I\; \backslash to\; I$ is a continuous function. We say that the number ''x'' is a ''periodic point of period m'' if ''f'' ''m''(''x'') = ''x'' (where ''f'' ''m'' denotes the composition of ''m'' copies of ''f'') and having ''least period m'' if furthermore ''f'' ''k''(''x'') ≠ ''x'' for all 0 < ''k'' < ''m''. We are interested in the possible periods of periodic points of ''f''. Consider the following ordering of the positive integers: :$\backslash begin$ 3 & 5 & 7 & 9 & 11 & \ldots & (2n+1)\cdot2^ & \ldots\\ 3\cdot2 & 5\cdot2 & 7\cdot2 & 9\cdot2 & 11\cdot2 & \ldots & (2n+1)\cdot2^ & \ldots\\ 3\cdot2^ & 5\cdot2^ & 7\cdot2^ & 9\cdot2^ & 11\cdot2^ & \ldots & (2n+1)\cdot2^ & \ldots\\ 3\cdot2^ & 5\cdot2^ & 7\cdot2^ & 9\cdot2^ & 11\cdot2^ & \ldots & (2n+1)\cdot2^ & \ldots\\ & \vdots\\ \ldots & 2^ & \ldots & 2^ & 2^ & 2^ & 2 & 1\end We start, that is, with the odd numbers in increasing order, then 2 times the odds, 4 times the odds, 8 times the odds, etc., and at the end we put the powers of two in decreasing order. Every positive integer appears exactly once somewhere on this list. Note that this ordering is not a well-ordering. Sharkovskii's theorem states that if ''f'' has a periodic point of least period ''m'' and ''m'' precedes ''n'' in the above ordering, then ''f'' has also a periodic point of least period ''n''. As a consequence, we see that if ''f'' has only finitely many periodic points, then they must all have periods which are powers of two. Furthermore, if there is a periodic point of period three, then there are periodic points of all other periods. Sharkovskii's theorem does not state that there are ''stable'' cycles of those periods, just that there are cycles of those periods. For systems such as the logistic map, the bifurcation diagram shows a range of parameter values for which apparently the only cycle has period 3. In fact, there must be cycles of all periods there, but they are not stable and therefore not visible on the computer generated picture. The assumption of continuity is important, as the discontinuous function $f\; :\; x\; \backslash mapsto\; (1-x)^$, for which every non-zero value has period 3, would otherwise be a counterexample. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Sharkovskii's theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.