翻訳と辞書 Words near each other ・ Complex Projects Contract ・ Complex quadratic polynomial ・ Complex question ・ Complex reflection group ・ Complex regional pain syndrome ・ Complex representation ・ Complex response ・ Complex sales ・ Complex seeing ・ Complex segregation analysis ・ Complex Simplicity ・ Complex society ・ Complex space ・ Complex spacetime ・ Complex Sportif Hérémakono ・ Complex squaring map ・ Complex structure ・ Complex structure theory in English law ・ Complex structured finance transactions ・ Complex system ・ Complex systems ・ Complex Systems (journal) ・ Complex systems biology ・ Complex text layout ・ Complex torus ・ Complex training ・ Complex variables ・ Complex vector bundle ・ Complex vertebral malformation ・ Complex volcano Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Complex squaring map ： ウィキペディア英語版 Complex squaring map In mathematics, the complex squaring map, a polynomial mapping of degree two, is a simple and accessible demonstration of chaos in dynamical systems. It can be constructed by performing the following steps: # Choose any complex number on the unit circle whose argument (complex angle) is not a rational fraction of π, # Repeatedly square that number. This repetition (iteration) produces a sequence of complex numbers that can be described alone by their complex angle. Any choice of starting angle that satisfies (1) above will produce an extremely complicated sequence of angles, that belies the simplicity of the steps. It can be shown that the sequence will be chaotic, i.e. it is sensitive to the detailed choice of starting angle. == Chaos and the complex squaring map == The informal reason why the iteration is chaotic is that the angle doubles on every iteration and doubling grows very quickly as the angle becomes ever larger, but angles which differ by multiples of 2π (radians) are identical. Thus, when the angle exceeds 2π, it must ''wrap'' to the remainder on division by 2π. Therefore the angle is transformed according to the dyadic transformation (also known as the 2x mod 1 map). As the initial value ''z''0 has been chosen so that its argument is not a rational multiple of π, the forward orbit of ''z''''n'' cannot repeat itself and become periodic. More formally, the iteration can be written as: :$\backslash qquad\; z\_\; =\; z\_n^2$ where $z\_n$ is the resulting sequence of complex numbers obtained by iterating the steps above, and $z\_0$ represents the initial starting number. We can solve this iteration exactly: :$\backslash qquad\; z\_n\; =\; z\_0^$ Starting with angle θ, we can write the initial term as $z\_0\; =\; \backslash exp(i\backslash theta)$ so that $z\_n\; =\; \backslash exp(i2^n\backslash theta)$. This makes the successive doubling of the angle clear. (This is equivalent to the relation $z\_n\; =\; \backslash cos(2^n\backslash theta)+i\; \backslash sin(2^n\backslash theta)$.) 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「Complex squaring map」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.