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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 Thomas' cyclically symmetric attractor ： ウィキペディア英語版 Thomas' cyclically symmetric attractor In the dynamical systems theory, Thomas' cyclically symmetric attractor is a 3D strange attractor originally proposed by René Thomas. It has a simple form which is cyclically symmetric in the x,y, and z variables and can be viewed as the trajectory of a frictionally dampened particle moving in a 3D lattice of forces. The simple form has made it a popular example. It is described by the differential equations :$\backslash frac\; =\; \backslash sin(y)-bx$ :$\backslash frac\; =\; \backslash sin(z)-by$ :$\backslash frac\; =\; \backslash sin(x)-bz$ where $b$ is a constant. $b$ corresponds to how dissipative the system is, and acts as a bifurcation parameter. For $b>1$ the origin is the single stable equilibrium. At $b=1$ it undergoes a pitchfork bifurcation, splitting into two attractive fixed points. As the parameter is decreased further they undergo a Hopf bifurcation at $b\backslash approx\; 0.32899$, creating a stable limit cycle. The limit cycle the undergoes a period doubling cascade and becomes chaotic at $b\backslash approx\; 0.208186$. Beyond this the attractor expands, undergoing a series of crises (up to six separate attractors can coexist for certain values). The fractal dimension of the attractor increases towards 3.〔 In the limit $b=0$ the system lacks dissipation and the trajectory ergodically wanders the entire space (with an exception for 1.67%, where it drifts parallel to one of the coordinate axes: this corresponds to quasiperiodic torii). The dynamics has been described as deterministic fractional Brownian motion, and exhibits anomalous diffusion.〔 ==References== 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「Thomas' cyclically symmetric attractor」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.