Symbolic dynamics and periodic orbits of the Lorenz attractor*
Symbolic dynamics and periodic orbits of the Lorenz attractor*
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DOI：
10.1088/0951-7715/16/3/314
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发表时间：
2003-05
期刊：
影响因子：
1.7
通讯作者：
D. Viswanath
中科院分区：
文献类型：
--
作者：
D. Viswanath
蝴蝶状的洛伦兹吸引子是最著名的混沌图像之一。本文的计算利用符号动力学和双曲理论的其他基本概念，使用周期轨道分解洛伦兹吸引子。我们计算了长度为 20 或更小的符号序列对应的所有 111011 个周期轨道、符号序列有数百个符号的周期轨道、洛伦兹吸引子的康托叶以及靠近原点鞍座的周期轨道。我们推导出一种计算周期性轨道的方法，该方法尽可能接近机器精度允许的洛伦兹吸引子上的给定点。该方法给出了双曲理论基本假设的算法实现，即双曲不变集中周期轨道的密度。所有周期轨道均采用 14 位精确数字计算。
The butterfly-like Lorenz attractor is one of the best known images of chaos. The computations in this paper exploit symbolic dynamics and other basic notions of hyperbolicity theory to take apart the Lorenz attractor using periodic orbits. We compute all 111011 periodic orbits corresponding to symbol sequences of length 20 or less, periodic orbits whose symbol sequences have hundreds of symbols, the Cantor leaves of the Lorenz attractor, and periodic orbits close to the saddle at the origin. We derive a method for computing periodic orbits as close as machine precision allows to a given point on the Lorenz attractor. This method gives an algorithmic realization of a basic hypothesis of hyperbolicity theory—namely, the density of periodic orbits in hyperbolic invariant sets. All periodic orbits are computed with 14 accurate digits.