混沌分段描述的马尔可夫链近似
A Markov Chain Approximation of a Segment Description of Chaos
摘要：
我们测试了一个马尔可夫链近似的帐篷映射，Ménon系统，Hénon映射和Lorenz系统上的混沌（和湍流）的分段描述（李，2007）。对于帐篷映射，我们计算段时间长度（迭代）$T = 1, 2, 3, 100$的段上的马尔可夫链的概率转移矩阵。该矩阵具有对应于$T = 1, 2, 3$的$1, 2, 4$帐篷;并且对于$T = 100$几乎是均匀的。作为$T \ra +\infty$，我们的猜想是矩阵将接近均匀矩阵（即每个条目都相同）。对于简单的不动点吸引子，雷诺平均的性能优于最大概率马尔可夫链和段连接。但对于Hénon映射中的奇异吸引子和Lorenz系统，Reynolds平均的表现非常差，甚至比最大概率马尔可夫链和分段链接更差。
英文摘要：
We test a Markov chain approximation to the segment description (Li, 2007) of chaos (and turbulence) on a tent map, the Minea system, the Hénon map, and the Lorenz system. For the tent map, we compute the probability transition matrix of the Markov chain on the segments for segment time length (iterations) $T = 1, 2, 3, 100$. The matrix has $1, 2, 4$ tents corresponding to $T = 1, 2, 3$; and is almost uniform for $T = 100$. As $T \ra +\infty$, our conjecture is that the matrix will approach a uniform matrix (i.e. every entry is the same). For the simple fixed point attractor in the Minea system, the Reynolds average performs excellently and better than the maximal probability Markov chain and segment linking. But for the strange attractors in the Hénon map, and the Lorenz system, the Reynolds average performs very poorly and worse than the maximal probability Markov chain and segment linking.
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