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洛伦兹吸引子 (查看源代码)

2021年11月14日 (日) 22:02的版本

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对系统的[[流函数]]（stream function）和温度建模的偏微分方程隶属于谱[[加勒金法（Galerkin method）|Galerkin approximation]]：水动力场以傅里叶级数展开，然后将其严格截断为一个流函数项和两个温度项。这就把模型方程简化为一系列三个耦合的、非线性的常微分方程。详细的推导可以在非线性动力学相关的文献中找到。<ref>{{harvtxt|Hilborn|2000}}, Appendix C; {{harvtxt|Bergé|Pomeau|Vidal|1984}}, Appendix D</ref> The Lorenz system is a reduced version of a larger system studied earlier by Barry Saltzman.<ref>{{harvtxt|Saltzman|1962}}</ref>

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== ~~Resolution of Smale~~'s ~~14th problem~~ ==

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== 对斯梅尔问题（Smale's problems）的解 ==

Smale's 14th problem says 'Do the properties of the Lorenz attractor exhibit that of a [[Attractor#Strange attractor|strange attractor]]?', it was answered affirmatively by [[Warwick Tucker]] in 2002.<ref name="Tucker 2002"/> To prove this result, Tucker used rigorous numerics methods like [[interval arithmetic]] and [[Normal form (dynamical systems)|normal forms]]. First, Tucker defined a cross section <math>\Sigma\subset \{x_3 = r - 1 \}</math> that is cut transversely by the flow trajectories. From this, one can define the first-return map <math>P</math>, which assigns to each <math>x\in\Sigma</math> the point <math>P(x)</math> where the trajectory of <math>x</math> first intersects <math>\Sigma</math>.

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斯梅尔问题的第14题这样问到：“洛伦兹吸引子是否表现出[[奇异吸引子]]的特性？”在2002年[[Warwick Tucker]]给出了肯定的答案。<ref name="Tucker 2002"/>为了证明这一结果，Tucker使用了严格的数值方法，例如：[[区间运算]]（interval arithmetic）和[[标准形]]（normal form）。首先，它定义了一个横断面<math>\Sigma\subset \{x_3 = r - 1 \}</math>，这一截面被流的轨迹横向切割。由此，我们可以定义首回归映象（first-return map）<math>P</math>，对每个<math>x\in\Sigma</math> 点<math>P(x)</math> 与<math>x</math>的轨迹在<math>\Sigma</math>首先相交。

Then the proof is split in three main points that are proved and imply the existence of a strange attractor.<ref name="Viana 2000">{{harvtxt|Viana|2000}}</ref> The three points are:

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* The return map admits a forward invariant cone field

* Vectors inside this invariant cone field are uniformly expanded by the derivative <math>DP</math> of the return map.

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之后，证明被分成了三部分。这三个部分被分别证明，并阐明了奇异吸引子的存在。<ref name="Viana 2000">{{harvtxt|Viana|2000}}</ref>这三部分分别是：

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*存在一个区域<math>N\subset\Sigma</math> 与首回归映象不相关，即<math>P(N)\subset N</math>

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*回归映象允许一个相前的不变锥形场

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*不变锥形场内的向量都被回归映象<math>DP</math>均匀的展开。

To prove the first point, we notice that the cross section <math>\Sigma</math> is cut by two arcs formed by <math>P(\Sigma)</math> (see <ref name="Viana 2000"/>). Tucker covers the location of these two arcs by small rectangles <math>R_i</math>, the union of these rectangles gives <math>N</math>. Now, the goal is to prove that for all points in <math>N</math>, the flow will bring back the points in <math>\Sigma</math>, in <math>N</math>. To do that, we take a plan <math>\Sigma'</math> below <math>\Sigma</math> at a distance <math>h</math> small, then by taking the center <math>c_i</math> of <math>R_i</math> and using Euler integration method, one can estimate where the flow will bring <math>c_i</math> in <math>\Sigma'</math> which gives us a new point <math>c_i'</math>. Then, one can estimate where the points in <math>\Sigma</math> will be mapped in <math>\Sigma'</math> using Taylor expansion, this gives us a new rectangle <math>R_i'</math> centered on <math>c_i</math>. Thus we know that all points in <math>R_i</math> will be mapped in <math>R_i'</math>. The goal is to do this method recursively until the flow comes back to <math>\Sigma</math> and we obtain a rectangle <math>Rf_i</math> in <math>\Sigma</math> such that we know that <math>P(R_i)\subset Rf_i</math>. The problem is that our estimation may become imprecise after several iterations, thus what Tucker does is to split <math>R_i'</math> into smaller rectangles <math>R_{i,j}</math> and then apply the process recursively.

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为了证明第一点，我们需要注意到横截面<math>\Sigma</math> 被<math>P(\Sigma)</math>所形成的两个弧切开(参考<ref name="Viana 2000"/>)。Tucker使用很多小矩形<math>R_i</math>覆盖了这两个弧线的位置。这些矩形的集合可以用<math>N</math>表示。现在，我们需要证明的是，对于<math>N</math>中的所有点，流将把<math>\Sigma</math>中的点带回到 <math>N</math>。要做到这一点，我们需要在距离<math>\Sigma</math><math>h</math>处取一个平面<math>\Sigma'</math> below <math>\Sigma</math>，然后通过取<math>R_i</math>的中心<math>c_i</math>和欧拉积分法可以估计出流将把<math>c_i</math>带到<math>Sigma'</math>上的位置。这样我们就得到了新的中心<math>c_i'</math>。之后我们可以用泰勒展开法估计<math>\Sigma</math> 中的点在<math>\Sigma'</math>中的映射位置，这样我们就得到了以<math>c_i</math>为中心的新矩形<math>R_i'</math>。这样，我们就知道，<math>R_i</math> 上的所有点都会映射到 <math>R_i'</math> 上。我们的目标是多次迭代这一过程，直到流回到<math>\Sigma</math>。这时我们就得到了<math>\Sigma</math>中的一个矩形<math>Rf_i</math>，我们知道<math>P(R_i)\subset Rf_i</math>。问题是，我们的估计在几次迭代后可能会变得不精确。因此Tucker将<math>R_i'</math>分割成更小的矩形<math>R_{i,j}</math>并不断的递归这个过程。

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Another problem is that as we are applying this algorithm, the flow becomes more 'horizontal' (see <ref name="Viana 2000"/>), leading to a dramatic increase in imprecision. To prevent this, the algorithm changes the orientation of the cross sections, becoming either horizontal or vertical.

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另一个问题是，当我们使用这个算法时，流会变得更加 "水平"（参考<ref name="Viana 2000"/>），这会导致精确度的极具降低。为了防止这种情况，这一算法改变了横截面的方向，使它既可以水平又可以垂直。

== Contributions ==

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