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→Resolution of Smale's 14th problem
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>
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>
== Resolution of Smale's 14th problem ==
== 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>.
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>.
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:
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:
* The return map admits a forward invariant cone field
* 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.
* Vectors inside this invariant cone field are uniformly expanded by the derivative <math>DP</math> of the return map.
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：
*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>
*回归映象允许一个相前的不变锥形场
*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.
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.
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。
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.
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.
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 ==
== Contributions ==