洛伦兹吸引子 - 版本历史

王盛嘉：/* 概述 */

2026-11-02T13:44:12Z

概述

2021年11月21日 (日) 11:04 Moyuefeng

2021-11-21T11:04:47Z

2021年11月21日 (日) 10:59 Moyuefeng

2021-11-21T10:59:41Z

Moyuefeng：/* References */

2021-11-21T10:18:58Z

References

2021年11月18日 (四) 14:45 Moyuefeng

2021-11-18T14:45:52Z

2021年11月18日 (四) 14:24 Moyuefeng

2021-11-18T14:24:21Z

Moyuefeng：/* 其它贡献人 */

2021-11-14T14:12:23Z

其它贡献人

Moyuefeng：/* Contributions */

2021-11-14T14:10:29Z

Contributions

Moyuefeng：/* Resolution of Smale's 14th problem */

2021-11-14T14:02:30Z

Resolution of Smale's 14th problem

2021年11月13日 (六) 16:07 Moyuefeng

2021-11-13T16:07:40Z

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 From a technical standpoint, the Lorenz system is [[nonlinearity|nonlinear]], non-periodic, three-dimensional and [[deterministic system (mathematics)|deterministic]]. The Lorenz equations have been the subject of hundreds of research articles, and at least one book-length study.<ref name="Sparrow 1982"/>
-从技术的角度看，洛伦兹系统是[[非线性|nonlinear]]、非周期性、三维和[[数学上确定的系统|deterministic]]。洛伦兹方程已经作为研究对象被数百篇论文讨论过，其中至少有一篇长篇研究。<ref name="Sparrow 1982"/>
+从技术的角度看，洛伦兹系统是[[非线性|非线性]]、非周期性、三维和[[数学上确定的系统|数学上的确定系统]]。洛伦兹方程已经作为研究对象被数百篇论文讨论过，其中至少有一篇长篇研究。<ref name="Sparrow 1982"/>
 ==分析==

←上一版本		2021年11月21日 (日) 11:04的版本
第9行：		第9行：


−	（关于这一词的由来需要参考书籍《The essence of ~~chaos》）引用原文如下：(未翻译)The~~ thing that has made the origin of the phrase a bit uncertainis a peculiarity of the first chaotic system that I studied in detail.Here an abbreviated graphical representation of a special collectionof states known as a “strange attractor” was subsequently foundto resemble a butterfly, and soon became known as the butterfly.In Figure 2 we see one butterfly; a representative of a closely relatedspecies appears on the inside cover of Gleick’s book. A number ofpeople with whom I have talked have assumed that the butterflyeffect was named after this attractor. Perhaps it was.Some correspondents have also called my attention to RayBradbury’s intriguing short story “A Sound of Thunder,” writtenlong before the Washington meeting. Here the death of a prehistoricbutterfly, and its consequent failure to reproduce, change theoutcome of a present-day presidential election.Before the Washington meeting I had sometimes used a sea gullas a symbol for sensitive dependence. The switch to a butterfly wasactually made by the session convenor, the meteorologist PhilipMerilees, who was unable to check with me when he had to submitthe program titles. Phil has recently assured me that he was notfamiliar with Bradbury’s story. Perhaps the butterfly, with itsseeming frailty and lack of power, is a natural choice for a symbolof the small that can produce the great.Other symbols have preceded the sea gull. In George R.Stewart’snovel Storm, a copy of which my sister gave me for Christmas whenshe first learned that I was to become a meteorology student, ameteorologist recalls his professor’s remark that a man sneezing inChina may set people to shoveling snow in New York. Stewart’sprofessor was simply echoing what some real-world meteorologistshad been saying for many years, sometimes facetiously, sometimesseriously<ref>{{cite book \| last1=Lorenz \| first1=Edward N. \| title=The Essence of Chaos \| publisher=UCL Press \| isbn=978-1-85-728454-6 \| year=1993}}</ref>	+	（关于这一词的由来需要参考书籍《The essence of chaos》）引用原文如下：The thing that has made the origin of the phrase a bit uncertainis a peculiarity of the first chaotic system that I studied in detail.Here an abbreviated graphical representation of a special collectionof states known as a “strange attractor” was subsequently foundto resemble a butterfly, and soon became known as the butterfly.In Figure 2 we see one butterfly; a representative of a closely relatedspecies appears on the inside cover of Gleick’s book. A number ofpeople with whom I have talked have assumed that the butterflyeffect was named after this attractor. Perhaps it was.Some correspondents have also called my attention to RayBradbury’s intriguing short story “A Sound of Thunder,” writtenlong before the Washington meeting. Here the death of a prehistoricbutterfly, and its consequent failure to reproduce, change theoutcome of a present-day presidential election.Before the Washington meeting I had sometimes used a sea gullas a symbol for sensitive dependence. The switch to a butterfly wasactually made by the session convenor, the meteorologist PhilipMerilees, who was unable to check with me when he had to submitthe program titles. Phil has recently assured me that he was notfamiliar with Bradbury’s story. Perhaps the butterfly, with itsseeming frailty and lack of power, is a natural choice for a symbolof the small that can produce the great.Other symbols have preceded the sea gull. In George R.Stewart’snovel Storm, a copy of which my sister gave me for Christmas whenshe first learned that I was to become a meteorology student, ameteorologist recalls his professor’s remark that a man sneezing inChina may set people to shoveling snow in New York. Stewart’sprofessor was simply echoing what some real-world meteorologistshad been saying for many years, sometimes facetiously, sometimesseriously<ref>{{cite book \| last1=Lorenz \| first1=Edward N. \| title=The Essence of Chaos \| publisher=UCL Press \| isbn=978-1-85-728454-6 \| year=1993}}</ref>

−	~~据洛伦兹本人在《混沌的本质（The~~ essence of chaos）》中的说法。“蝴蝶效应”这一词最早可以追溯到Ray Bradbury的短篇故事《雷声（A Sound of Thunder）》在1972年美国科学促进会举办的年会上，会议召集人、气象学家Philip Merilees将洛伦兹报告的副标题改成了蝴蝶，尽管他本人并不知晓Ray Bradbury的故事。在此之前洛伦兹通常用海鸥来称呼这一现象。<ref>{{cite book \| last1=Lorenz \| first1=Edward N. \| title=The Essence of Chaos \| publisher=UCL Press \| isbn=978-1-85-728454-6 \| year=1993}}</ref>	+	（对以上内容的简述）据洛伦兹本人在《混沌的本质（The essence of chaos）》中的说法。“蝴蝶效应”这一词最早可以追溯到Ray Bradbury的短篇故事《雷声（A Sound of Thunder）》在1972年美国科学促进会举办的年会上，会议召集人、气象学家Philip Merilees将洛伦兹报告的副标题改成了蝴蝶，尽管他本人并不知晓Ray Bradbury的故事。在此之前洛伦兹通常用海鸥来称呼这一现象。<ref>{{cite book \| last1=Lorenz \| first1=Edward N. \| title=The Essence of Chaos \| publisher=UCL Press \| isbn=978-1-85-728454-6 \| year=1993}}</ref>

@@ 第7行： / 第7行： @@
 “洛伦兹系统”是由[[爱德华-诺顿-洛伦兹|爱德华-洛伦兹]]最先提出的一种由[[常微分方程]]构成的系统。值得注意的是，在一些特定参数和初始条件下，它的解是[[混沌理论|混沌]]的。更确切的说，“洛伦兹吸引子”其实是洛伦兹系统所有的混沌解。人们通常所说的"[[蝴蝶效应]]"源于洛伦兹吸引子的现实意义：即在任何物理系统中，在对初始条件不完全了解的情况下，即使是蝴蝶拍打翅膀引起的微小扰动都可以让我们无法成功的预测系统的未来进程。对物理系统来说，即使它可以被完整的测定，但人们仍然不能完全预测它（即使排除了量子效应也不行）。洛伦兹吸引子绘制成的图形看起来就像一只蝴蝶。这也是“蝴蝶效应”这一词的由来。
-（关于这一词的由来需要参考书籍《The essence of chaos》）引用原文如下：(未翻译)The thing that has made the origin of the phrase a bit uncertain
-is a peculiarity of the first chaotic system that I studied in detail.
-Here an abbreviated graphical representation of a special collection
+（关于这一词的由来需要参考书籍《The essence of chaos》）引用原文如下：(未翻译)The thing that has made the origin of the phrase a bit uncertainis a peculiarity of the first chaotic system that I studied in detail.Here an abbreviated graphical representation of a special collectionof states known as a “strange attractor” was subsequently foundto resemble a butterfly, and soon became known as the butterfly.In Figure 2 we see one butterfly; a representative of a closely relatedspecies appears on the inside cover of Gleick’s book. A number ofpeople with whom I have talked have assumed that the butterflyeffect was named after this attractor. Perhaps it was.Some correspondents have also called my attention to RayBradbury’s intriguing short story “A Sound of Thunder,” writtenlong before the Washington meeting. Here the death of a prehistoricbutterfly, and its consequent failure to reproduce, change theoutcome of a present-day presidential election.Before the Washington meeting I had sometimes used a sea gullas a symbol for sensitive dependence. The switch to a butterfly wasactually made by the session convenor, the meteorologist PhilipMerilees, who was unable to check with me when he had to submitthe program titles. Phil has recently assured me that he was notfamiliar with Bradbury’s story. Perhaps the butterfly, with itsseeming frailty and lack of power, is a natural choice for a symbolof the small that can produce the great.Other symbols have preceded the sea gull. In George R.Stewart’snovel Storm, a copy of which my sister gave me for Christmas whenshe first learned that I was to become a meteorology student, ameteorologist recalls his professor’s remark that a man sneezing inChina may set people to shoveling snow in New York. Stewart’sprofessor was simply echoing what some real-world meteorologistshad been saying for many years, sometimes facetiously, sometimesseriously<ref>{{cite book | last1=Lorenz | first1=Edward N. | title=The Essence of Chaos | publisher=UCL Press | isbn=978-1-85-728454-6 | year=1993}}</ref>
-of states known as a “strange attractor” was subsequently found
-to resemble a butterfly, and soon became known as the butterfly.
+据洛伦兹本人在《混沌的本质（The essence of chaos）》中的说法。“蝴蝶效应”这一词最早可以追溯到Ray Bradbury的短篇故事《雷声（A Sound of Thunder）》在1972年美国科学促进会举办的年会上，会议召集人、气象学家Philip Merilees将洛伦兹报告的副标题改成了蝴蝶，尽管他本人并不知晓Ray Bradbury的故事。在此之前洛伦兹通常用海鸥来称呼这一现象。<ref>{{cite book | last1=Lorenz | first1=Edward N. | title=The Essence of Chaos | publisher=UCL Press | isbn=978-1-85-728454-6 | year=1993}}</ref>
-In Figure 2 we see one butterfly; a representative of a closely related
 ==概述==
@@ 第269行： / 第245行： @@
 == Gallery ==
 <gallery>
-File:Lorenz system r28 s10 b2-6666.png|A solution in the Lorenz attractor plotted at high resolution in the x-z plane.
+File:Lorenz system r28 s10 b2-6666.png|A solution in the Lorenz attractor plotted at high resolution in the x-z plane. x-z平面上对洛伦兹吸引子其中一个解的高清图。
-File:Lorenz attractor.svg|A solution in the Lorenz attractor rendered as an SVG.
+File:Lorenz attractor.svg|A solution in the Lorenz attractor rendered as an SVG.以SVG形式呈现的洛伦兹吸引子的解。
-File:A Lorenz system.ogv|An animation showing trajectories of multiple solutions in a Lorenz system.
+File:A Lorenz system.ogv|An animation showing trajectories of multiple solutions in a Lorenz system.动画显示了洛伦兹系统中多个解的轨迹。
-File:Lorenzstill-rubel.png|A solution in the Lorenz attractor rendered as a metal wire to show direction and [[Three-dimensional space|3D]] structure.
+File:Lorenzstill-rubel.png|A solution in the Lorenz attractor rendered as a metal wire to show direction and [[Three-dimensional space|3D]] structure.洛伦兹吸引子的一个解被渲染成金属线，以显示方向和[[三维空间|3D]]结构。
-File:Lorenz.ogv|An animation showing the divergence of nearby solutions to the Lorenz system.
+File:Lorenz.ogv|An animation showing the divergence of nearby solutions to the Lorenz system.动画显示出在洛伦兹系统的解附近会出现发散的情况。
-File:Intermittent Lorenz Attractor - Chaoscope.jpg|A visualization of the Lorenz attractor near an intermittent cycle.
+File:Intermittent Lorenz Attractor - Chaoscope.jpg|A visualization of the Lorenz attractor near an intermittent cycle.洛伦兹吸引子在一个间歇循环附近。
-File:Lorenz apparition small.gif|Two streamlines in a Lorenz system, from rho=0 to rho=28 (sigma=10, beta=8/3)
+File:Lorenz apparition small.gif|Two streamlines in a Lorenz system, from rho=0 to rho=28 (sigma=10, beta=8/3)洛伦兹系统中的两条流线，从rho=0到rho=28（sigma=10，beta=8/3）。
-File:Lorenz(rho).gif| Animation of a Lorenz System with rho- dependence
+File:Lorenz(rho).gif| Animation of a Lorenz System with rho- dependence 动画展示出洛伦兹系统具有rho-依赖性。
 </gallery>

@@ 第318行： / 第318行： @@
 * {{cite journal |last1=Viana |first1=Marcelo |title=What's new on Lorenz strange attractors? |journal=The Mathematical Intelligencer |date=2000 |volume=22 |issue=3 |pages=6–19|doi=10.1007/BF03025276 }}
 * {{cite journal|last1=Lorenz|first1=Edward N.|author1-link=Edward N. Lorenz|title=The statistical prediction of solutions of dynamic equations.|journal=Symposium on Numerical Weather Prediction in Tokyo|year=1960|url=http://eaps4.mit.edu/research/Lorenz/The_Statistical_Prediction_of_Solutions_1962.pdf}}
-* {{cite book | last1=N.Lorenz | first1=Edward | title=The Essence of Chaos | publisher=UCL Press | isbn=978-1-85-728454-6 | year=1993}}
+* {{cite book | last1=Lorenz | first1=Edward N. | title=The Essence of Chaos | publisher=UCL Press | isbn=978-1-85-728454-6 | year=1993}}
 == Further reading ==

@@ 第5行： / 第5行： @@
 The '''Lorenz system''' is a system of [[ordinary differential equation]]s first studied by [[Edward Norton Lorenz|Edward Lorenz]]. It is notable for having [[Chaos theory|chaotic]] solutions for certain parameter values and initial conditions. In particular, the '''Lorenz attractor''' is a set of chaotic solutions of the Lorenz system. In popular media the "[[butterfly effect]]" stems from the real-world implications of the Lorenz attractor, i.e. that in any physical system, in the absence of perfect knowledge of the initial conditions (even the minuscule disturbance of the air due to a butterfly flapping its wings), our ability to predict its future course will always fail. This underscores that physical systems can be completely deterministic and yet still be inherently unpredictable even in the absence of quantum effects. The shape of the Lorenz attractor itself, when plotted graphically, may also be seen to resemble a butterfly.
-“洛伦兹系统”是由[[爱德华-诺顿-洛伦兹|爱德华-洛伦兹]]最先提出的一种由[[常微分方程]]构成的系统。值得注意的是，在一些特定参数和初始条件下，它的解是[[混沌理论|混沌]]的。更确切的说，“洛伦兹吸引子”其实是洛伦兹系统所有的混沌解。人们通常所说的"[[蝴蝶效应]]"源于洛伦兹吸引子的现实意义：即在任何物理系统中，在对初始条件不完全了解的情况下，即使是蝴蝶拍打翅膀引起的微小扰动都可以让我们无法成功的预测系统的未来进程。对物理系统来说，即使它可以被完整的测定，但人们仍然不能完全预测它（即使排除了量子效应也不行）。洛伦兹吸引子绘制成的图形看起来就像一只蝴蝶。这也是“蝴蝶效应”这一词的由来。（关于这一词的由来需要参考书籍《The essence of chaos》）
+“洛伦兹系统”是由[[爱德华-诺顿-洛伦兹|爱德华-洛伦兹]]最先提出的一种由[[常微分方程]]构成的系统。值得注意的是，在一些特定参数和初始条件下，它的解是[[混沌理论|混沌]]的。更确切的说，“洛伦兹吸引子”其实是洛伦兹系统所有的混沌解。人们通常所说的"[[蝴蝶效应]]"源于洛伦兹吸引子的现实意义：即在任何物理系统中，在对初始条件不完全了解的情况下，即使是蝴蝶拍打翅膀引起的微小扰动都可以让我们无法成功的预测系统的未来进程。对物理系统来说，即使它可以被完整的测定，但人们仍然不能完全预测它（即使排除了量子效应也不行）。洛伦兹吸引子绘制成的图形看起来就像一只蝴蝶。这也是“蝴蝶效应”这一词的由来。
 ==概述==
@@ 第287行： / 第318行： @@
 * {{cite journal |last1=Viana |first1=Marcelo |title=What's new on Lorenz strange attractors? |journal=The Mathematical Intelligencer |date=2000 |volume=22 |issue=3 |pages=6–19|doi=10.1007/BF03025276 }}
 * {{cite journal|last1=Lorenz|first1=Edward N.|author1-link=Edward N. Lorenz|title=The statistical prediction of solutions of dynamic equations.|journal=Symposium on Numerical Weather Prediction in Tokyo|year=1960|url=http://eaps4.mit.edu/research/Lorenz/The_Statistical_Prediction_of_Solutions_1962.pdf}}
 == Further reading ==

@@ 第2行： / 第2行： @@
 {{Short description|System of ordinary differential equations with chaotic solutions}}
 {{Distinguish|Lorenz curve|Lorentz distribution}}
-[[File:A Trajectory Through Phase Space in a Lorenz Attractor.gif|frame|right|A sample solution in the Lorenz attractor when ρ = 28, σ = 10, and β = 8/3]]
+[[File:A Trajectory Through Phase Space in a Lorenz Attractor.gif|frame|right|当 ρ = 28, σ = 10, β = 8/3时洛伦兹吸引子的一个解 A sample solution in the Lorenz attractor when ρ = 28, σ = 10, and β = 8/3]]
 The '''Lorenz system''' is a system of [[ordinary differential equation]]s first studied by [[Edward Norton Lorenz|Edward Lorenz]]. It is notable for having [[Chaos theory|chaotic]] solutions for certain parameter values and initial conditions. In particular, the '''Lorenz attractor''' is a set of chaotic solutions of the Lorenz system. In popular media the "[[butterfly effect]]" stems from the real-world implications of the Lorenz attractor, i.e. that in any physical system, in the absence of perfect knowledge of the initial conditions (even the minuscule disturbance of the air due to a butterfly flapping its wings), our ability to predict its future course will always fail. This underscores that physical systems can be completely deterministic and yet still be inherently unpredictable even in the absence of quantum effects. The shape of the Lorenz attractor itself, when plotted graphically, may also be seen to resemble a butterfly.
@@ 第85行： / 第84行： @@
 {|class="wikitable" width=777px
 |-
-! colspan=2|Example solutions of the Lorenz system for different values of ρ
+! colspan=2|Example solutions of the Lorenz system for different values of ρ举例：当ρ取不同值时洛伦兹系统的解
 |-
 |align="center"|[[Image:Lorenz Ro14 20 41 20-200px.png]]
@@ 第99行： / 第98行： @@
 |align="center"|'''''ρ'' = 28, ''σ'' = 10, ''β'' = 8/3''' [[:Image:Lorenz Ro28.png|(Enlarge)]]
 |-
-|align="center" colspan=2| For small values of ''ρ'', the system is stable and evolves to one of two fixed point attractors. When ρ is larger than 24.74, the fixed points become repulsors and the trajectory is repelled by them in a very complex way.
+|align="center" colspan=2| For small values of ''ρ'', the system is stable and evolves to one of two fixed point attractors. When ρ is larger than 24.74, the fixed points become repulsors and the trajectory is repelled by them in a very complex way.当ρ取值较小时系统较为稳定，并能演化出一种拥有两个固定点的吸引子。当ρ大于24.74时，固定点将会移动，而互相平行的轨迹会再次以一种复杂的方式相交。
 |}
@@ 第114行： / 第113行： @@
 |align="center"|[[Image:Lorenz caos3-175.png]]
 |-
-|align="center" colspan=3| These figures — made using ''ρ'' = 28, ''σ'' = 10 and ''β'' = 8/3 — show three time segments of the 3-D evolution of two trajectories (one in blue, the other in yellow) in the Lorenz attractor starting at two initial points that differ only by 10<sup>−5</sup> in the ''x''-coordinate. Initially, the two trajectories seem coincident (only the yellow one can be seen, as it is drawn over the blue one) but, after some time, the divergence is obvious.
+|align="center" colspan=3| These figures — made using ''ρ'' = 28, ''σ'' = 10 and ''β'' = 8/3 — show three time segments of the 3-D evolution of two trajectories (one in blue, the other in yellow) in the Lorenz attractor starting at two initial points that differ only by 10<sup>−5</sup> in the ''x''-coordinate. Initially, the two trajectories seem coincident (only the yellow one can be seen, as it is drawn over the blue one) but, after some time, the divergence is obvious.这些图片中''ρ'' = 28, ''σ'' = 10 还有 ''β'' = 8/3。图中展示了三维图像中，三个不同时间段下两个轨迹（一蓝一黄）的情况。这两个轨迹的初始点的''x''坐标相差10<sup>−5</sup>。刚开始，这两个轨迹看起来是相关的（图中我们只能看到黄色的轨迹，因为它覆盖在蓝色轨迹的上方）。但一段时间以后，我们能清楚的看到两者的差异。
 |}
 ==与帐篷映射的关系==
-[[File:Lorenz_Map.png|thumb|A recreation of Lorenz's results created on [[Mathematica]]. Points above the red line correspond to the system switching lobes.|upright=1.3]]
+[[File:Lorenz_Map.png|thumb|A recreation of Lorenz's results created on [[Mathematica]]. Points above the red line correspond to the system switching lobes.在[[Mathematica]]上对洛伦兹的结果进行重现。处于红线上方的点与系统波段（lobe）转换相关。|upright=1.3 ]]
 In Figure 4 of his paper,<ref name=lorenz /> Lorenz plotted the relative maximum value in the z direction achieved by the system against the previous relative maximum in the z direction. This procedure later became known as a Lorenz map (not to be confused with a [[Poincaré plot]], which plots the intersections of a trajectory with a prescribed surface). The resulting plot has a shape very similar to the [[tent map]]. Lorenz also found that when the maximum z value is above a certain cut-off, the system will switch to the next lobe. Combining this with the chaos known to be exhibited by the tent map, he showed that the system switches between the two lobes chaotically.
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 === 使用 Mathematica 进行模拟===
 Standard way:
 <syntaxhighlight lang="mathematica">
@@ 第158行： / 第158行： @@
 Less verbose:
 <syntaxhighlight lang="mathematica">

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 == 其它贡献人 ==
 Lorenz acknowledges the contributions from [[Ellen Fetter]] in his paper, who is responsible for the numerical simulations and figures.<ref name="lorenz"/> Also, [[Margaret Hamilton (software engineer)|Margaret Hamilton]] helped in the initial, numerical computations leading up to the findings of the Lorenz model.<ref>{{harvtxt|Lorenz|1960}}</ref>
-在他论文中提及了[[Ellen Fetter]]所作的贡献。他负责数值模拟和图像。<ref name="lorenz"/> 另外，[[Margaret Hamilton]]在最初的数值计算中提供了帮助，使洛伦兹模型得以被发现。<ref>{{harvtxt|Lorenz|1960}}</ref>
 == Gallery ==

@@ 第203行： / 第203行： @@
 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>.
 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.
 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.
 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.
 == Contributions ==

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 |}
-==Connection to tent map==
+==与帐篷映射的关系==
 [[File:Lorenz_Map.png|thumb|A recreation of Lorenz's results created on [[Mathematica]]. Points above the red line correspond to the system switching lobes.|upright=1.3]]
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 In Figure 4 of his paper,<ref name=lorenz /> Lorenz plotted the relative maximum value in the z direction achieved by the system against the previous relative maximum in the z direction. This procedure later became known as a Lorenz map (not to be confused with a [[Poincaré plot]], which plots the intersections of a trajectory with a prescribed surface). The resulting plot has a shape very similar to the [[tent map]]. Lorenz also found that when the maximum z value is above a certain cut-off, the system will switch to the next lobe. Combining this with the chaos known to be exhibited by the tent map, he showed that the system switches between the two lobes chaotically.
-==Simulations==
+在论文的图4中，<ref name=lorenz />洛伦兹将系统在z方向上取得的相对最大值与之前在z方向上的相对最大值作了对比。通过这一步骤我们得到了洛伦兹图（不要与[[彭加莱图]]（Poincaré plot）相混淆，彭加莱图绘制的是轨迹与规定曲面的交点）。这一图型的形状与[[帐篷映射]]（tent map）非常相似。洛伦兹还发现，当Z的最大值值高于某个截断时，系统就会切换到下一个波段（lobe）。通过结合这一点和帐篷映射所表现出的混沌性，洛伦兹阐明了这一系统在两个波段（lobe）之间地切换是混沌的。
-===MATLAB simulation===
+==模拟==
 <syntaxhighlight lang="matlab">
 % Solve over time interval [0,100] with initial conditions [1,1,1]
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 </syntaxhighlight>
-=== Mathematica simulation ===
+=== 使用 Mathematica 进行模拟===
 Standard way:
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 </syntaxhighlight>
-=== Python simulation ===
+=== 使用 Python 进行模拟 ===
 <syntaxhighlight lang="python">
 import numpy as np
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 == Derivation of the Lorenz equations as a model for atmospheric convection ==
 The Lorenz equations are derived from the [[Boussinesq approximation (buoyancy)|Oberbeck–Boussinesq approximation]] to the equations describing fluid circulation in a shallow layer of fluid, heated uniformly from below and cooled uniformly from above.<ref name="lorenz"/> This fluid circulation is known as [[Rayleigh–Bénard convection]]. The fluid is assumed to circulate in two dimensions (vertical and horizontal) with periodic rectangular boundary conditions.
 The partial differential equations modeling the system's [[stream function]] and temperature are subjected to a spectral [[Galerkin method|Galerkin approximation]]: the hydrodynamic fields are expanded in Fourier series, which are then severely truncated to a single term for the stream function and two terms for the temperature. This reduces the model equations to a set of three coupled, nonlinear ordinary differential equations. A detailed derivation may be found, for example, in nonlinear dynamics texts.<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 ==