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[[File:洛伦兹吸引子通过相空间的轨迹.gif|frame|right|当 ρ = 28, σ = 10, β = 8/3时洛伦兹吸引子的一个解]]
[[File:洛伦兹吸引子通过相空间的轨迹.gif|frame|right|当 ρ = 28, σ = 10, β = 8/3时洛伦兹吸引子的一个解]]
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.
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.
==Overview==
“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》）
==概述==
In 1963, [[Edward Norton Lorenz|Edward Lorenz]], with the help of [[Ellen Fetter]], developed a simplified mathematical model for [[atmospheric convection]].<ref name=lorenz>{{harvtxt|Lorenz|1963}}</ref> The model is a system of three ordinary differential equations now known as the Lorenz equations:
In 1963, [[Edward Norton Lorenz|Edward Lorenz]], with the help of [[Ellen Fetter]], developed a simplified mathematical model for [[atmospheric convection]].<ref name=lorenz>{{harvtxt|Lorenz|1963}}</ref> The model is a system of three ordinary differential equations now known as the Lorenz equations:
1963年，[[爱德华-诺顿-洛伦兹|爱德华-洛伦兹]]在[[艾伦—费特]]的帮助下，为[[大气对流]]构建了一个简化的数学模型。<ref name=lorenz>{{harvtxt|Lorenz|1963}}</ref>该模型是一个由三个常微分方程组成的系统，这些公式现在被称为Lorenz方程:
: <math> \begin{align}
: <math> \begin{align}
The equations relate the properties of a two-dimensional fluid layer uniformly warmed from below and cooled from above. In particular, the equations describe the rate of change of three quantities with respect to time: <math>x</math> is proportional to the rate of convection, <math>y</math> to the horizontal temperature variation, and <math>z</math> to the vertical temperature variation.<ref name="Sparrow 1982">{{harvtxt|Sparrow|1982}}</ref> The constants <math>\sigma</math>, <math>\rho</math>, and <math>\beta</math> are system parameters proportional to the [[Prandtl number]], [[Rayleigh number]], and certain physical dimensions of the layer itself.<ref name="Sparrow 1982"/>
The equations relate the properties of a two-dimensional fluid layer uniformly warmed from below and cooled from above. In particular, the equations describe the rate of change of three quantities with respect to time: <math>x</math> is proportional to the rate of convection, <math>y</math> to the horizontal temperature variation, and <math>z</math> to the vertical temperature variation.<ref name="Sparrow 1982">{{harvtxt|Sparrow|1982}}</ref> The constants <math>\sigma</math>, <math>\rho</math>, and <math>\beta</math> are system parameters proportional to the [[Prandtl number]], [[Rayleigh number]], and certain physical dimensions of the layer itself.<ref name="Sparrow 1982"/>
这些方程被用来描述与一个下面升温而上面同时降温的二维流体层。特别的是，这些方程描述了三个量相对于时间的变化率。<math>x</math>与流速度成正比，<math>y</math>与水平温度变化成正比，而<math>z</math>与垂直温度变化成正比。 <ref name="Sparrow 1982">{{harvtxt|Sparrow|1982}}</ref> 常数<math>sigma</math>、<math>rho</math>和<math>beta</math>是与[[普朗特数]]、[[瑞利数]]和层本身的某些物理尺寸相称的系统参数。
The Lorenz equations also arise in simplified models for [[laser]]s,<ref>{{harvtxt|Haken|1975}}</ref> [[electrical generator|dynamos]],<ref>{{harvtxt|Knobloch|1981}}</ref> [[thermosyphon]]s,<ref>{{harvtxt|Gorman|Widmann|Robbins|1986}}</ref> brushless [[DC motor]]s,<ref>{{harvtxt|Hemati|1994}}</ref> [[electric circuit]]s,<ref>{{harvtxt|Cuomo|Oppenheim|1993}}</ref> [[chemical reaction]]s<ref>{{harvtxt|Poland|1993}}</ref> and [[forward osmosis]].<ref>{{harvtxt|Tzenov|2014}}{{citation needed|date=June 2017<!--doesn't point anywhere-->}}</ref> The Lorenz equations are also the governing equations in Fourier space for the [[Malkus waterwheel]].<ref>{{harvtxt|Kolář|Gumbs|1992}}</ref><ref>{{harvtxt|Mishra|Sanghi|2006}}</ref> The Malkus waterwheel exhibits chaotic motion where instead of spinning in one direction at a constant speed, its rotation will speed up, slow down, stop, change directions, and oscillate back and forth between combinations of such behaviors in an unpredictable manner.
The Lorenz equations also arise in simplified models for [[laser]]s,<ref>{{harvtxt|Haken|1975}}</ref> [[electrical generator|dynamos]],<ref>{{harvtxt|Knobloch|1981}}</ref> [[thermosyphon]]s,<ref>{{harvtxt|Gorman|Widmann|Robbins|1986}}</ref> brushless [[DC motor]]s,<ref>{{harvtxt|Hemati|1994}}</ref> [[electric circuit]]s,<ref>{{harvtxt|Cuomo|Oppenheim|1993}}</ref> [[chemical reaction]]s<ref>{{harvtxt|Poland|1993}}</ref> and [[forward osmosis]].<ref>{{harvtxt|Tzenov|2014}}{{citation needed|date=June 2017<!--doesn't point anywhere-->}}</ref> The Lorenz equations are also the governing equations in Fourier space for the [[Malkus waterwheel]].<ref>{{harvtxt|Kolář|Gumbs|1992}}</ref><ref>{{harvtxt|Mishra|Sanghi|2006}}</ref> The Malkus waterwheel exhibits chaotic motion where instead of spinning in one direction at a constant speed, its rotation will speed up, slow down, stop, change directions, and oscillate back and forth between combinations of such behaviors in an unpredictable manner.