更改

[[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.

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。更具体的说，''洛伦兹吸引子''由洛伦兹系统中的一系列混沌解所组成。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。这说明就算一个物理系统可以被完全的测定，但即使排除了量子效应，人们仍然不能完全预测它。洛伦兹吸引子绘制成图形形状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》）

 

==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:

: <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"/>

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.