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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.
洛伦兹方程也出现在[[激光]]、<ref>{{harvtxt|Haken|1975}}</ref>[[发电机|dynamos]]、<ref>{{harvtxt|Knobloch|1981}}</ref>[[热虹吸管]]、 <ref>{harvtxt|Gorman|Widmann|Robbins|1986}</ref>无刷[[直流电动机]]、<ref>{harvtxt|Hemati|1994}</ref>[[电路]]、 <ref>{harvtxt|Cuomo|Oppenheim|1993}}</ref>[[化学反应]]、<ref>{harvtxt|Poland|1993}}</ref>和[[正向渗透]] <ref>{{harvtxt|Tzenov|2014}}{{citation needed|date=June 2017<!--doesn't point anywhere-->}}</ref>等其它领域的简化模型中。洛伦兹方程也是[[马尔库斯水车]]在傅里叶空间的控制方程<ref>{{harvtxt|Kolář|Gumbs|1992}}</ref><ref>{{harvtxt|Mishra|Sanghi|2006}}</ref>马尔库斯水车能够表现出混沌运动。它不是以一个恒定的速度朝固定的方向旋转，而是在旋转的过程中经历加速、减速、停止、改变方向等过程。并且这些过程以不可预测的方式在互相转变。
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"/>
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"/>
==Analysis==
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"/>
==分析==
One normally assumes that the parameters <math>\sigma</math>, <math>\rho</math>, and <math>\beta</math> are positive. Lorenz used the values <math>\sigma = 10</math>, <math>\beta = 8/3</math> and <math>\rho = 28 </math>. The system exhibits chaotic behavior for these (and nearby) values.<ref>{{harvtxt|Hirsch|Smale|Devaney|2003}}, pp. 303–305</ref>
One normally assumes that the parameters <math>\sigma</math>, <math>\rho</math>, and <math>\beta</math> are positive. Lorenz used the values <math>\sigma = 10</math>, <math>\beta = 8/3</math> and <math>\rho = 28 </math>. The system exhibits chaotic behavior for these (and nearby) values.<ref>{{harvtxt|Hirsch|Smale|Devaney|2003}}, pp. 303–305</ref>
通常情况下默认参数<math>\sigma</math>, <math>\rho</math>, and <math>\beta</math>取正值。洛伦兹将这三个参数设定为：<math>\sigma = 10</math>, <math>\beta = 8/3</math> and <math>\rho = 28 </math>。在参数取这些值（或近似值）的时候系统会体现出混沌行为。<ref>{{harvtxt|Hirsch|Smale|Devaney|2003}}, pp. 303–305</ref>
If <math>\rho < 1</math> then there is only one equilibrium point, which is at the origin. This point corresponds to no convection. All orbits converge to the origin, which is a global [[attractor]], when <math>\rho < 1</math>.<ref>{{harvtxt|Hirsch|Smale|Devaney|2003}}, pp. 306+307</ref>
If <math>\rho < 1</math> then there is only one equilibrium point, which is at the origin. This point corresponds to no convection. All orbits converge to the origin, which is a global [[attractor]], when <math>\rho < 1</math>.<ref>{{harvtxt|Hirsch|Smale|Devaney|2003}}, pp. 306+307</ref>
A [[pitchfork bifurcation]] occurs at <math>\rho = 1</math>, and for <math>\rho > 1 </math> two additional critical points appear at: <math>\left( \sqrt{\beta(\rho-1)}, \sqrt{\beta(\rho-1)}, \rho-1 \right) </math> and <math>\left( -\sqrt{\beta(\rho-1)}, -\sqrt{\beta(\rho-1)}, \rho-1 \right). </math>
如果<math>\rho < 1</math>，在原点只存在一个动态平衡点。这一点与任何对流都不契合。当<math>\rho < 1</math>时，所有的轨道都会收敛于原点，这一原点是全局的[[吸引子]]。<ref>{{harvtxt|Hirsch|Smale|Devaney|2003}}, pp. 306+307</ref>
These correspond to steady convection. This pair of equilibrium points is stable only if
A [[pitchfork bifurcation]] occurs at <math>\rho = 1</math>, and for <math>\rho > 1 </math> two additional critical points appear at: <math>\left( \sqrt{\beta(\rho-1)}, \sqrt{\beta(\rho-1)}, \rho-1 \right) </math> and <math>\left( -\sqrt{\beta(\rho-1)}, -\sqrt{\beta(\rho-1)}, \rho-1 \right). </math> These correspond to steady convection.
当<math>\rho = 1</math>会出现一个[[叉式分岔]]，而当<math>\rho > 1 </math>两个重要的点将会出现在<math>\left( \sqrt{\beta(\rho-1)}, \sqrt{\beta(\rho-1)}, \rho-1 \right) </math> 和 <math>\left( -\sqrt{\beta(\rho-1)}, -\sqrt{\beta(\rho-1)}, \rho-1 \right)</math> 处。这两点与稳定的对流相契合。
This pair of equilibrium points is stable only if
:<math>\rho < \sigma\frac{\sigma+\beta+3}{\sigma-\beta-1}, </math>
:<math>\rho < \sigma\frac{\sigma+\beta+3}{\sigma-\beta-1}, </math>
which can hold only for positive <math>\rho</math> if <math>\sigma > \beta+1</math>. At the critical value, both equilibrium points lose stability through a subcritical [[Hopf bifurcation]].<ref>{{harvtxt|Hirsch|Smale|Devaney|2003}}, pp. 307+308</ref>
which can hold only for positive <math>\rho</math> if <math>\sigma > \beta+1</math>. At the critical value, both equilibrium points lose stability through a subcritical [[Hopf bifurcation]].<ref>{{harvtxt|Hirsch|Smale|Devaney|2003}}, pp. 307+308</ref>
仅当参数满足公式<math>\rho < \sigma\frac{\sigma+\beta+3}{\sigma-\beta-1} </math>时这两个动态平衡点才会达到稳定。当<math>\sigma > \beta+1</math>时<math>\rho</math>需要取正值才能使这两个点稳定。在取临界值时，一个亚临界的[[霍普夫分岔]]会使这两个动态平衡点失去稳定性。
When <math>\rho = 28</math>, <math>\sigma = 10</math>, and <math>\beta = 8/3</math>, the Lorenz system has chaotic solutions (but not all solutions are chaotic). Almost all initial points will tend to an invariant set{{snd}}the Lorenz attractor{{snd}}a [[Attractor#Strange attractor|strange attractor]], a [[fractal]], and a [[Hidden attractor#Self-excited attractors|self-excited attractor]] with respect to all three equilibria. Its [[Hausdorff dimension]] is estimated from above by the [[Lyapunov dimension|Lyapunov dimension (Kaplan-Yorke dimension)]] as 2.06 ± 0.01,<ref name=Kuznetsov-2020-ND>{{Cite journal |
When <math>\rho = 28</math>, <math>\sigma = 10</math>, and <math>\beta = 8/3</math>, the Lorenz system has chaotic solutions (but not all solutions are chaotic). Almost all initial points will tend to an invariant set{{snd}}the Lorenz attractor{{snd}}a [[Attractor#Strange attractor|strange attractor]], a [[fractal]], and a [[Hidden attractor#Self-excited attractors|self-excited attractor]] with respect to all three equilibria. Its [[Hausdorff dimension]] is estimated from above by the [[Lyapunov dimension|Lyapunov dimension (Kaplan-Yorke dimension)]] as 2.06 ± 0.01,<ref name=Kuznetsov-2020-ND>{{Cite journal |