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doi=10.1007/s11071-020-05856-4| url=https://link.springer.com/article/10.1007/s11071-020-05856-4|doi-access=free }}
doi=10.1007/s11071-020-05856-4| url=https://link.springer.com/article/10.1007/s11071-020-05856-4|doi-access=free }}
</ref> and the [[correlation dimension]] is estimated to be 2.05 ± 0.01.<ref>{{harvtxt|Grassberger|Procaccia|1983}}</ref>
</ref> and the [[correlation dimension]] is estimated to be 2.05 ± 0.01.<ref>{{harvtxt|Grassberger|Procaccia|1983}}</ref>
当<math>\rho = 28</math>、<math>\sigma = 10</math>且<math>\beta = 8/3</math>时，洛伦兹系统存在混沌的解（但不是所有的解都是混沌的）。处于混沌态时几乎所有的初始点都会倾向于一个不变的集合，这个集合正是洛伦兹吸引子。它是一个奇异吸引子[[Attractor#Strange attractor|strange attractor]]，是一个[[分形]]，还是一个自激吸引子[[Hidden attractor#Self-excited attractors|self-excited attractor]]且拥有三个平衡点。它的[[Hausdorff维度]]由Lyapunov维度（Kaplan-Yorke维度）[[Lyapunov dimension|Lyapunov dimension (Kaplan-Yorke dimension)]] 估计为2.06±0.01，
<ref name=Kuznetsov-2020-ND>{{Cite journal |
first1=N.V. |last1=Kuznetsov| first2=T.N. |last2=Mokaev | first3=O.A. |last3=Kuznetsova | first4=E.V. |last4=Kudryashova|
title=The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension|
journal= Nonlinear Dynamics|year=2020 |
doi=10.1007/s11071-020-05856-4| url=https://link.springer.com/article/10.1007/s11071-020-05856-4|doi-access=free }}
</ref>而[[相关维度]]估计为2.05±0.01。<ref>{{harvtxt|Grassberger|Procaccia|1983}}</ref>
The exact Lyapunov dimension formula of the global attractor can be found analytically under classical restrictions on the parameters:<ref>{{harvtxt|Leonov|Kuznetsov|Korzhemanova|Kusakin|2016}}</ref><ref name=Kuznetsov-2020-ND/><ref name=2020-KuznetsovR>{{cite book | first1= Nikolay | last1=Kuznetsov |
The exact Lyapunov dimension formula of the global attractor can be found analytically under classical restrictions on the parameters:<ref>{{harvtxt|Leonov|Kuznetsov|Korzhemanova|Kusakin|2016}}</ref><ref name=Kuznetsov-2020-ND/><ref name=2020-KuznetsovR>{{cite book | first1= Nikolay | last1=Kuznetsov |
first2=Volker | last2=Reitmann | year = 2020| title = Attractor Dimension Estimates for Dynamical Systems: Theory and Computation|
first2=Volker | last2=Reitmann | year = 2020| title = Attractor Dimension Estimates for Dynamical Systems: Theory and Computation|
publisher = Springer| location = Cham|url=https://www.springer.com/gp/book/9783030509866}}</ref>
publisher = Springer| location = Cham|url=https://www.springer.com/gp/book/9783030509866}}</ref>
全局吸引子Lyapunov 维度的精确公式可以通过经典方法分析、限制参数得到:<ref>{{harvtxt|Leonov|Kuznetsov|Korzhemanova|Kusakin|2016}}</ref><ref name=Kuznetsov-2020-ND/><ref name=2020-KuznetsovR>{{cite book | first1= Nikolay | last1=Kuznetsov |
first2=Volker | last2=Reitmann | year = 2020| title = Attractor Dimension Estimates for Dynamical Systems: Theory and Computation|
publisher = Springer| location = Cham|url=https://www.springer.com/gp/book/9783030509866}}</ref>
<math> 3 - \frac{2 (\sigma + \beta + 1)}{\sigma + 1 + \sqrt{(\sigma-1)^2 + 4 \sigma \rho}}. </math>
<math> 3 - \frac{2 (\sigma + \beta + 1)}{\sigma + 1 + \sqrt{(\sigma-1)^2 + 4 \sigma \rho}}. </math>
The Lorenz attractor is difficult to analyze, but the action of the differential equation on the attractor is described by a fairly simple geometric model.<ref>{{Cite journal|title = Structural stability of Lorenz attractors|journal = Publications Mathématiques de l'Institut des Hautes Études Scientifiques|date = 1979-12-01|issn = 0073-8301|pages = 59–72|volume = 50|issue = 1|doi = 10.1007/BF02684769|first = John|last = Guckenheimer|first2 = R. F.|last2 = Williams|url = http://www.numdam.org/item/PMIHES_1979__50__59_0/}}</ref> Proving that this is indeed the case is the fourteenth problem on the list of [[Smale's problems]]. This problem was the first one to be resolved, by [[Warwick Tucker]] in 2002.<ref name="Tucker 2002">{{harvtxt|Tucker|2002}}</ref>
The Lorenz attractor is difficult to analyze, but the action of the differential equation on the attractor is described by a fairly simple geometric model.<ref>{{Cite journal|title = Structural stability of Lorenz attractors|journal = Publications Mathématiques de l'Institut des Hautes Études Scientifiques|date = 1979-12-01|issn = 0073-8301|pages = 59–72|volume = 50|issue = 1|doi = 10.1007/BF02684769|first = John|last = Guckenheimer|first2 = R. F.|last2 = Williams|url = http://www.numdam.org/item/PMIHES_1979__50__59_0/}}</ref> Proving that this is indeed the case is the fourteenth problem on the list of [[Smale's problems]]. This problem was the first one to be resolved, by [[Warwick Tucker]] in 2002.<ref name="Tucker 2002">{{harvtxt|Tucker|2002}}</ref>
尽管描述洛伦兹吸引子的微分方程通过一个相当简单的几何模型就能描述，但洛伦兹吸引子实际上很难分析。<ref>{{Cite journal|title = Structural stability of Lorenz attractors|journal = Publications Mathématiques de l'Institut des Hautes Études Scientifiques|date = 1979-12-01|issn = 0073-8301|pages = 59–72|volume = 50|issue = 1|doi = 10.1007/BF02684769|first = John|last = Guckenheimer|first2 = R. F.|last2 = Williams|url = http://www.numdam.org/item/PMIHES_1979__50__59_0/}}</ref> 这也使得相关问题被列入[[斯迈尔问题（Smale's problems）]]中的第14个。[[Warwick Tucker]]于2002年解决了这一问题，这也是第一个被解决的斯迈尔问题。<ref name="Tucker 2002">{{harvtxt|Tucker|2002}}</ref>
For other values of <math>\rho</math>, the system displays knotted periodic orbits. For example, with <math>\rho = 99.96</math> it becomes a ''T''(3,2) [[torus knot]].
For other values of <math>\rho</math>, the system displays knotted periodic orbits. For example, with <math>\rho = 99.96</math> it becomes a ''T''(3,2) [[torus knot]].
当<math>\rho</math>取其它值时，系统会表现出一个结状的周期轨道。例如，当<math>\rho = 99.96</math>时，系统会变为一个''T''(3,2) [[环面纽结（torus knot）]]
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