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{{Short description|System of ordinary differential equations with chaotic solutions}}

{{Short description|System of ordinary differential equations with chaotic solutions}}

{{Distinguish|Lorenz curve|Lorentz distribution}}

{{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]]

[[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.

{|class="wikitable" width=777px

{|class="wikitable" width=777px

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! 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

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|align="center"|[[Image:Lorenz Ro14 20 41 20-200px.png]]

|align="center"|[[Image:Lorenz Ro14 20 41 20-200px.png]]

|align="center"|'''''ρ'' = 28, ''σ'' = 10, ''β'' = 8/3''' [[:Image:Lorenz Ro28.png|(Enlarge)]]

|align="center"|'''''ρ'' = 28, ''σ'' = 10, ''β'' = 8/3''' [[:Image:Lorenz Ro28.png|(Enlarge)]]

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|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。

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|align="center"|[[Image:Lorenz caos3-175.png]]

|align="center"|[[Image:Lorenz caos3-175.png]]

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|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 10⁻⁵ 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 10⁻⁵ 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10⁻⁵。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。

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==与帐篷映射的关系==

==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.

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.

=== 使用 Mathematica 进行模拟===

=== 使用 Mathematica 进行模拟===

Standard way:

Standard way:

<syntaxhighlight lang="mathematica">

<syntaxhighlight lang="mathematica">

Less verbose:

Less verbose:

<syntaxhighlight lang="mathematica">

<syntaxhighlight lang="mathematica">