更改

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==Connection to tent map==

==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.|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.

==Simulations==

在论文的图4中，<ref name=lorenz />洛伦兹将系统在z方向上取得的相对最大值与之前在z方向上的相对最大值作了对比。通过这一步骤我们得到了洛伦兹图（不要与[[彭加莱图]]（Poincaré plot）相混淆，彭加莱图绘制的是轨迹与规定曲面的交点）。这一图型的形状与[[帐篷映射]]（tent map）非常相似。洛伦兹还发现，当Z的最大值值高于某个截断时，系统就会切换到下一个波段（lobe）。通过结合这一点和帐篷映射所表现出的混沌性，洛伦兹阐明了这一系统在两个波段（lobe）之间地切换是混沌的。

===MATLAB simulation===

==模拟==

 

===使用 MATLAB 进行模拟===

<syntaxhighlight lang="matlab">

<syntaxhighlight lang="matlab">

% Solve over time interval [0,100] with initial conditions [1,1,1]

% Solve over time interval [0,100] with initial conditions [1,1,1]

</syntaxhighlight>

</syntaxhighlight>

=== Mathematica simulation ===

=== 使用 Mathematica 进行模拟===

Standard way:

Standard way:

</syntaxhighlight>

</syntaxhighlight>

=== Python simulation ===

=== 使用 Python 进行模拟 ===

<syntaxhighlight lang="python">

<syntaxhighlight lang="python">

import numpy as np

import numpy as np

== Derivation of the Lorenz equations as a model for atmospheric convection ==

== Derivation of the Lorenz equations as a model for atmospheric convection ==

The Lorenz equations are derived from the [[Boussinesq approximation (buoyancy)|Oberbeck–Boussinesq approximation]] to the equations describing fluid circulation in a shallow layer of fluid, heated uniformly from below and cooled uniformly from above.<ref name="lorenz"/> This fluid circulation is known as [[Rayleigh–Bénard convection]]. The fluid is assumed to circulate in two dimensions (vertical and horizontal) with periodic rectangular boundary conditions.

The Lorenz equations are derived from the [[Boussinesq approximation (buoyancy)|Oberbeck–Boussinesq approximation]] to the equations describing fluid circulation in a shallow layer of fluid, heated uniformly from below and cooled uniformly from above.<ref name="lorenz"/> This fluid circulation is known as [[Rayleigh–Bénard convection]]. The fluid is assumed to circulate in two dimensions (vertical and horizontal) with periodic rectangular boundary conditions.

The partial differential equations modeling the system's [[stream function]] and temperature are subjected to a spectral [[Galerkin method|Galerkin approximation]]: the hydrodynamic fields are expanded in Fourier series, which are then severely truncated to a single term for the stream function and two terms for the temperature. This reduces the model equations to a set of three coupled, nonlinear ordinary differential equations. A detailed derivation may be found, for example, in nonlinear dynamics texts.<ref>{{harvtxt|Hilborn|2000}}, Appendix C; {{harvtxt|Bergé|Pomeau|Vidal|1984}}, Appendix D</ref> The Lorenz system is a reduced version of a larger system studied earlier by Barry Saltzman.<ref>{{harvtxt|Saltzman|1962}}</ref>

The partial differential equations modeling the system's [[stream function]] and temperature are subjected to a spectral [[Galerkin method|Galerkin approximation]]: the hydrodynamic fields are expanded in Fourier series, which are then severely truncated to a single term for the stream function and two terms for the temperature. This reduces the model equations to a set of three coupled, nonlinear ordinary differential equations. A detailed derivation may be found, for example, in nonlinear dynamics texts.<ref>{{harvtxt|Hilborn|2000}}, Appendix C; {{harvtxt|Bergé|Pomeau|Vidal|1984}}, Appendix D</ref> The Lorenz system is a reduced version of a larger system studied earlier by Barry Saltzman.<ref>{{harvtxt|Saltzman|1962}}</ref>

== Resolution of Smale's 14th problem ==

== Resolution of Smale's 14th problem ==