更改

=== Strange attractor 奇异吸引子===<!-- This section is linked from [[Lorenz attractor]] 本节链接自[[洛伦兹吸引子]]-->

=== Strange attractor 奇异吸引子===<!-- This section is linked from [[Lorenz attractor]] 本节链接自[[洛伦兹吸引子]]-->

[[File:Lorenz attractor yb.svg|thumb|200px|right|A plot of Lorenz's strange attractor for values&nbsp;''ρ''&nbsp;=&nbsp;28,&nbsp;''σ''&nbsp;=&nbsp;10,&nbsp;''β''&nbsp;=&nbsp;8/3]]

[[File:Lorenz attractor yb.svg|thumb|200px|right|A plot of Lorenz's strange attractor for values&nbsp;''ρ''&nbsp;=&nbsp;28,&nbsp;''σ''&nbsp;=&nbsp;10,&nbsp;''β''&nbsp;=&nbsp;8/3]]

A plot of Lorenz's strange attractor for values&nbsp;ρ&nbsp;=&nbsp;28,&nbsp;σ&nbsp;=&nbsp;10,&nbsp;β&nbsp;=&nbsp;8/3

A plot of Lorenz's strange attractor for values&nbsp;ρ&nbsp;=&nbsp;28,&nbsp;σ&nbsp;=&nbsp;10,&nbsp;β&nbsp;=&nbsp;8/3

An attractor is called strange if it has a fractal structure. This is often the case when the dynamics on it are chaotic, but strange nonchaotic attractors also exist.  If a strange attractor is chaotic, exhibiting sensitive dependence on initial conditions, then any two arbitrarily close alternative initial points on the attractor, after any of various numbers of iterations, will lead to points that are arbitrarily far apart (subject to the confines of the attractor), and after any of various other numbers of iterations will lead to points that are arbitrarily close together. Thus a dynamic system with a chaotic attractor is locally unstable yet globally stable: once some sequences have entered the attractor, nearby points diverge from one another but never depart from the attractor.

An attractor is called strange if it has a fractal structure. This is often the case when the dynamics on it are chaotic, but strange nonchaotic attractors also exist.  If a strange attractor is chaotic, exhibiting sensitive dependence on initial conditions, then any two arbitrarily close alternative initial points on the attractor, after any of various numbers of iterations, will lead to points that are arbitrarily far apart (subject to the confines of the attractor), and after any of various other numbers of iterations will lead to points that are arbitrarily close together. Thus a dynamic system with a chaotic attractor is locally unstable yet globally stable: once some sequences have entered the attractor, nearby points diverge from one another but never depart from the attractor.

The term strange attractor was coined by David Ruelle and Floris Takens to describe the attractor resulting from a series of bifurcations of a system describing fluid flow. Strange attractors are often differentiable in a few directions, but some are like a Cantor dust, and therefore not differentiable. Strange attractors may also be found in the presence of noise, where they may be shown to support invariant random probability measures of Sinai–Ruelle–Bowen type.

The term strange attractor was coined by David Ruelle and Floris Takens to describe the attractor resulting from a series of bifurcations of a system describing fluid flow. Strange attractors are often differentiable in a few directions, but some are like a Cantor dust, and therefore not differentiable. Strange attractors may also be found in the presence of noise, where they may be shown to support invariant random probability measures of Sinai–Ruelle–Bowen type.

大卫 · 鲁尔和弗洛里斯 · 塔肯斯提出了奇异吸引子一词，用来描述一个描述流体流动的系统的一系列分岔所产生的吸引子。奇异吸引子通常在几个方向上是可微的，但有些吸引子就像康托尘埃，因此是不可微的。在噪声存在的情况下也可以发现奇异吸引子，它们可能支持 Sinai-Ruelle-Bowen 型的不变随机概率测度。

术语“奇异吸引子”是由[[David Ruelle]]和[[Floris Takens]]提出，用来描述一个描述流体流动的系统的一系列[[分岔理论|分叉]]所产生的吸引子。<ref>{{cite journal |last=Ruelle |first=David |last2=Takens |first2=Floris |date=1971 |title=On the nature of turbulence |url=http://projecteuclid.org/euclid.cmp/1103857186 |journal=Communications in Mathematical Physics |volume=20 |issue=3 |pages=167–192 |doi=10.1007/bf01646553}}</ref> 奇异吸引子通常在几个方向上[[可微函数|可微]]，但有些吸引子是[[同胚|样]]一个[[康托尘埃]]，因此不可微。在存在噪声的情况下，也可以发现奇异的吸引子，它们可以证明支持Sinai-Ruelle-Bowen型的不变随机概率测度。<ref name="Stochastic climate dynamics: Random attractors and time-dependent invariant measures">{{cite journal|author1=Chekroun M. D. |author2=Simonnet E. |author3=Ghil M. |author-link3=Michael Ghil |name-list-style=amp|

year = 2011 |

year = 2011 |

Examples of strange attractors include the double-scroll attractor, Hénon attractor, Rössler attractor, and Lorenz attractor.

Examples of strange attractors include the double-scroll attractor, Hénon attractor, Rössler attractor, and Lorenz attractor.

journal = Physica D |

journal = Physica D |

Bifurcation diagram of the [[logistic map. The attractor(s) for any value of the parameter r are shown on the ordinate in the domain <math>0<x<1</math>. The colour of a point indicates how often the point <math>(r, x)</math> is visited over the course of 10⁶ iterations: frequently encountered values are coloured in blue, less frequently encountered values are yellow. A bifurcation appears around <math>r\approx3.0</math>, a second bifurcation (leading to four attractor values) around <math>r\approx3.5</math>. The behaviour is increasingly complicated for <math>r>3.6</math>, interspersed with regions of simpler behaviour (white stripes).]]

Bifurcation diagram of the [[logistic map. The attractor(s) for any value of the parameter r are shown on the ordinate in the domain <math>0<x<1</math>. The colour of a point indicates how often the point <math>(r, x)</math> is visited over the course of 10⁶ iterations: frequently encountered values are coloured in blue, less frequently encountered values are yellow. A bifurcation appears around <math>r\approx3.0</math>, a second bifurcation (leading to four attractor values) around <math>r\approx3.5</math>. The behaviour is increasingly complicated for <math>r>3.6</math>, interspersed with regions of simpler behaviour (white stripes).]]

[逻辑地图]的分枝图。参数 r 的任何值的吸引子都显示在域 < math > 0 < x < 1 </math > 的纵坐标上。点的颜色表示在10 < sup > 6 </sup > 迭代过程中访问点 < math > (r，x) </math > 的频率: 经常遇到的值为蓝色，较少遇到的值为黄色。在 < math > r 左右出现分岔，在 < math > r 左右出现第二个分岔(导致四个吸引子值)。[参考译文]由于[ math > r > 3.6] ，这种行为变得越来越复杂，其中还穿插了一些行为较为简单的区域(白色条纹)

[[逻辑映射的分岔图。参数r的任何值的吸引子显示在区间<math>0<x<1</math>的纵坐标上。点的颜色表示在10⁶次迭代过程中访问点<math>(r, x)</math>的频率：经常遇到的值用蓝色表示，不太常见的值用黄色表示。在<math>r\approx3.0</math>附近出现分叉，在<math>r\approx3.5</math>附近出现第二个分叉（导致四个吸引子值）。当<math>r>3.6</math>时，行为变得越来越复杂，中间穿插着行为更简单的区域（白色条纹）。]]

 

doi = 10.1016/j.physd.2011.06.005|citeseerx=10.1.1.156.5891 }}</ref>

doi = 10.1016/j.physd.2011.06.005|citeseerx=10.1.1.156.5891 }}</ref>

The parameters of a dynamic equation evolve as the equation is iterated, and the specific values may depend on the starting parameters.  An example is the well-studied logistic map,  <math>x_{n+1}=rx_n(1-x_n)</math>, whose basins of attraction for various values of the parameter r are shown in the figure. If <math>r=2.6</math>, all starting x values of <math>x<0</math> will rapidly lead to function values that go to negative infinity; starting x values of <math>x>0</math> will go to infinity. But for <math>0<x<1</math> the x values rapidly converge to <math>x\approx0.615</math>, i.e. at this value of r, a single value of x is an attractor for the function's behaviour. For other values of r, more than one value of x may be visited: if r is 3.2, starting values of <math>0<x<1</math> will lead to function values that alternate between <math>x\approx0.513</math> and <math>x\approx0.799</math>. At some values of r, the attractor is a single point (a "fixed point"), at other values of r two values of x are visited in turn (a period-doubling bifurcation); at yet other values of r, any given number of values of x are visited in turn; finally, for some values of r, an infinitude of points are visited. Thus one and the same dynamic equation can have various types of attractors, depending on its starting parameters.

The parameters of a dynamic equation evolve as the equation is iterated, and the specific values may depend on the starting parameters.  An example is the well-studied logistic map,  <math>x_{n+1}=rx_n(1-x_n)</math>, whose basins of attraction for various values of the parameter r are shown in the figure. If <math>r=2.6</math>, all starting x values of <math>x<0</math> will rapidly lead to function values that go to negative infinity; starting x values of <math>x>0</math> will go to infinity. But for <math>0<x<1</math> the x values rapidly converge to <math>x\approx0.615</math>, i.e. at this value of r, a single value of x is an attractor for the function's behaviour. For other values of r, more than one value of x may be visited: if r is 3.2, starting values of <math>0<x<1</math> will lead to function values that alternate between <math>x\approx0.513</math> and <math>x\approx0.799</math>. At some values of r, the attractor is a single point (a "fixed point"), at other values of r two values of x are visited in turn (a period-doubling bifurcation); at yet other values of r, any given number of values of x are visited in turn; finally, for some values of r, an infinitude of points are visited. Thus one and the same dynamic equation can have various types of attractors, depending on its starting parameters.

动力学方程的参数随着方程的迭代而演化，具体值可能取决于起始参数。一个例子是研究得很好的 logistic 映射，< math > x _ n + 1} = rx _ n (1-x _ n) </math > ，其对参数 r 的各种值的吸引盆如图所示。如果 < math > r = 2.6 </math > ，所有 < math > x < 0 </math > 的起始 x 值将迅速导致函数值变为负无穷大; < math > x > 0 </math > 的起始 x 值将变为无穷大。但是对于 < math > 0 < x < 1 </math > x 值迅速收敛到 < math > x 大约0.615 </math > ，即。在 r 的这个值上，x 的一个单值是函数行为的吸引子。对于 r 的其他值，可以访问一个以上的 x 值: 如果 r 为3.2，那么 < math > 0 < x < 1 </math > 的起始值将导致函数值在 < math > x 大约0.513 </math > 和 < math > x 大约0.799 </math > 之间交替。在 r 的某些值，吸引子是一个单点(一个“不动点”) ，在 r 的其他值，x 的两个值依次访问(一个週期加倍分岔) ; 在 r 的其他值，x 的任意给定数目的值依次访问; 最后，对于 r 的某些值，访问无穷多个点。因此，同一个动力学方程可以有不同类型的吸引子，这取决于它的起始参数。

动力学方程的参数随着方程的迭代而变化，具体值可能取决于初始参数。一个例子是研究得很好的logistic地图，<math>x{n+1}=rx}n（1-xun）</math>，图中显示了参数r的各种值的吸引域。如果<math>r=2.6</math>，则<math>x<0</math>的所有起始x值将迅速导致函数值变为负无穷大；<math>x>0</math>的起始x值将变为无穷大。但是对于<math>0<x<1</math>，x值迅速收敛到<math>x\approx0.615</math>，也就是说，在这个r值下，x的单个值是函数行为的吸引子。对于r的其他值，可以访问x的多个值：如果r为3.2，<math>0<x<1</math>的起始值将导致函数值在<math>x\approx0.513</math>和<math>x\approx0.799</math>之间交替。在r的某些值处，吸引子是一个单点（“不动点”），在r的其他值处，依次访问x的两个值（倍周期分岔）；在r的其他值处，依次访问任意数量的x值；最后，对于r的某些值，访问无穷多个点。因此，同一个动力学方程可以有不同类型的吸引子，这取决于它的起始参数。

Examples of strange attractors include the [[Double scroll attractor|double-scroll attractor]], [[Hénon map|Hénon attractor]], [[Rössler attractor]], and [[Lorenz attractor]].

Examples of strange attractors include the [[Double scroll attractor|double-scroll attractor]], [[Hénon map|Hénon attractor]], [[Rössler attractor]], and [[Lorenz attractor]].

==Attractors characterize the evolution of a system==

==Attractors characterize the evolution of a system==