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文件:HenonMap.svg
Hénon attractor for a = 1.4 and b = 0.3
文件:Henon Multifractal Map movie.gif
Hénon attractor for a = 1.4 and b = 0.3

The Hénon map , sometimes called Hénon-Pomeau attractor/map,[1] is a discrete-time dynamical system. It is one of the most studied examples of dynamical systems that exhibit chaotic behavior. The Hénon map takes a point (xnyn) in the plane and maps it to a new point

thumb|Hénon attractor for a = 1.4 and b = 0.3 thumb|Hénon attractor for a = 1.4 and b = 0.3 The Hénon map , sometimes called Hénon-Pomeau attractor/map,Section 13.3.2; Hsu, Chieh Su. Cell-to-cell mapping: a method of global analysis for nonlinear systems. Vol. 64. Springer Science & Business Media, 2013

is a discrete-time dynamical system. It is one of the most studied examples of dynamical systems that exhibit chaotic behavior. The Hénon map takes a point (xn, yn) in the plane and maps it to a new point

1.4 and b = 0.3 thumb | Hénon attraction for a = 1.4 and b = 0.3 The Hénon map,sometimes called Hénon-au attractor/map,Section 13.3.2; Hsu,cheeh Su.胞胞映射: 非线性系统全局分析的一种方法。第一卷。64.斯普林格科学与商业媒体,2013年是一个离散时间的动力系统。这是一个最研究的例子,动态系统表现出混沌行为。Hénon 映射取平面上的一个点(xn,yn)并将其映射到一个新点

[math]\displaystyle{ \begin{cases}x_{n+1} = 1 - a x_n^2 + y_n\\y_{n+1} = b x_n.\end{cases} }[/math]
\begin{cases}x_{n+1} = 1 - a x_n^2 + y_n\\y_{n+1} = b x_n.\end{cases}
begin { cases } x { n + 1} = 1-a x _ n ^ 2 + y _ n y _ { n + 1} = b x _ n. end { cases }

The map depends on two parameters, a and b, which for the classical Hénon map have values of a = 1.4 and b = 0.3. For the classical values the Hénon map is chaotic. For other values of a and b the map may be chaotic, intermittent, or converge to a periodic orbit. An overview of the type of behavior of the map at different parameter values may be obtained from its orbit diagram.

The map depends on two parameters, a and b, which for the classical Hénon map have values of a = 1.4 and b = 0.3. For the classical values the Hénon map is chaotic. For other values of a and b the map may be chaotic, intermittent, or converge to a periodic orbit. An overview of the type of behavior of the map at different parameter values may be obtained from its orbit diagram.

这个映射依赖于两个参数 a 和 b,对于经典的 Hénon 映射,这两个参数的值分别为 a = 1.4和 b = 0.3。对于经典值,Hénon 映射是混沌的。对于 a 和 b 的其他值,映射可能是混沌的,间歇的,或收敛到一个周期轨道。从它的轨道图可以得到在不同参数值下地图行为类型的概述。

The map was introduced by Michel Hénon as a simplified model of the Poincaré section of the Lorenz model. For the classical map, an initial point of the plane will either approach a set of points known as the Hénon strange attractor, or diverge to infinity. The Hénon attractor is a fractal, smooth in one direction and a Cantor set in another. Numerical estimates yield a correlation dimension of 1.25 ± 0.02[2] and a Hausdorff dimension of 1.261 ± 0.003[3] for the attractor of the classical map.

The map was introduced by Michel Hénon as a simplified model of the Poincaré section of the Lorenz model. For the classical map, an initial point of the plane will either approach a set of points known as the Hénon strange attractor, or diverge to infinity. The Hénon attractor is a fractal, smooth in one direction and a Cantor set in another. Numerical estimates yield a correlation dimension of 1.25 ± 0.02 and a Hausdorff dimension of 1.261 ± 0.003 for the attractor of the classical map.

这张地图是由 Michel Hénon 介绍的,是洛伦兹模型庞加莱截面的简化模型。对于经典映射,平面的一个初始点要么接近一组被称为 Hénon 奇怪吸引子的点,要么向无穷远发散。Hénon 吸引子是一个分形,一个方向光滑,另一个方向是 Cantor 集。数值估计产生的关联维数为1.25 ± 0.02,豪斯多夫维数为1.261 ± 0.003。

Attractor

文件:Henon bifurcation map b=0.3.png
Orbit diagram for the Hénon map with b=0.3. Higher density (darker) indicates increased probability of the variable x acquiring that value for the given value of a. Notice the satellite regions of chaos and periodicity around a=1.075 -- these can arise depending upon initial conditions for x and y.

The Hénon map maps two points into themselves: these are the invariant points. For the classical values of a and b of the Hénon map, one of these points is on the attractor:

thumb|right|Orbit diagram for the Hénon map with b=0.3. Higher density (darker) indicates increased probability of the variable x acquiring that value for the given value of a. Notice the satellite regions of chaos and periodicity around a=1.075 -- these can arise depending upon initial conditions for x and y. The Hénon map maps two points into themselves: these are the invariant points. For the classical values of a and b of the Hénon map, one of these points is on the attractor:

= = 吸引子 = = 拇指 | 右 | b = 0.3的 Hénon 地图的轨道图。较高的密度(较暗)表示变量 x 获取给定值 a 的值的概率增加。注意 a = 1.075周围的混沌和周期性的卫星区域——这些可能根据 x 和 y 的初始条件而产生。赫农映射将两个点映射到它们自己: 这些是不变点。对于 Hénon 映射的经典值 a 和 b,其中一点在吸引子上:

[math]\displaystyle{ x = \frac{\sqrt{609}-7}{28} \approx 0.631354477, }[/math]
[math]\displaystyle{ y = \frac{3\left(\sqrt{609}-7\right)}{280} \approx 0.189406343. }[/math]
x = \frac{\sqrt{609}-7}{28} \approx 0.631354477,
y = \frac{3\left(\sqrt{609}-7\right)}{280} \approx 0.189406343.

28} approx 0.631354477,: y = frac {3 left (sqrt {609}-7 right)}{280} approx 0.189406343.

This point is unstable. Points close to this fixed point and along the slope 1.924 will approach the fixed point and points along the slope -0.156 will move away from the fixed point. These slopes arise from the linearizations of the stable manifold and unstable manifold of the fixed point. The unstable manifold of the fixed point in the attractor is contained in the strange attractor of the Hénon map.

This point is unstable. Points close to this fixed point and along the slope 1.924 will approach the fixed point and points along the slope -0.156 will move away from the fixed point. These slopes arise from the linearizations of the stable manifold and unstable manifold of the fixed point. The unstable manifold of the fixed point in the attractor is contained in the strange attractor of the Hénon map.

这一点是不稳定的。接近这个固定点的点和沿斜坡1.924将接近固定点,而沿斜坡0.156的点将离开固定点。这些斜率来自于不动点的稳定流形和不稳定流形的线性化。吸引子中不动点的不稳定流形包含在 Hénon 映射的奇异吸引子中。

The Hénon map does not have a strange attractor for all values of the parameters a and b. For example, by keeping b fixed at 0.3 the bifurcation diagram shows that for a = 1.25 the Hénon map has a stable periodic orbit as an attractor.

The Hénon map does not have a strange attractor for all values of the parameters a and b. For example, by keeping b fixed at 0.3 the bifurcation diagram shows that for a = 1.25 the Hénon map has a stable periodic orbit as an attractor.

Hénon 映射对参数 a 和 b 的所有值都没有奇异吸引子。例如,通过将 b 固定在0.3,分枝图表明对于 a = 1.25,Hénon 映射有一个稳定的周期轨道作为吸引子。

Cvitanović et al. have shown how the structure of the Hénon strange attractor can be understood in terms of unstable periodic orbits within the attractor.

Cvitanović et al. have shown how the structure of the Hénon strange attractor can be understood in terms of unstable periodic orbits within the attractor.

Cvitanović et al.已经展示了如何用吸引子内不稳定周期轨道来理解 Hénon 奇怪吸引子的结构。

文件:Henon map.gif
Classical Hénon map (15 iterations). Sub-iterations calculated using three steps decomposition.

frame|right|Classical Hénon map (15 iterations). Sub-iterations calculated using three steps decomposition.

帧 | 右 | 经典的 Hénon 映射(15个迭代)。子迭代计算使用三个步骤分解。

Decomposition

The Hénon map may be decomposed into an area-preserving bend:

[math]\displaystyle{ (x_1, y_1) = (x, 1 - ax^2 + y)\, }[/math],

a contraction in the x direction:

[math]\displaystyle{ (x_2, y_2) = (bx_1, y_1)\, }[/math],

and a reflection in the line y = x:

[math]\displaystyle{ (x_3, y_3) = (y_2, x_2)\, }[/math].

The Hénon map may be decomposed into an area-preserving bend:

(x_1, y_1) = (x, 1 - ax^2 + y)\,,

a contraction in the x direction:

(x_2, y_2) = (bx_1, y_1)\,,

and a reflection in the line y = x:

(x_3, y_3) = (y_2, x_2)\,.

= = = 分解 = = Hénon 映射可分解为: (x _ 1,y _ 1) = (x,1-ax ^ 2 + y) ,x 方向的收缩: : (x _ 2,y _ 2) = (bx _ 1,y _ 1) ,以及 y = x: (x _ 3,y _ 3) = (y _ 2,x _ 2)中的反射。

One Dimensional Decomposition

The Hénon map may also be deconstructed into a one-dimensional map, defined similarly to the Fibonacci Sequence.

[math]\displaystyle{ x_{n+1} = 1-a x_n^2 + b x_{n-1} }[/math]

The Hénon map may also be deconstructed into a one-dimensional map, defined similarly to the Fibonacci Sequence.

x_{n+1} = 1-a x_n^2 + b x_{n-1}

= = 一维分解 = = Hénon 映射也可分解为一维映射,定义类似于斐波那契数列。: x { n + 1} = 1-a x _ n ^ 2 + b x _ { n-1}

Special Cases and Low Period Orbits

If one solves the One Dimensional Hénon Map for the special case:

[math]\displaystyle{ X = x_{n-1} = x_n = x_{n+1} }[/math]

One arrives at the simple quadradic:

[math]\displaystyle{ X = 1-a X^2 + b X }[/math]

Or

[math]\displaystyle{ 0 = -a X^2 + (b-1) X + 1 }[/math]

If one solves the One Dimensional Hénon Map for the special case:

X = x_{n-1} = x_n = x_{n+1}

One arrives at the simple quadradic:

X = 1-a X^2 + b X

Or

0 = -a X^2 + (b-1) X + 1

= = = 特殊情况和低周期轨道 = = 若某人解决了特殊情况下的一维 Hénon 映射: : x = x { n-1} = x _ n = x _ { n + 1}一个到达简单的四边形 c: : x = 1-a x ^ 2 + b x 或: 0 =-a x ^ 2 + (b-1) x + 1

The quadratic formula yields:

[math]\displaystyle{ X = {b-1 \pm \sqrt{b^2-2b+1+4a} \over 2a} }[/math]

The quadratic formula yields:

X = {b-1 \pm \sqrt{b^2-2b+1+4a} \over 2a}

二次方程产生: x = { b-1 pm sqrt { b ^ 2-2b + 1 + 4 a }/2 a }

In the special case b=1, this is simplified to

[math]\displaystyle{ X = {\pm \sqrt{a} \over a} }[/math]

In the special case b=1, this is simplified to

X = {\pm \sqrt{a} \over a}

在特殊情况下 b = 1,简化为: x = { pm sqrt { a } over a }

If, in addition, a is in the form [math]\displaystyle{ {1 \over c^n} }[/math] the formula is further simplified to

[math]\displaystyle{ X = \pm c^{n/2} }[/math]

If, in addition, a is in the form {1 \over c^n} the formula is further simplified to

X = \pm c^{n/2}

此外,如果 a 的形式是{1/c ^ n } ,则公式进一步简化为: x = pm c ^ { n/2}

In practice the starting point (X,X) will follow a 4-point loop in two dimensions passing through all quadrants.

In practice the starting point (X,X) will follow a 4-point loop in two dimensions passing through all quadrants.

在实践中,起始点(x,x)将遵循一个四点循环在二维通过所有象限。

[math]\displaystyle{ (X,X) = (X,-X) }[/math]
[math]\displaystyle{ (X,-X) = (-X,-X) }[/math]
[math]\displaystyle{ (-X,-X) = (-X,X) }[/math]
[math]\displaystyle{ (-X,X) = (X,X) }[/math]
(X,X) = (X,-X)
(X,-X) = (-X,-X)
(-X,-X) = (-X,X)
(-X,X) = (X,X)
(X,X) = (X,-X)
(X,-X) = (-X,-X)
(-X,-X) = (-X,X)
(-X,X) = (X,X)

History

In 1976 France, the Lorenz attractor is analyzed by the physicist Yves Pomeau who performs a series of numerical calculations with J.L. Ibanez.[4] The analysis produces a kind of complement to the work of Ruelle (and Lanford) presented in 1975. It is the Lorenz attractor, that is to say, the one corresponding to the original differential equations, and its geometric structure that interest them. Pomeau and Ibanez combine their numerical calculations with the results of mathematical analysis, based on the use of Poincaré sections. Stretching, folding, sensitivity to initial conditions are naturally brought in this context in connection with the Lorenz attractor. If the analysis is ultimately very mathematical, Pomeau and Ibanez follow, in a sense, a physicist approach, experimenting with the Lorenz system numerically.

In 1976 France, the Lorenz attractor is analyzed by the physicist Yves Pomeau who performs a series of numerical calculations with J.L. Ibanez. The analysis produces a kind of complement to the work of Ruelle (and Lanford) presented in 1975. It is the Lorenz attractor, that is to say, the one corresponding to the original differential equations, and its geometric structure that interest them. Pomeau and Ibanez combine their numerical calculations with the results of mathematical analysis, based on the use of Poincaré sections. Stretching, folding, sensitivity to initial conditions are naturally brought in this context in connection with the Lorenz attractor. If the analysis is ultimately very mathematical, Pomeau and Ibanez follow, in a sense, a physicist approach, experimenting with the Lorenz system numerically.

在1976年的法国,物理学家 Yves Pomeau 对洛伦兹吸引子进行了分析,他和 j.l. 进行了一系列的数值计算。女名女子名。这一分析对1975年提出的鲁尔(和兰福德)的著作提供了一种补充。这就是洛伦兹吸引子,也就是说,对应于原始微分方程的吸引子,以及吸引它们的几何结构。和 Ibanez 将他们的数值计算和基于 Poincaré 切片的数学分析结合起来。拉伸,折叠,对初始条件的敏感性在这种情况下自然地与洛伦兹吸引子联系在一起。如果这个分析最终是非常数学化的,那么从某种意义上来说,Pomeau 和 Ibanez 遵循一种物理学方法,用洛伦兹系统进行数值实验。

Two openings are brought specifically by these experiences. They make it possible to highlight a singular behavior of the Lorenz system: there is a transition, characterized by a critical value of the parameters of the system, for which the system switches from a strange attractor position to a configuration in a limit cycle. The importance will be revealed by Pomeau himself (and a collaborator, Paul Manneville) through the "scenario" of Intermittency, proposed in 1979.

Two openings are brought specifically by these experiences. They make it possible to highlight a singular behavior of the Lorenz system: there is a transition, characterized by a critical value of the parameters of the system, for which the system switches from a strange attractor position to a configuration in a limit cycle. The importance will be revealed by Pomeau himself (and a collaborator, Paul Manneville) through the "scenario" of Intermittency, proposed in 1979.

这些经历带来了两个特别的机会。它们使得强调 Lorenz 系统的奇异行为成为可能: 存在一个过渡,即系统参数的一个临界拥有属性,对此系统从一个奇怪的吸引子位置切换到一个极限环中的构型。欧博美本人(及其合作者保罗 · 曼奈维尔)将透过一九七九年提出的「阵发」剧本,揭示其重要性。

The second path suggested by Pomeau and Ibanez is the idea of realizing dynamical systems even simpler than that of Lorenz, but having similar characteristics, and which would make it possible to prove more clearly "evidences" brought to light by numerical calculations. Since the reasoning is based on Poincaré's section, he proposes to produce an application of the plane in itself, rather than a differential equation, imitating the behavior of Lorenz and its strange attractor. He builds one in an ad hoc manner which allows him to better base his reasoning.

The second path suggested by Pomeau and Ibanez is the idea of realizing dynamical systems even simpler than that of Lorenz, but having similar characteristics, and which would make it possible to prove more clearly "evidences" brought to light by numerical calculations. Since the reasoning is based on Poincaré's section, he proposes to produce an application of the plane in itself, rather than a differential equation, imitating the behavior of Lorenz and its strange attractor. He builds one in an ad hoc manner which allows him to better base his reasoning.

和 Ibanez 提出的第二条路径是实现比 Lorenz 更简单但具有相似特征的动力系统,这使得通过数值计算证明更清晰的“证据”成为可能。由于推理是基于庞加莱的截面,他建议模仿洛伦兹及其奇怪吸引子的行为,创造一个平面本身的应用,而不是一个微分方程。他以一种特殊的方式构建一个程序,这使他能够更好地进行推理。

In January 1976, Pomeau presented his work during a seminar given at the Côte d'Azur Observatory, attended by Michel Hénon. Michel Hénon uses Pomeau’s suggestion to obtain a simple system with a strange attractor.[5][6]

In January 1976, Pomeau presented his work during a seminar given at the Côte d'Azur Observatory, attended by Michel Hénon. Michel Hénon uses Pomeau’s suggestion to obtain a simple system with a strange attractor.

1976年1月,au 在蔚蓝海岸天文台的一个研讨会上展示了他的作品,Michel Hénon 也参加了这个研讨会。米歇尔 · 赫农利用博美奥的建议,获得了一个具有奇怪吸引子的简单系统。

See also

  • Horseshoe map
  • Takens' theorem

= = =

  • 马蹄映射
  • Takens 定理

Notes

  1. Section 13.3.2; Hsu, Chieh Su. Cell-to-cell mapping: a method of global analysis for nonlinear systems. Vol. 64. Springer Science & Business Media, 2013
  2. P. Grassberger; I. Procaccia (1983). "Measuring the strangeness of strange attractors". Physica. 9D (1–2): 189–208. Bibcode:1983PhyD....9..189G. doi:10.1016/0167-2789(83)90298-1.
  3. D.A. Russell; J.D. Hanson; E. Ott (1980). "Dimension of strange attractors". Physical Review Letters. 45 (14): 1175. Bibcode:1980PhRvL..45.1175R. doi:10.1103/PhysRevLett.45.1175.
  4. "Pomeau_Ibanez 1976".
  5. "L'attracteur de Hénon".
  6. "Deux exemples français : Yves Pomeau et Michel Hénon".

References

  • . Reprinted in: Chaos and Fractals, A Computer Graphical Journey: Ten Year Compilation of Advanced Research (Ed. C. A. Pickover). Amsterdam, Netherlands: Elsevier, pp. 69–71, 1998

References

  • .年再版: 混沌与分形,计算机图形之旅: 十年高级研究汇编(教育部)。C. a. Pickover).荷兰阿姆斯特丹: Elsevier,pp。69–71, 1998

External links

  • Interactive Henon map and Henon attractor in Chaotic Maps
  • Another interactive iteration of the Henon Map by A. Luhn
  • Orbit Diagram of the Hénon Map by C. Pellicer-Lostao and R. Lopez-Ruiz after work by Ed Pegg Jr, The Wolfram Demonstrations Project.
  • Matlab code for the Hénon Map by M.Suzen
  • Simulation of Hénon map in javascript (experiences.math.cnrs.fr) by Marc Monticelli.

= = 外部链接 = =

  • 混沌地图中的交互式 Henon 映射和 Henon 吸引子
  • Henon 映射的另一个交互式迭代 a. Luhn
  • 由 c. Pellicer-Lostao 和 r. Lopez-Ruiz 绘制的 Hénon 映射的轨道图,完成 Ed Pegg Jr,Wolfram 演示项目的工作后。
  • m. suzen 的 Hénon Map 的 Matlab 代码
  • Marc Monticelli 用 javascript ( experiences.math.cnrs.fr )模拟 Hénon 地图。

模板:Chaos theory


Category:Chaotic maps

类别: 混沌地图


This page was moved from wikipedia:en:Hénon map. Its edit history can be viewed at 厄农映射/edithistory