# 厄农映射

Hénon attractor for a = 1.4 and b = 0.3

Hénon attractor for a = 1.4 and b = 0.3

The Hénon map , sometimes called Hénon-Pomeau attractor/map, 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

$\displaystyle{ \begin{cases}x_{n+1} = 1 - a x_n^2 + y_n\\y_{n+1} = b x_n.\end{cases} }$
\begin{cases}x_{n+1} = 1 - a x_n^2 + y_n\\y_{n+1} = b x_n.\end{cases}
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.

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.

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.

## Attractor

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:

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

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。

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.

## Decomposition

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

$\displaystyle{ (x_1, y_1) = (x, 1 - ax^2 + y)\, }$,

a contraction in the x direction:

$\displaystyle{ (x_2, y_2) = (bx_1, y_1)\, }$,

and a reflection in the line y = x:

$\displaystyle{ (x_3, y_3) = (y_2, x_2)\, }$.

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.

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

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:

$\displaystyle{ X = x_{n-1} = x_n = x_{n+1} }$

$\displaystyle{ X = 1-a X^2 + b X }$

Or

$\displaystyle{ 0 = -a X^2 + (b-1) X + 1 }$

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

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

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:

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

The quadratic formula yields:

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

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

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

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

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

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

$\displaystyle{ X = \pm c^{n/2} }$

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

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.

$\displaystyle{ (X,X) = (X,-X) }$
$\displaystyle{ (X,-X) = (-X,-X) }$
$\displaystyle{ (-X,-X) = (-X,X) }$
$\displaystyle{ (-X,X) = (X,X) }$
(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)
(-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. 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.

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.

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.

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.

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。