# 吸引子

## 数学定义

f(z) = z2 + c某一特定参数的吸引3-周期循环及其直接吸引池。最暗的三个点是3-周期循环的点，它们依次引出，从吸引域中的任何一点迭代都会（通常是渐近的）收敛到这三个点的序列。

$\displaystyle{ f(t,(x,v))=(x+tv,v).\ }$

• “A”是“f”中的“前向不变”：如果“A”是“A”的元素，则对于所有“t”>0，“f”（“t”，“A”）也是。
• 存在一个“A”的邻域称为“A”的“吸引域”，表示为“B”（“A”），它由所有“B”点组成，这些点“B”在极限t → ∞"时“进入”A“。更正式地说，“B”（“A”）是相空间中所有点“B”的集合，具有以下特性：

• “A”中不存在具有前两个属性的正确（非空）子集。

## 吸引子的类型

### 驻点

[[文件：临界轨道3d.png |右|拇指|根据复二次多项式演化的复数的弱吸引不动点。相空间是水平复平面；纵轴测量访问复平面中的点的频率。复平面中峰值频率正下方的点是不动点吸引子。]]

### 极限环

[[文件：VanDerPolPhaseSpace.png|center| 250px |拇指|

Van der Pol相位肖像：吸引极限环

]] Van der Pol phase portrait: an attracting limit cycle]] 范德波尔相图: 一个吸引极限环 ]]

### 奇异吸引子

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

## 吸引子表征系统的演化

[[文件：逻辑图分岔图，Matplotlib.svg|350px |拇指|右|分岔图逻辑图。参数“r”所有的吸引子显示在区间$\displaystyle{ 0\lt x\lt 1 }$的纵坐标上。点的颜色表示在10次6次迭代过程中访问点$\displaystyle{ （r，x） }$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$\displaystyle{ r\approx3.0 }$附近出现分岔，在$\displaystyle{ r\approx3.5 }$附近出现第二个分岔（导致四个吸引子值）。当$\displaystyle{ r\gt 3.6\lt math\gt 时，行为变得越来越复杂，中间穿插着简单行为区域（白色条纹）。]] The parameters of a dynamic equation evolve as the equation is iterated, and the specific values may depend on the starting parameters. An example is the well-studied [[logistic map]], \lt math\gt x_{n+1}=rx_n(1-x_n) }$, whose basins of attraction for various values of the parameter r are shown in the figure. If $\displaystyle{ r=2.6 }$, all starting x values of $\displaystyle{ x\lt 0 }$ will rapidly lead to function values that go to negative infinity; starting x values of $\displaystyle{ x\gt 0 }$ will go to infinity. But for $\displaystyle{ 0\lt x\lt 1 }$ the x values rapidly converge to $\displaystyle{ x\approx0.615 }$, 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 $\displaystyle{ 0\lt x\lt 1 }$ will lead to function values that alternate between $\displaystyle{ x\approx0.513 }$ and $\displaystyle{ x\approx0.799 }$. 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.

## 吸引池

Similar features apply to linear differential equations. The scalar equation $\displaystyle{ dx/dt =ax }$ causes all initial values of x except zero to diverge to infinity if a > 0 but to converge to an attractor at the value 0 if a < 0, making the entire number line the basin of attraction for 0. And the matrix system $\displaystyle{ dX/dt=AX }$ gives divergence from all initial points except the vector of zeroes if any eigenvalue of the matrix A is positive; but if all the eigenvalues are negative the vector of zeroes is an attractor whose basin of attraction is the entire phase space.

### Linear equation or system线性方程或系统

A single-variable (univariate) linear difference equation of the homogeneous form $\displaystyle{ x_t=ax_{t-1} }$ diverges to infinity if |a| > 1 from all initial points except 0; there is no attractor and therefore no basin of attraction. But if |a| < 1 all points on the number line map asymptotically (or directly in the case of 0) to 0; 0 is the attractor, and the entire number line is the basin of attraction.

Equations or systems that are nonlinear can give rise to a richer variety of behavior than can linear systems. One example is Newton's method of iterating to a root of a nonlinear expression. If the expression has more than one real root, some starting points for the iterative algorithm will lead to one of the roots asymptotically, and other starting points will lead to another. The basins of attraction for the expression's roots are generally not simple—it is not simply that the points nearest one root all map there, giving a basin of attraction consisting of nearby points. The basins of attraction can be infinite in number and arbitrarily small. For example, for the function $\displaystyle{ f(x)=x^3-2x^2-11x+12 }$, the following initial conditions are in successive basins of attraction:

Likewise, a linear matrix difference equation in a dynamic vector X, of the homogeneous form $\displaystyle{ X_t=AX_{t-1} }$ in terms of square matrix A will have all elements of the dynamic vector diverge to infinity if the largest eigenvalue of A is greater than 1 in absolute value; there is no attractor and no basin of attraction. But if the largest eigenvalue is less than 1 in magnitude, all initial vectors will asymptotically converge to the zero vector, which is the attractor; the entire n-dimensional space of potential initial vectors is the basin of attraction.

Basins of attraction in the complex plane for using Newton's method to solve x5 − 1 = 0. Points in like-colored regions map to the same root; darker means more iterations are needed to converge.

Similar features apply to linear differential equations. The scalar equation $\displaystyle{ dx/dt =ax }$ causes all initial values of x except zero to diverge to infinity if a > 0 but to converge to an attractor at the value 0 if a < 0, making the entire number line the basin of attraction for 0. And the matrix system $\displaystyle{ dX/dt=AX }$ gives divergence from all initial points except the vector of zeroes if any eigenvalue of the matrix A is positive; but if all the eigenvalues are negative the vector of zeroes is an attractor whose basin of attraction is the entire phase space.

2.35287527 converges to 4;

2.35287527汇聚到4;

### Nonlinear equation or system非线性方程或系统

2.35284172 converges to −3;

2.35284172 收敛到 −3;

2.35283735 converges to 4;

2.35283735收敛到4;

Equations or systems that are nonlinear can give rise to a richer variety of behavior than can linear systems. One example is Newton's method of iterating to a root of a nonlinear expression. If the expression has more than one real root, some starting points for the iterative algorithm will lead to one of the roots asymptotically, and other starting points will lead to another. The basins of attraction for the expression's roots are generally not simple—it is not simply that the points nearest one root all map there, giving a basin of attraction consisting of nearby points. The basins of attraction can be infinite in number and arbitrarily small. For example,[10] for the function $\displaystyle{ f(x)=x^3-2x^2-11x+12 }$, the following initial conditions are in successive basins of attraction:

2.352836327 converges to −3;

2.352836323 converges to 1.

2.352836323汇聚为1。

Basins of attraction in the complex plane for using Newton's method to solve x5 − 1 = 0. Points in like-colored regions map to the same root; darker means more iterations are needed to converge.

Newton's method can also be applied to complex functions to find their roots. Each root has a basin of attraction in the complex plane; these basins can be mapped as in the image shown. As can be seen, the combined basin of attraction for a particular root can have many disconnected regions. For many complex functions, the boundaries of the basins of attraction are fractals.

2.35287527 converges to 4;
2.35284172 converges to −3;
2.35283735 converges to 4;

Parabolic partial differential equations may have finite-dimensional attractors. The diffusive part of the equation damps higher frequencies and in some cases leads to a global attractor. The Ginzburg–Landau, the Kuramoto–Sivashinsky, and the two-dimensional, forced Navier–Stokes equations are all known to have global attractors of finite dimension.

2.352836327 converges to −3;
2.352836323 converges to 1.

For the three-dimensional, incompressible Navier–Stokes equation with periodic boundary conditions, if it has a global attractor, then this attractor will be of finite dimensions.

Newton's method can also be applied to complex functions to find their roots. Each root has a basin of attraction in the complex plane; these basins can be mapped as in the image shown. As can be seen, the combined basin of attraction for a particular root can have many disconnected regions. For many complex functions, the boundaries of the basins of attraction are fractals.

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Chaotic hidden attractor (green domain) in Chua's system. 蔡氏系统中的混沌隐藏吸引子（绿域）。

Trajectories with initial data in a neighborhood of two saddle points (blue) tend (red arrow) to infinity or tend (black arrow) to stable zero equilibrium point (orange).

]]

From a computational point of view, attractors can be naturally regarded as self-excited attractors or

• [双曲线集][]

Category:Limit sets

This page was moved from wikipedia:en:Attractor. Its edit history can be viewed at 吸引子/edithistory

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14. Leonov G.A.; Vagaitsev V.I.; Kuznetsov N.V. (August 2006 journal = Physica D). [http://www.ams.org/notices/200607/what-is-ruelle.pdf Http://www.ams.org/notices/200607/what-is-ruelle.pdf url = http://www.math.spbu.ru/user/nk/PDF/2012-Physica-D-Hidden-attractor-Chua-circuit-smooth.pdf "什么是... 奇异吸引子？ year = 2012"] Check |url= value (help) (PDF). 美国数学学会公告 title = Hidden attractor in smooth Chua systems. 53 53 volume = 241 (7 第7期 issue = 18): 764–765. Retrieved 16 January 2008. Missing pipe in: |date= (help); Unknown parameter |页= ignored (help); Missing pipe in: |issue= (help); Missing pipe in: |journal= (help); Missing pipe in: |url= (help); Missing pipe in: |volume= (help); Missing pipe in: |title= (help); line feed character in |date= at position 12 (help); line feed character in |journal= at position 9 (help); line feed character in |volume= at position 3 (help); line feed character in |title= at position 14 (help); line feed character in |url= at position 53 (help); line feed character in |issue= at position 2 (help); Check date values in: |date= (help) 16 January 2008}} doi = 10.1016/j.physd.2012.05.016}}
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