# 吸引子

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In the mathematical field of dynamical systems, an **attractor** is a set of numerical values toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed.

In the mathematical field of dynamical systems, an attractor is a set of numerical values toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed.

在动力系统的数学领域中，吸引子 attractor是系统在众多初始条件下所趋向的一组数值。即使稍微受到干扰，与吸引子的值接近的系统值仍然能够保证近似性。

In finite-dimensional systems, the evolving variable may be represented algebraically as an *n*-dimensional vector. The attractor is a region in *n*-dimensional space. In physical systems, the *n* dimensions may be, for example, two or three positional coordinates for each of one or more physical entities; in economic systems, they may be separate variables such as the inflation rate and the unemployment rate.

In finite-dimensional systems, the evolving variable may be represented algebraically as an n-dimensional vector. The attractor is a region in n-dimensional space. In physical systems, the n dimensions may be, for example, two or three positional coordinates for each of one or more physical entities; in economic systems, they may be separate variables such as the inflation rate and the unemployment rate.

在有限维系统finite-dimensional systems中，演化变量可用代数表示为 n 维向量。吸引子是 n 维空间中的一个区域。在物理系统中，n 维可以是一个或多个物理实体的两个或三个位置坐标; 在经济系统中，它们可以是单独的变量，如通货膨胀率 inflation rate和失业率unemployment rate。

If the evolving variable is two- or three-dimensional, the attractor of the dynamic process can be represented geometrically in two or three dimensions, (as for example in the three-dimensional case depicted to the right). An attractor can be a point, a finite set of points, a curve, a manifold, or even a complicated set with a fractal structure known as a *strange attractor* (see strange attractor below). If the variable is a scalar, the attractor is a subset of the real number line. Describing the attractors of chaotic dynamical systems has been one of the achievements of chaos theory.

If the evolving variable is two- or three-dimensional, the attractor of the dynamic process can be represented geometrically in two or three dimensions, (as for example in the three-dimensional case depicted to the right). An attractor can be a point, a finite set of points, a curve, a manifold, or even a complicated set with a fractal structure known as a strange attractor (see strange attractor below). If the variable is a scalar, the attractor is a subset of the real number line. Describing the attractors of chaotic dynamical systems has been one of the achievements of chaos theory.

如果演化变量是二维或三维的，则动态过程的吸引子可以用几何方式表示为二维或三维(例如右图所示的三维情况)。一个吸引子可以是一个点，一个有限的点集，一条曲线，一个流形，甚至是一个具有分形结构的复杂集合——我们称之为奇异吸引子 strange attractor (见下面的奇异吸引子)。如果变量是标量，那么吸引子就是实数线的子集。描述 混沌动力学系统chaotic dynamical systems的吸引子是混沌理论的重要成果之一。

A trajectory of the dynamical system in the attractor does not have to satisfy any special constraints except for remaining on the attractor, forward in time. The trajectory may be periodic or chaotic. If a set of points is periodic or chaotic, but the flow in the neighborhood is away from the set, the set is not an attractor, but instead is called a **repeller** (or *repellor*).

A trajectory of the dynamical system in the attractor does not have to satisfy any special constraints except for remaining on the attractor, forward in time. The trajectory may be periodic or chaotic. If a set of points is periodic or chaotic, but the flow in the neighborhood is away from the set, the set is not an attractor, but instead is called a repeller (or repellor).

动力系统在吸引子中的轨迹除了需要保持时间向前，不必满足任何特殊的约束条件。轨迹可能是周期性的，也可能是混沌的。如果一组点是周期性的或混沌的，但其附近的流远离该集合，则该集合不是吸引子，而是 排斥点（或斥点）repeller (or repellor)

## Motivation of attractors 吸引子的动力机制

A dynamical system is generally described by one or more differential or difference equations. The equations of a given dynamical system specify its behavior over any given short period of time. To determine the system's behavior for a longer period, it is often necessary to integrate the equations, either through analytical means or through iteration, often with the aid of computers.

A dynamical system is generally described by one or more differential or difference equations. The equations of a given dynamical system specify its behavior over any given short period of time. To determine the system's behavior for a longer period, it is often necessary to integrate the equations, either through analytical means or through iteration, often with the aid of computers.

我们通常用一个或多个微分方程或差分方程描述动力系统。一个给定动力系统的方程可以表明它在任何给定短时间内的行为。为了确定系统在较长时间内的行为，我们往往需要通过分析手段或 迭代Iteration(通常借助于计算机)来对方程进行积分。

Dynamical systems in the physical world tend to arise from dissipative systems: if it were not for some driving force, the motion would cease. (Dissipation may come from internal friction, thermodynamic losses, or loss of material, among many causes.) The dissipation and the driving force tend to balance, killing off initial transients and settle the system into its typical behavior. The subset of the phase space of the dynamical system corresponding to the typical behavior is the **attractor**, also known as the attracting section or attractee.

Dynamical systems in the physical world tend to arise from dissipative systems: if it were not for some driving force, the motion would cease. (Dissipation may come from internal friction, thermodynamic losses, or loss of material, among many causes.) The dissipation and the driving force tend to balance, killing off initial transients and settle the system into its typical behavior. The subset of the phase space of the dynamical system corresponding to the typical behavior is the attractor, also known as the attracting section or attractee.

物理世界中的动力系统往往产生于耗散系统dissipative system: 如果没有某种驱动力，运动就会停止。(耗散可能来自内部摩擦，热力学损失，材料损失等许多原因。)当耗散和驱动力趋于平衡时，初始瞬态Initial transients会被消除，使系统进入其典型状态。与典型状态相对应的动力系统相空间的子集是吸引子——也称为吸引部分。

Invariant sets and limit sets are similar to the attractor concept. An *invariant set* is a set that evolves to itself under the dynamics.^{[3]} Attractors may contain invariant sets. A *limit set* is a set of points such that there exists some initial state that ends up arbitrarily close to the limit set (i.e. to each point of the set) as time goes to infinity. Attractors are limit sets, but not all limit sets are attractors: It is possible to have some points of a system converge to a limit set, but different points when perturbed slightly off the limit set may get knocked off and never return to the vicinity of the limit set.

Invariant sets and limit sets are similar to the attractor concept. An invariant set is a set that evolves to itself under the dynamics. Attractors may contain invariant sets. A limit set is a set of points such that there exists some initial state that ends up arbitrarily close to the limit set (i.e. to each point of the set) as time goes to infinity. Attractors are limit sets, but not all limit sets are attractors: It is possible to have some points of a system converge to a limit set, but different points when perturbed slightly off the limit set may get knocked off and never return to the vicinity of the limit set.

不变集和极限集的概念与吸引子类似。不变集是在动力学作用下向自身演化的集合。^{[4]} 不变集可能包含于吸引子。极限集是一组点，这些点存在一定的初始状态，但是随着最终时间趋近无穷远时将任意接近极限集(即收敛到集合的每个点）。吸引子是极限集，但不是所有的极限集都是吸引子: 系统的某些点可能会收敛到极限集，但是稍微偏离极限集的点可能会被敲掉，永远不会回到极限集附近。

For example, the damped pendulum has two invariant points: the point x_{0} of minimum height and the point x_{1} of maximum height. The point x_{0} is also a limit set, as trajectories converge to it; the point x_{1} is not a limit set. Because of the dissipation due to air resistance, the point x_{0} is also an attractor. If there was no dissipation, x_{0} would not be an attractor. Aristotle believed that objects moved only as long as they were pushed, which is an early formulation of a dissipative attractor.

For example, the damped pendulum has two invariant points: the point of minimum height and the point of maximum height. The point is also a limit set, as trajectories converge to it; the point is not a limit set. Because of the dissipation due to air resistance, the point is also an attractor. If there was no dissipation, would not be an attractor. Aristotle believed that objects moved only as long as they were pushed, which is an early formulation of a dissipative attractor.

例如，阻尼摆damping ratio|damped有两个不变点: 最小高度点x_{0}和最大高度点x_{1}。点x_{0}也是一个极限集,因为轨迹向它收敛；点 x_{1}不是一个极限集。由于空气阻力的耗散，点x_{0}也是吸引子。如果没有耗散，x_{0}就不会出现吸引子。亚里士多德 Aristotle认为物体只有在被推动时才会移动——这是耗散吸引子 dissipative attractor的早期表述。

Some attractors are known to be chaotic (see #Strange attractor), in which case the evolution of any two distinct points of the attractor result in exponentially diverging trajectories, which complicates prediction when even the smallest noise is present in the system.^{[5]}

Some attractors are known to be chaotic (see #Strange attractor), in which case the evolution of any two distinct points of the attractor result in exponentially diverging trajectories, which complicates prediction when even the smallest noise is present in the system.

有些吸引子是混沌的（参见#奇异吸引子），在这种情况下，吸引子的任意两个不同点的演化都会触发指数发散轨迹。此时即使系统中有一点噪声，预测也会因此而变得复杂。^{[6]}

## Mathematical definition数学定义

Let *t* represent time and let *f*(*t*, •) be a function which specifies the dynamics of the system. That is, if *a* is a point in an *n*-dimensional phase space, representing the initial state of the system, then *f*(0, *a*) = *a* and, for a positive value of *t*, *f*(*t*, *a*) is the result of the evolution of this state after *t* units of time. For example, if the system describes the evolution of a free particle in one dimension then the phase space is the plane **R**^{2} with coordinates (*x*,*v*), where *x* is the position of the particle, *v* is its velocity, *a* = (*x*,*v*), and the evolution is given by

Let t represent time and let f(t, •) be a function which specifies the dynamics of the system. That is, if a is a point in an n-dimensional phase space, representing the initial state of the system, then f(0, a) = a and, for a positive value of t, f(t, a) is the result of the evolution of this state after t units of time. For example, if the system describes the evolution of a free particle in one dimension then the phase space is the plane R^{2} with coordinates (x,v), where x is the position of the particle, v is its velocity, a = (x,v), and the evolution is given by

设 t 表示时间，设 f (t，•)为指定系统动力学的函数。也就是说，如果 a 是 n 维相空间中一个表示系统初始状态的点，那么 f (0，a) = a。对于 t 的正值，f (t，a)是该状态在 t 个时间单位之后演化的结果。例如，如果系统描述了自由粒子在一维空间中的演化，那么相空间是坐标为(x，v)的平面 **R**^{2} ，其中 x 是粒子的位置，v 是粒子的速度，a = (x,v),由以下给出

右|拇指|吸引周期-3旋回及其对“f”（“z”）参数化的直接吸引域。三个最暗的点是3循环的点，它们按顺序相互连接，从吸引域中的任何点迭代都会（通常是渐进的）收敛到这三个点的序列。

Attracting period-3 cycle and its immediate basin of attraction for a certain parametrization of f(z) = z^{2} + c. The three darkest points are the points of the 3-cycle, which lead to each other in sequence, and iteration from any point in the basin of attraction leads to (usually asymptotic) convergence to this sequence of three points.

f(z) = z^{2} + 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。

- [math]\displaystyle{ f(t,(x,v))=(x+tv,v).\ }[/math]

```
[math]\displaystyle{ f(t,(x,v))=(x+tv,v).\ }[/math]
```

F (t，(x，v)） = (x + tv，v)

An **attractor** is a subset *A* of the phase space characterized by the following three conditions:

An attractor is a subset A of the phase space characterized by the following three conditions:

吸引子是相空间 phase space 的子集 subset A——这需要具有以下三个条件：

*A*is*forward invariant*under*f*: if*a*is an element of*A*then so is*f*(*t*,*a*), for all*t*> 0.

- “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。

- There exists a neighborhood of
*A*, called the**basin of attraction**for*A*and denoted*B*(*A*), which consists of all points*b*that "enter*A*in the limit*t*→ ∞". More formally,*B*(*A*) is the set of all points*b*in the phase space with the following property:

- 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：

- For any open neighborhood
*N*of*A*, there is a positive constant*T*such that*f*(*t*,*b*) ∈*N*for all real*t*>*T*.

- For any open neighborhood

For any open neighborhood N of A, there is a positive constant T such that f(t,b) ∈ N for all real t > T.

●对于“A”的任何开邻域“N”，存在一个正常数“T”，使得对所有实数“T”>“T”，有*f*(*t*,*b*) ∈ *N*，。

- There is no proper (non-empty) subset of
*A*having the first two properties.

- “A”中不存在具有前两个属性的正确（非空）子集。

Since the basin of attraction contains an open set containing *A*, every point that is sufficiently close to *A* is attracted to *A*. The definition of an attractor uses a metric on the phase space, but the resulting notion usually depends only on the topology of the phase space. In the case of **R**^{n}, the Euclidean norm is typically used.

Since the basin of attraction contains an open set containing A, every point that is sufficiently close to A is attracted to A. The definition of an attractor uses a metric on the phase space, but the resulting notion usually depends only on the topology of the phase space. In the case of R^{n}, the Euclidean norm is typically used.

由于吸引域包含一个含有 a 的开集合，所以每一个足够接近 a 的点都会被 a 吸引。吸引子的定义使用了相空间上的一个度量，但得到的结果通常只取决于相空间的拓扑结构。在R^{n}的情况下，我们通常会使用欧氏范数Euclidean norm。

Many other definitions of attractor occur in the literature. For example, some authors require that an attractor have positive measure (preventing a point from being an attractor), others relax the requirement that *B*(*A*) be a neighborhood. ^{[7]}

Many other definitions of attractor occur in the literature. For example, some authors require that an attractor have positive measure (preventing a point from being an attractor), others relax the requirement that B(A) be a neighborhood.

在文献中有吸引子的其他定义出现。例如，一些作者要求吸引子具有正测度(防止一个点成为吸引子) ，另一些作者则弱化了B(A)作为一个邻域的要求。^{[8]}

## Types of attractors 吸引子的类型

Attractors are portions or subsets of the phase space of a dynamical system. Until the 1960s, attractors were thought of as being simple geometric subsets of the phase space, like points, lines, surfaces, and simple regions of three-dimensional space. More complex attractors that cannot be categorized as simple geometric subsets, such as topologically wild sets, were known of at the time but were thought to be fragile anomalies. Stephen Smale was able to show that his horseshoe map was robust and that its attractor had the structure of a Cantor set.

Attractors are portions or subsets of the phase space of a dynamical system. Until the 1960s, attractors were thought of as being simple geometric subsets of the phase space, like points, lines, surfaces, and simple regions of three-dimensional space. More complex attractors that cannot be categorized as simple geometric subsets, such as topologically wild sets, were known of at the time but were thought to be fragile anomalies. Stephen Smale was able to show that his horseshoe map was robust and that its attractor had the structure of a Cantor set.

吸引子是动力系统的相空间的一部分或子集。直到20世纪60年代，吸引子被认为是相空间的简单几何子集——像点、线、面和简单的三维空间。更复杂的吸引子，不能被归类为简单的几何子集，如 拓扑野生集topologically wild sets,——虽然在当时是已知的，但却被认为是脆弱的异常事物。斯蒂芬·斯梅尔Stephen Smale能够证明他的马蹄映射是稳定的，它的吸引子具有康托尔集Cantor set结构。

Two simple attractors are a fixed point and the limit cycle. Attractors can take on many other geometric shapes (phase space subsets). But when these sets (or the motions within them) cannot be easily described as simple combinations (e.g. intersection and union) of fundamental geometric objects (e.g. lines, surfaces, spheres, toroids, manifolds), then the attractor is called a *strange attractor*.

Two simple attractors are a fixed point and the limit cycle. Attractors can take on many other geometric shapes (phase space subsets). But when these sets (or the motions within them) cannot be easily described as simple combinations (e.g. intersection and union) of fundamental geometric objects (e.g. lines, surfaces, spheres, toroids, manifolds), then the attractor is called a strange attractor.

两个简单的吸引子是一个不动点fixed point 和一个极限环 limit cycle 。吸引子可以呈现出许多几何形状（相空间子集）。但当这些集合（或其中的运动）不能简单地描述为基本几何对象（例如，直线，曲面，球体，环面，流形）的简单组合（例如，交集和并集）时，这个吸引子就被称为“奇异吸引子”。

### Fixed point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。]]

A fixed point of a function or transformation is a point that is mapped to itself by the function or transformation. If we regard the evolution of a dynamical system as a series of transformations, then there may or may not be a point which remains fixed under each transformation. The final state that a dynamical system evolves towards corresponds to an attracting fixed point of the evolution function for that system, such as the center bottom position of a damped pendulum, the level and flat water line of sloshing water in a glass, or the bottom center of a bowl contain a rolling marble. But the fixed point(s) of a dynamic system is not necessarily an attractor of the system. For example, if the bowl containing a rolling marble was inverted and the marble was balanced on top of the bowl, the center bottom (now top) of the bowl is a fixed state, but not an attractor. This is equivalent to the difference between stable and unstable equilibria. In the case of a marble on top of an inverted bowl (a hill), that point at the top of the bowl (hill) is a fixed point (equilibrium), but not an attractor (stable equilibrium).

A fixed point of a function or transformation is a point that is mapped to itself by the function or transformation. If we regard the evolution of a dynamical system as a series of transformations, then there may or may not be a point which remains fixed under each transformation. The final state that a dynamical system evolves towards corresponds to an attracting fixed point of the evolution function for that system, such as the center bottom position of a damped pendulum, the level and flat water line of sloshing water in a glass, or the bottom center of a bowl contain a rolling marble. But the fixed point(s) of a dynamic system is not necessarily an attractor of the system. For example, if the bowl containing a rolling marble was inverted and the marble was balanced on top of the bowl, the center bottom (now top) of the bowl is a fixed state, but not an attractor. This is equivalent to the difference between stable and unstable equilibria. In the case of a marble on top of an inverted bowl (a hill), that point at the top of the bowl (hill) is a fixed point (equilibrium), but not an attractor (stable equilibrium).

函数或变换的不动点是能够通过函数或变换映射到自身的点。如果我们把动力系统的演化看作是一系列的转变，那么在每一转变中，都可能会有一个点是固定的——当然也可能没有。动力系统的最终状态与该系统演化函数的吸引固定点相对应，例如阻尼摆的中心底部位置，玻璃杯中晃动水的水平线和平坦线，在碗的底部中心滚动的大理石。但是动态系统的不动点不一定是系统的吸引子。例如，如果装有滚动大理石的碗被倒置，大理石在碗的顶部达到平衡状态，碗的中心底部(现在是顶部)就是一个固定的状态但不是一个吸引子。这等价于稳定平衡点和不稳定平衡点之差。如果一个大理石在一个倒碗(山)的顶部，这个在碗(山)的顶部的点是一个固定点(平衡) ，但不是一个吸引子(稳定的平衡)。

In addition, physical dynamic systems with at least one fixed point invariably have multiple fixed points and attractors due to the reality of dynamics in the physical world, including the nonlinear dynamics of stiction, friction, surface roughness, deformation (both elastic and plasticity), and even quantum mechanics.^{[9]} In the case of a marble on top of an inverted bowl, even if the bowl seems perfectly hemispherical, and the marble's spherical shape, are both much more complex surfaces when examined under a microscope, and their shapes change or deform during contact. Any physical surface can be seen to have a rough terrain of multiple peaks, valleys, saddle points, ridges, ravines, and plains.^{[10]} There are many points in this surface terrain (and the dynamic system of a similarly rough marble rolling around on this microscopic terrain) that are considered stationary or fixed points, some of which are categorized as attractors.

In addition, physical dynamic systems with at least one fixed point invariably have multiple fixed points and attractors due to the reality of dynamics in the physical world, including the nonlinear dynamics of stiction, friction, surface roughness, deformation (both elastic and plasticity), and even quantum mechanics. In the case of a marble on top of an inverted bowl, even if the bowl seems perfectly hemispherical, and the marble's spherical shape, are both much more complex surfaces when examined under a microscope, and their shapes change or deform during contact. Any physical surface can be seen to have a rough terrain of multiple peaks, valleys, saddle points, ridges, ravines, and plains. There are many points in this surface terrain (and the dynamic system of a similarly rough marble rolling around on this microscopic terrain) that are considered stationary or fixed points, some of which are categorized as attractors.

此外，由于物理世界动力学的现实性——包括非线性动力学的粘滞，摩擦，表面粗糙度，变形(弹性和塑性) ，甚至量子力学quantum mechanics——至少拥有一个固定点的物理动力系统总是有多个固定点和吸引子^{[9]}。回到倒置碗顶上的大理石这一例子，即使碗看起来是完美的半球形，大理石是规范的球形，在显微镜下观察时它们的表面实际上都十分复杂，它们的形状在接触过程中改变。任何物理表面都可以被视作一个由多个山峰、山谷、鞍点、山脊、峡谷和平原组成的崎岖地形^{[10]}。在这个表面地形中有许多点(以及在这个微观地形上滚动的同样粗糙的大理石的动力系统)被认为是静止的或不动的，其中一些被归类为吸引子。

### Finite number of points有限点数

In a discrete-time system, an attractor can take the form of a finite number of points that are visited in sequence. Each of these points is called a periodic point. This is illustrated by the logistic map, which depending on its specific parameter value can have an attractor consisting of 2^{n} points, 3×2^{n} points, etc., for any value of *n*.

In a discrete-time system, an attractor can take the form of a finite number of points that are visited in sequence. Each of these points is called a periodic point. This is illustrated by the logistic map, which depending on its specific parameter value can have an attractor consisting of 2^{n} points, 3×2^{n} points, etc., for any value of n.

在一个离散时间系统中，吸引子可以以有限数量点的形式出现——这些数量点可以被依次访问。其中每个点都被称为周期点。逻辑图说明了这一点，根据其特定参数值，对于任何“n”值，可以有由2^{n}点、3×2^{n}点等组成的吸引子。

### Limit cycle 极限环

A limit cycle is a periodic orbit of a continuous dynamical system that is isolated. Examples include the swings of a pendulum clock, and the heartbeat while resting. (The limit cycle of an ideal pendulum is not an example of a limit cycle attractor because its orbits are not isolated: in the phase space of the ideal pendulum, near any point of a periodic orbit there is another point that belongs to a different periodic orbit, so the former orbit is not attracting).

A limit cycle is a periodic orbit of a continuous dynamical system that is isolated. Examples include the swings of a pendulum clock, and the heartbeat while resting. (The limit cycle of an ideal pendulum is not an example of a limit cycle attractor because its orbits are not isolated: in the phase space of the ideal pendulum, near any point of a periodic orbit there is another point that belongs to a different periodic orbit, so the former orbit is not attracting).

极限环是连续动力系统的周期轨道，它是孤立点。例如时钟的摆动，以及休息时的心跳。（理想摆的极限环不是极限环吸引子的一个例子，因为它的轨道不是孤立的：在理想摆的相空间中，在一个周期轨道的任何一个点附近都有另一个点属于不同的周期轨道，因此前一个轨道不具有吸引力）。

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

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

### Limit torus 极限环

There may be more than one frequency in the periodic trajectory of the system through the state of a limit cycle. For example, in physics, one frequency may dictate the rate at which a planet orbits a star while a second frequency describes the oscillations in the distance between the two bodies. If two of these frequencies form an irrational fraction (i.e. they are incommensurate), the trajectory is no longer closed, and the limit cycle becomes a limit torus. This kind of attractor is called an *N*_{t} -torus if there are N_{t} incommensurate frequencies. For example, here is a 2-torus:

There may be more than one frequency in the periodic trajectory of the system through the state of a limit cycle. For example, in physics, one frequency may dictate the rate at which a planet orbits a star while a second frequency describes the oscillations in the distance between the two bodies. If two of these frequencies form an irrational fraction (i.e. they are incommensurate), the trajectory is no longer closed, and the limit cycle becomes a limit torus. This kind of attractor is called an -torus if there are incommensurate frequencies. For example, here is a 2-torus:

在处于极限循环状态的系统的周期轨迹中可能存在多个频率。例如，在物理学中，一个频率可以决定一颗行星围绕恒星运行的速率，而第二个频率则描述了两个天体之间的距离振荡。如果其中两个频率形成无理分数 irrational fraction （即它们是非公度），则轨迹不再闭合，极限循环变成极限环。如果存在 N_{t}非公度频率，这种吸引子被称为*N*_{t} 环面。例如这个2环面体：

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A time series corresponding to this attractor is a quasiperiodic series: A discretely sampled sum of N_{t} periodic functions (not necessarily sine waves) with incommensurate frequencies. Such a time series does not have a strict periodicity, but its power spectrum still consists only of sharp lines.

A time series corresponding to this attractor is a quasiperiodic series: A discretely sampled sum of periodic functions (not necessarily sine waves) with incommensurate frequencies. Such a time series does not have a strict periodicity, but its power spectrum still consists only of sharp lines.

与这个吸引子对应的时间序列是一个准周期序列: 具有非公度频率的周期函数(不一定是正弦波)的离散采样和。这样的时间序列不具有严格的周期性，但其功率谱仍然只包含锐线。

### Strange attractor 奇异吸引子

thumb | 200px | right | 洛伦兹奇异吸引子的图， *ρ* = 28, *σ* = 10, *β* = 8/3

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

关于洛伦兹奇异吸引子 ρ = 28，σ = 10，β = 8/3的图

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.^{[11]}

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.

如果吸引子具有分形结构，则称为“奇异”。这种情况通常发生在在当它的动力学系统符合混沌理论时，但是奇异的非混沌吸引子也存在。如果一个奇异吸引子是混沌的，表现出对初始条件的敏感依赖性，那么在吸引子上两个任意接近的备选初始点经过多次迭代后，都会指向任意相距很远的点（受吸引子的限制），而在经历其他次数的迭代之后，会指向任意接近的点。因此，具有混沌吸引子的动态系统是局部不稳定但全局稳定的：一旦一些序列进入吸引子，附近的点就会发散，但不会离开。^{[12]}

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.^{[13]} 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.引用错误：没有找到与`</ref>`

对应的`<ref>`

标签奇异吸引子通常在几个方向上可微，但有些吸引子与康托尘类似，因此不可微。在噪声条件下，人们也能发现奇异吸引子，这可以用来支持Sinai-Ruelle-Bowen型的不变随机概率测度。^{[14]}

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]\displaystyle{ 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]\displaystyle{ r=2.6 }[/math], all starting x values of [math]\displaystyle{ x\lt 0 }[/math] will rapidly lead to function values that go to negative infinity; starting x values of [math]\displaystyle{ x\gt 0 }[/math] will go to infinity. But for [math]\displaystyle{ 0\lt x\lt 1 }[/math] the x values rapidly converge to [math]\displaystyle{ 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]\displaystyle{ 0\lt x\lt 1 }[/math] will lead to function values that alternate between [math]\displaystyle{ x\approx0.513 }[/math] and [math]\displaystyle{ 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变化，具体值可能取决于初始参数。其中一个例子是得到深入研究的逻辑图，[math]\displaystyle{ x{n+1}=rx}n（1-xun） }[/math]，图中显示了参数r各种值的吸引域。如果[math]\displaystyle{ r=2.6 }[/math]，则[math]\displaystyle{ x\lt 0 }[/math]的所有起始x值将迅速使函数值变为负无穷大；[math]\displaystyle{ x\gt 0 }[/math]的起始x值将变为正无穷大。但是对于[math]\displaystyle{ 0\lt x\lt 1 }[/math]，x值会迅速收敛到[math]\displaystyle{ 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，x可以取多个值：如果r为3.2，[math]\displaystyle{ 0\lt x\lt 1 }[/math]的起始值将导致函数值在[math]\displaystyle{ x\approx0.513 }[/math]和[math]\displaystyle{ 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, Hénon attractor, Rössler attractor, and Lorenz attractor.

奇异吸引子的例子包括双涡旋吸引子、 Hénon吸引子、Rössler吸引子和Lorenz吸引子。

## Attractors characterize the evolution of a system吸引子表征系统的演化

An attractor's basin of attraction is the region of the phase space, over which iterations are defined, such that any point (any initial condition) in that region will asymptotically be iterated into the attractor. For a stable linear system, every point in the phase space is in the basin of attraction. However, in nonlinear systems, some points may map directly or asymptotically to infinity, while other points may lie in a different basin of attraction and map asymptotically into a different attractor; other initial conditions may be in or map directly into a non-attracting point or cycle.

吸引子的吸引域是相空间的区域，迭代在这个区域上得到定义，使得该区域中的任何点（任何初始条件）都将渐近地迭代到吸引子中。对于一个稳定的线性系统，相空间中的每一点都在吸引域中。然而，在非线性系统中，有些点可能直接或渐近地映射到无穷大，而另一些点可能位于不同的吸引域中并渐近映射到不同的吸引子；其他初始条件则可能位于或直接映射到非吸引点或循环中。

[[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[math]\displaystyle{ 0\lt x\lt 1 }[/math]的纵坐标上。点的颜色表示在10次^{6次迭代过程中访问点[math]\displaystyle{ （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]\displaystyle{ r\approx3.0 }[/math]附近出现分岔，在[math]\displaystyle{ r\approx3.5 }[/math]附近出现第二个分岔（导致四个吸引子值）。当[math]\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) }[/math], whose basins of attraction for various values of the parameter r are shown in the figure. If [math]\displaystyle{ r=2.6 }[/math], all starting x values of [math]\displaystyle{ x\lt 0 }[/math] will rapidly lead to function values that go to negative infinity; starting x values of [math]\displaystyle{ x\gt 0 }[/math] will go to infinity. But for [math]\displaystyle{ 0\lt x\lt 1 }[/math] the x values rapidly converge to [math]\displaystyle{ 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]\displaystyle{ 0\lt x\lt 1 }[/math] will lead to function values that alternate between [math]\displaystyle{ x\approx0.513 }[/math] and [math]\displaystyle{ 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变化，具体值可能取决于初始参数。一个例子是得到深入研究的逻辑图，[math]\displaystyle{ x{n+1}=rx\u n（1-xün） }[/math]，图中显示了参数r各种值的吸引域。如果[math]\displaystyle{ r=2.6 }[/math]，则[math]\displaystyle{ x\lt 0 }[/math]的所有x值将迅速使函数值变为负无穷大；[math]\displaystyle{ x\gt 0的起始x值将变为正无穷大。但是对于\lt math\gt 0\lt x\lt 1 }[/math]，x值会迅速收敛到[math]\displaystyle{ 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，x可以取多个值：如果r为3.2，[math]\displaystyle{ 0\lt x\lt 1 }[/math]的起始值将导致函数值在[math]\displaystyle{ x\approx0.513 }[/math]和[math]\displaystyle{ 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。

A single-variable (univariate) linear difference equation of the homogeneous form [math]\displaystyle{ x_t=ax_{t-1} }[/math] 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.

齐次形式的单变量（单变量）线性差分方程[math]\displaystyle{ x_t=ax{t-1} }[/math]从除0以外的所有初始点| a|>1发散到无穷大；没有吸引子，因此没有吸引池。但是如果| a |<1，则数线图上的所有点渐进地（或在0的情况下直接）趋向0；0是吸引子，整个数线是吸引域。

## Basins of attraction吸引池

Likewise, a linear matrix difference equation in a dynamic vector X, of the homogeneous form [math]\displaystyle{ X_t=AX_{t-1} }[/math] 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.

同样地，动态向量X中的线性矩阵差分方程，如果a的最大特征值绝对值大于1，则动态向量X中的所有元素[math]\displaystyle{ X_t=AX_{t-1} }[/math] 都将发散到无穷大；不存在吸引子和吸引池。但如果最大特征值小于1，则所有初始向量将渐近收敛于零向量，即零为吸引子；潜在初始向量的整个n维空间就是吸引池 basin of attraction 。

An attractor's **basin of attraction** is the region of the phase space, over which iterations are defined, such that any point (any initial condition) in that region will asymptotically be iterated into the attractor. For a stable linear system, every point in the phase space is in the basin of attraction. However, in nonlinear systems, some points may map directly or asymptotically to infinity, while other points may lie in a different basin of attraction and map asymptotically into a different attractor; other initial conditions may be in or map directly into a non-attracting point or cycle.^{[15]}

吸引子的吸引域是相空间的区域，迭代在这个区域上得到定义，使得该区域中的任何点（任何初始条件）都将渐近地迭代到吸引子中。对于一个稳定的线性系统，相空间中的每一点都在吸引域中。然而，在非线性系统中，有些点可能直接或渐近地映射到无穷大，而另一些点可能位于不同的吸引域中并渐近映射到不同的吸引子；其他初始条件则可能位于或直接映射到非吸引点或循环中。^{[16]}

Similar features apply to linear differential equations. The scalar equation [math]\displaystyle{ dx/dt =ax }[/math] 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 [math]\displaystyle{ dX/dt=AX }[/math] 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.

类似的特征也适用于线性微分方程。标量方程[math]\displaystyle{ dx/dt =ax }[/math] 使得除0以外的所有 x 的初始值在 a > 0时发散到无穷大，但在 a < 0时收敛到吸引子，使整条数线成为0的吸引池。如果矩阵 A 的任何特征值是正的，则该矩阵系统[math]\displaystyle{ dX/dt=AX }[/math]从除零向量以外的所有初始点发散; 但如果所有特征值都是负的，则零向量就是吸引池为整个相空间的吸引子。

### Linear equation or system线性方程或系统

A single-variable (univariate) linear difference equation of the homogeneous form [math]\displaystyle{ x_t=ax_{t-1} }[/math] 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.

如果除了0以外的所有初始点|“A”>>1，单变量线性齐次方程[math]\displaystyle{ x|t=ax{t-1} }[/math]（差分方程）发散到无穷大；没有吸引子，因此没有吸引池。但如果 |*a*| < 1，则数线图上的所有点渐进（或在0的情况下直接映射）到0；0是吸引子，整个数线是吸引池。

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 [math]\displaystyle{ f(x)=x^3-2x^2-11x+12 }[/math], the following initial conditions are in successive basins of attraction:

与线性系统相比，非线性方程或系统可以产生更多行为。一个例子是非线性表达式根的牛顿迭代法。如果表达式有多个实根，则迭代算法的某些起始点会渐近地靠近其中一个根，而其他起始点会得出另一个根。表达式根的吸引池通常并不简单，最接近某一个根的点都被映射到那里，从而形成由附近点组成的吸引区。吸引池在数值上可以是无限的，可以任意小。例如，对于函数[math]\displaystyle{ f(x)=x^3-2x^2-11x+12 }[/math]，以下初始条件在连续的吸引池中：

Likewise, a linear matrix difference equation in a dynamic vector *X*, of the homogeneous form [math]\displaystyle{ X_t=AX_{t-1} }[/math] 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.

同样的，针对在动态向量*X*中的线性矩阵差分方程（在平方矩阵*a'中表现为齐次形式[math]\displaystyle{ X|t=AX{t-1} }[/math]），如果“a”的最大特征值在绝对值上大于1，则动态向量的所有元素将发散到无穷大；没有吸引子，也没有吸引池。但如果最大特征值小于1，则所有初始向量将渐近收敛于零向量，即零为吸引子；潜在初始向量的整个n维空间就是吸引池。*

Basins of attraction in the complex plane for using Newton's method to solve x^{5} − 1 = 0. Points in like-colored regions map to the same root; darker means more iterations are needed to converge.

用牛顿法求解 x^{5} − 1 = 0。相似颜色区域中的点映射到同一个根; 颜色较深意味着需要更多的迭代来收敛。

Similar features apply to linear differential equations. The scalar equation [math]\displaystyle{ dx/dt =ax }[/math] 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 [math]\displaystyle{ dX/dt=AX }[/math] 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.

类似的特征也适用于线性微分方程。标量方程[math]\displaystyle{ dx/dt=ax }[/math]会导致“x”的所有初始值（除了0）发散到无穷大，如果“a”<0，则收敛到值为0的吸引子，使整条数线成为0的吸引域。如果矩阵“A”的任何特征值都为正，则矩阵系统的dX/dt=AX会从除零向量外所有的初始点发散；但如果所有特征值都为负，则零点向量是一个吸引子，其吸引域是整个相空间。

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,^{[17]} for the function [math]\displaystyle{ f(x)=x^3-2x^2-11x+12 }[/math], the following initial conditions are in successive basins of attraction:

与线性系统相比，非线性方程或系统可以产生更多行为。一个例子是非线性表达式根的牛顿迭代法。如果表达式有多个实根，则迭代算法的某些起始点会渐近地靠近其中一个根，而其他起始点会得出另一个根。表达式根的吸引池通常并不简单，最接近某一个根的点都被映射到那里，从而形成由附近点组成的吸引区。吸引池在数值上可以是无限的，可以任意小。例如^{[18]}对于函数[math]\displaystyle{ f(x)=x^3-2x^2-11x+12 }[/math]，以下初始条件在连续的吸引池中
2.352836327 converges to −3;

2.352836327 converges to −3;

2.352836323 converges to 1.

2.352836323汇聚为1。

拇指|复杂平面中的吸引盆地，用于使用牛顿法求解“x”^{5}—1 ；=0。相同颜色区域中的点映射到同一根；较暗表示需要更多迭代才能收敛。

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.

牛顿法也可以应用于求复变函数的根。在复杂的平面上，每个根部都有一个吸引池; 这些区域可以如图所示绘制出来。可以看出，为一个特定的根而组合成的吸引池可以有许多不相连的地区。对于许多复杂函数来说，吸引池的边界为分形。

- 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.

抛物型偏微分方程可能具有有限维吸引子。在某些情况下，方程的扩散部分会阻抑更高的频率，触发一个全局吸引子。金兹堡－朗道方程 Ginzburg-Landau equations 、 K-S方程 Kuramoto-Sivashinsky equations 和二维强迫纳维-斯托克斯方程forced Navier–Stokes equation都具有有限维的全局吸引子。

- 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.

对于具有周期边界条件的三维不可压缩方程 Navier-Stokes equation，如果它有一个全局吸引子那么这个吸引子将是有限维的。

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.

-->

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).

带有初始数据的一个邻近鞍点saddle points (蓝色)的轨迹趋向(红色箭头)无穷大或趋向(黑色箭头)稳定的零平衡点(橙色)。

]]

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

从计算的角度来看，吸引子可以分为“自激吸引子”或*隐藏吸引子*^{[19]}^{[20]}^{[21]}^{[22]} Self-excited attractors can be localized numerically by standard computational procedures, in which after a transient sequence, a trajectory starting from a point on an unstable manifold in a small neighborhood of an unstable equilibrium reaches an attractor, such as the classical attractors in the Van der Pol, Belousov–Zhabotinsky, Lorenz, and many other dynamical systems. In contrast, the basin of attraction of a hidden attractor does not contain neighborhoods of equilibria, so the hidden attractor cannot be localized by standard computational procedures.

自激吸引子可以用标准的计算程序进行数值定域，在一个瞬态序列之后，从不稳定平衡小邻域中不稳定流形上的点出发的轨迹将到达一个吸引子，例如范德波尔振荡器中的经典吸引子Van der Pol oscillator，别洛乌索夫·扎伯廷斯基反应 Belousov–Zhabotinsky reaction，洛伦兹吸引子 Lorenz attractor，以及其他许多动力系统。相比之下，隐藏吸引子的吸引池不包含平衡邻域，因此隐藏吸引子不能用标准的计算程序进行局部化。 -->

## See also 请参阅

- [双曲线集][]

Category:Limit sets

类别: 极限集

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

- ↑ The figure shows the attractor of a second order 3-D Sprott-type polynomial, originally computed by Nicholas Desprez using the Chaoscope freeware (cf. http://www.chaoscope.org/gallery.htm and the linked project files for parameters).
- ↑ The figure shows the attractor of a second order 3-D Sprott-type polynomial, originally computed by Nicholas Desprez using the Chaoscope freeware (cf. http://www.chaoscope.org/gallery.htm and the linked project files for parameters).
- ↑ Carvalho, A.; Langa, J.A.; Robinson, J. (2012).
*Attractors for infinite-dimensional non-autonomous dynamical systems*.**182**. Springer. - ↑ Carvalho, A.; Langa, J.A.; Robinson, J. (2012).
*Attractors for infinite-dimensional non-autonomous dynamical systems*.**182**. Springer. - ↑ Kantz, H.; Schreiber, T. (2004).
*Nonlinear time series analysis*. Cambridge university press. - ↑ Kantz, H.; Schreiber, T. (2004).
*Nonlinear time series analysis*. Cambridge university press. - ↑ John Milnor (1985). "On the concept of attractor".
*Communications in Mathematical Physics*.**99**(2): 177–195. doi:10.1007/BF01212280. - ↑ John Milnor (1985). "On the concept of attractor".
*Communications in Mathematical Physics*.**99**(2): 177–195. doi:10.1007/BF01212280. - ↑
^{9.0}^{9.1}Greenwood, J. A.; J. B. P. Williamson (6 December 1966). "Contact of Nominally Flat Surfaces".*Proceedings of the Royal Society*.**295**(1442): 300–319. doi:10.1098/rspa.1966.0242. - ↑
^{10.0}^{10.1}Vorberger, T. V. (1990).*Surface Finish Metrology Tutorial*. U.S. Department of Commerce, National Institute of Standards (NIST). p. 5. https://www.nist.gov/calibrations/upload/89-4088.pdf. - ↑ Grebogi Celso, Ott Edward, Yorke James A (1987). "Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics".
*Science*.**238**(4827): 632–638. Bibcode:1987Sci...238..632G. doi:10.1126/science.238.4827.632. PMID 17816542.CS1 maint: multiple names: authors list (link) - ↑ Grebogi Celso, Ott Edward, Yorke James A (1987). "Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics".
*Science*.**238**(4827): 632–638. Bibcode:1987Sci...238..632G. doi:10.1126/science.238.4827.632. PMID 17816542.CS1 maint: multiple names: authors list (link) - ↑ Ruelle, David; Takens, Floris (1971). "On the nature of turbulence".
*Communications in Mathematical Physics*.**20**(3): 167–192. doi:10.1007/bf01646553. - ↑ Chekroun M. D.; Simonnet E.; Ghil M. (2011). "Stochastic climate dynamics: Random attractors and time-dependent invariant measures".
**240**(21): 1685–1700. CiteSeerX 10.1.1.156.5891. Unknown parameter`|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]\displaystyle{ 0\lt x\lt 1 }[/math]. The colour of a point indicates how often the point [math]\displaystyle{ (r, x) }[/math] is visited over the course of 10`

ignored (help); Unknown parameter^{6}iterations: frequently encountered values are coloured in blue, less frequently encountered values are yellow. A bifurcation appears around [math]\displaystyle{ r\approx3.0 }[/math], a second bifurcation (leading to four attractor values) around [math]\displaystyle{ r\approx3.5 }[/math]. The behaviour is increasingly complicated for [math]\displaystyle{ r\gt 3.6 }[/math], interspersed with regions of simpler behaviour (white stripes).]] 逻辑映射的分岔图。参数r所有的吸引子显示在区间[math]\displaystyle{ 0\lt x\lt 1 }[/math]的纵坐标上。点的颜色表示在10^{6}次迭代过程中访问点[math]\displaystyle{ (r, x) }[/math]的频率：用蓝色表示经常遇到的值，用黄色表示不太常见的值。在[math]\displaystyle{ r\approx3.0 }[/math]附近出现分叉，在[math]\displaystyle{ r\approx3.5 }[/math]附近出现第二个分叉（导致四个吸引子值）。当[math]\displaystyle{ r\gt 3.6 }[/math]时，行为变得越来越复杂，中间则穿插着简单行为区域（白色条纹）。]] doi=`|Examples of strange attractors include the double-scroll attractor, Hénon attractor, Rössler attractor, and Lorenz attractor. 奇异吸引子的例子包括<font color=`

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(help) - ↑ Strelioff, C.; Hübler, A. (2006). "Medium-Term Prediction of Chaos".
*Phys. Rev. Lett*.**96**(4): 044101. doi:10.1103/PhysRevLett.96.044101. PMID 16486826. - ↑ Strelioff, C.; Hübler, A. (2006). "Medium-Term Prediction of Chaos".
*Phys. Rev. Lett*.**96**(4): 044101. doi:10.1103/PhysRevLett.96.044101. PMID 16486826. - ↑ Dence, Thomas, "Cubics, chaos and Newton's method",
*Mathematical Gazette*81, November 1997, 403–408. - ↑ dance，Thomas，“Cubics，chaos and Newton's method”，“mathematic Gazette”811997年11月，403–408。
- ↑ Leonov G.A.; Vagaitsev V.I.; Kuznetsov N.V. (2011). "Localization of hidden Chua's attractors" (PDF).
*Physics Letters A*.**375**(23): 2230–2233. doi:10.1016/j.physleta.2011.04.037. - ↑ Ruelle, David
第一名: David; Vagaitsev V.I.; Kuznetsov N.V.; Leonov G.A. (2011). "Algorithms for Finding Hidden Oscillations in Nonlinear Systems. The Aizerman and Kalman Conjectures and Chua's Circuits" (PDF).
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*International Journal of Bifurcation and Chaos*.**23**(1): art. no. 1330002. doi:10.1142/S0218127413300024.

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