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吸引子 (查看源代码)

2020年10月25日 (日) 17:55的版本

删除2,641字节 、 2020年10月25日 (日) 17:55
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Moved page from wikipedia:en:Attractor (history)
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此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。
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此词条暂由彩云小译翻译，翻译字数共3237，未经人工整理和审校，带来阅读不便，请见谅。
    
{{short description|Concept in dynamical systems}}
 
{{short description|Concept in dynamical systems}}
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{{more footnotes|date=March 2013}}
 
{{more footnotes|date=March 2013}}
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[[File:Attractor Poisson Saturne.jpg|right|333px|thumb|Visual representation of a strange attractor]]
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[[File:Attractor Poisson Saturne.jpg|right|333px|thumb|Visual representation of a [[#Strange_attractor|strange attractor]]<ref>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).</ref>.]]
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Visual representation of a strange attractor
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strange attractor.]]
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奇异吸引子的可视化表示
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[奇怪的吸引子]
    
In the [[mathematics|mathematical]] field of [[dynamical system]]s, 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 [[mathematics|mathematical]] field of [[dynamical system]]s, 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.
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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.
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在动力系统的数学领域中，吸引子是系统在各种起始条件下趋于演化的一组数值。系统的价值观，得到足够接近吸引器的价值保持接近，即使轻微扰动。
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在动力系统的数学领域中，吸引子是系统在各种起始条件下趋于演化的一组数值。系统的价值观，得到足够接近的吸引器价值仍然接近，即使轻微扰动。
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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.
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如果演化变量是二维或三维的，则动态过程的吸引子可以几何地表示为二维或三维(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，甚至一个具有分形结构的复杂集合，称为奇怪吸引子(见下面的奇怪吸引子)。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的吸引子是混沌理论的重要成果之一。
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如果演化变量是二维或三维的，则动态过程的吸引子可以几何地表示为二维或三维(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，甚至是一个复杂的集合，具有一个分形结构称为奇怪吸引子(见下面的奇怪吸引子)。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的吸引子是混沌理论的重要成果之一。
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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).
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动力系统在吸引子中的轨迹不需要满足任何特殊的约束条件，除了保持在吸引子上，在时间上前进。轨迹可能是周期性的或混沌的。如果一组点是周期性的或混沌的，但是邻近的流离开了这组点，这组点不是吸引子，而是被称为排斥者(或排斥者)。
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在吸引子中的动力系统轨迹不需要满足任何特殊的约束条件，除了在时间上保持在吸引子上。轨迹可能是周期性的或混沌的。如果一组点是周期性的或混沌的，但是邻近的流离开了这组点，这组点不是一个吸引子，而是被称为排斥者(或排斥者)。
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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.
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动力系统通常由一个或多个微分方程或差分方程描述。一个给定动力系统的方程确定了它在任何给定的短时间内的行为。为了确定系统在较长时间内的行为，往往需要通过分析手段或通过迭代(通常借助于计算机)对方程进行积分。
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动力系统通常由一个或多个微分方程或差分方程描述。一个给定动力系统的方程表明了它在任何给定的短时间内的行为。为了确定系统在较长时间内的行为，往往需要通过分析手段或通过迭代(通常借助于计算机)对方程进行积分。
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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.
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物理世界中的动力系统往往产生于耗散系统: 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，或材料损失等许多原因。)耗散和驱动力趋于平衡，消除初始瞬态，使系统进入典型状态。与典型行为相对应的动力系统相空间的子集是吸引子，也称为吸引部分或吸引子。
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物理世界中的动力系统往往产生于耗散系统: 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，或材料损失等许多原因。)耗散和驱动力趋于平衡，消除初始瞬变，使系统进入其典型状态。与典型行为相对应的动力系统相空间的子集是吸引子，也称为吸引部分或吸引子。
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Invariant sets and [[limit set]]s 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.
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Invariant sets and [[limit set]]s are similar to the attractor concept. An ''invariant set'' is a set that evolves to itself under the dynamics.<ref>{{cite book|author1=Carvalho, A.|author2=Langa, J.A.|author3=Robinson, J.|year=2012|title=Attractors for infinite-dimensional non-autonomous dynamical systems|volume=182|publisher=Springer|p=109}}</ref> 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.
 
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.
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不变集和极限集类似于吸引子的概念。不变集是在动力学下向自身演化的集合。吸引子可能包含不变集。极限集是一组点，这些点存在一些初始状态，这些初始状态最终任意接近极限集(即。随着时间推移到无穷远。吸引子是极限集，但不是所有的极限集都是吸引子: 系统的某些点收敛到极限集是可能的，但是稍微偏离极限集的不同点可能会被敲掉，而且永远不会回到极限集的附近。
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不变集和极限集类似于吸引子的概念。不变集是在动力学下向自身演化的集合。吸引子可能包含不变集。极限集是一组点，这些点存在一些初始状态，这些初始状态最终任意接近极限集(即。随着时间的推移到无穷远。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不同点可能会被敲掉，永远不会回到极限集附近。
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Exponential [[chaos theory|divergence of trajectories]] complicates detailed predictions, but the world is knowable due to the existence of robust attractors.
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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 [[chaos theory|diverging trajectories]], which complicates prediction when even the smallest noise is present in the system.<ref>{{cite book|author1=Kantz, H.|author2=Schreiber, T.|year=2004|title=Nonlinear time series analysis|publisher=Cambridge university press}}</ref>
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Exponential divergence of trajectories complicates detailed predictions, but the world is knowable due to the existence of robust attractors.
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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.
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轨迹的指数分散使详细的预测变得复杂，但由于存在强大的吸引子，世界是可知的。
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一些吸引子被认为是混沌的(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导致了指数发散轨迹，当系统中甚至存在最小的噪声时，这使预测变得复杂。
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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<sup>2</sup> with coordinates (x,v), where x is the position of the particle, v is its velocity, a&nbsp;=&nbsp;(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<sup>2</sup> with coordinates (x,v), where x is the position of the particle, v is its velocity, a&nbsp;=&nbsp;(x,v), and the evolution is given by
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设 t 表示时间，设 f (t，•)是指定系统动力学的函数。也就是说，如果 a 是 n 维相空间中的一个点，表示系统的初始状态，那么 f (0，a) a，对于 t 的正值，f (t，a)是这个状态在 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，那么相空间就是平面 r sup 2 / sup 与坐标(x，v) ，其中 x 是粒子的位置，v 是粒子的速度，a (x，v) ，演化过程由
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设 t 表示时间，设 f (t，•)是指定系统动力学的函数。也就是说，如果 a 是 n 维相空间中的一个点，表示系统的初始状态，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，那么相空间就是平面 r < sup > 2 </sup > 和坐标(x，v) ，其中 x 是粒子的位置，v 是粒子的速度，a = (x，v)
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Attracting period-3 cycle and its immediate basin of attraction for a certain parametrization of f(z)&nbsp;=&nbsp;z<sup>2</sup>&nbsp;+&nbsp;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.
 
Attracting period-3 cycle and its immediate basin of attraction for a certain parametrization of f(z)&nbsp;=&nbsp;z<sup>2</sup>&nbsp;+&nbsp;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.
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F (z) zsup 2 / sup + c 参数化的吸引周期 -3旋回及其直接吸引盆。 三个最黑暗的点是三个循环中的点，它们按顺序相互连接，从吸力盆地中的任意点迭代到这三个点的顺序(通常是渐近的)收敛。
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F (z) = z < sup > 2 </sup > + c 某一参数的吸引周期 -3旋回及其直接吸引盆。三个最黑暗的点是三个循环中的点，它们按顺序相互连接，从吸力盆地中的任意点迭代到这三个点的顺序(通常是渐近的)收敛。
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  <math> f(t,(x,v))=(x+tv,v).\ </math>
 
  <math> f(t,(x,v))=(x+tv,v).\ </math>
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Math f (t，(x，v))(x + tv，v) .  / math
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F (t，(x，v) = (x + tv，v)
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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:
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吸引子是相空间 a 的子集，它具有以下三个拥有属性:
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吸引子是相空间 a 的一个子集，它具有以下三个拥有属性:
    
* ''A'' is ''forward invariant'' under ''f'': if ''a'' is an element of ''A'' then so is ''f''(''t'',''a''), for all&nbsp;''t''&nbsp;>&nbsp;0.
 
* ''A'' is ''forward invariant'' under ''f'': if ''a'' is an element of ''A'' then so is ''f''(''t'',''a''), for all&nbsp;''t''&nbsp;>&nbsp;0.
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  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 N of A, there is a positive constant T such that f(t,b) ∈ N for all real t > T.
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对于 a 的任意开邻域 n，存在一个正常数 t，使得 f (t，b)∈ n 对于所有实 t。
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对于 a 的任意开邻域 n，存在一个正常数 t 使得 f (t，b)∈ n 对于所有实 t > t。
    
* There is no proper (non-empty) subset of ''A'' having the first two properties.
 
* There is no proper (non-empty) subset of ''A'' having the first two properties.
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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<sup>n</sup>, 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<sup>n</sup>, the Euclidean norm is typically used.
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由于吸引盆包含一个含有 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依赖于相空间的拓扑结构。在 r sup n / sup 的情况下，通常使用欧氏范数。
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由于吸引盆包含一个含有 a 的开集，所以每一个离 a 足够近的点都会被 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依赖于相空间的拓扑结构。在 r < sup > n </sup > 的情况下，通常使用欧氏范数。
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Many other definitions of attractor occur in the literature.  For example, some authors require that an attractor have positive [[measure (mathematics)|measure]] (preventing a point from being an attractor), others relax the requirement that ''B''(''A'') be a neighborhood. <ref>Milnor, J. (1985). "On the Concept of Attractor." Comm. Math. Phys 99: 177–195.
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Many other definitions of attractor occur in the literature.  For example, some authors require that an attractor have positive [[measure (mathematics)|measure]] (preventing a point from being an attractor), others relax the requirement that ''B''(''A'') be a neighborhood. <ref>{{cite journal | author=John Milnor | author-link=John Milnor | title= On the concept of attractor | journal=Communications in Mathematical Physics | year=1985 | volume=99 | pages=177–195| doi= 10.1007/BF01212280 | issue=2}}</ref>
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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. <ref>Milnor, J. (1985). "On the Concept of Attractor." Comm. Math. Phys 99: 177–195.
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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.  
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吸引子的许多其他定义出现在文献中。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) ，另一些作者放松了 b (a)是一个邻域的要求。裁判米尔诺，j。(1985).“关于吸引子的概念。”委员会。数学。体育99:177-195。
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吸引子的许多其他定义出现在文献中。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) ，另一些作者放松了 b (a)是一个邻域的要求。
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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.
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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，在当时是已知的，但被认为是脆弱的异常。斯蒂芬 · 斯梅尔能够证明他的马蹄地图是健壮的，它的吸引子具有康托集的结构。
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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，吸引子都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区域。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，在当时是已知的，但被认为是脆弱的异常。斯蒂芬 · 斯梅尔能够证明他的马蹄地图是健壮的，它的吸引子具有康托集的结构。
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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).
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函数或变换的不动点是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，那么在每一个转变下，可能会有一个点保持不变，也可能没有。动力系统的最终状态对应于该系统演化函数的吸引固定点，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，或碗的底部中心含有滚动的大理石。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，大理石平衡在碗的顶部，碗的中心底部(现在是顶部)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，这个点在碗(山)的顶部是一个固定点(平衡) ，但不是一个吸引子(稳定的平衡)。
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函数或变换的不动点是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，那么在每一个转变下，可能会有一个点是固定的，也可能没有。动力系统的最终状态对应于该系统演化函数的吸引固定点，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，或碗的底部中心含有滚动的大理石。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，大理石平衡在碗的顶部，碗的中心底部(现在是顶部)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，这个点在碗(山)的顶部是一个固定点(平衡) ，但不是一个吸引子(稳定的平衡)。
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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.
 
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.
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此外，至少有一个固定点的物理动力系统，由于物理世界中动力学的现实性，包括粘滞、摩擦、表面粗糙度、变形(弹性和塑性)甚至量子力学的非线性动力学，总是有多个固定点和吸引子。在倒置碗顶部的大理石上，即使碗看起来完美的半球形，和大理石的球形，在显微镜下观察时都是更复杂的表面，它们的形状在接触过程中改变或变形。任何物理表面都可以看到一个由多个山峰、山谷、鞍点、山脊、峡谷和平原组成的崎岖地形。在这个表面地形中有许多点(以及类似的粗糙大理石在这个微观地形上滚动的动力系统)被认为是静止或固定的点，其中一些被归类为吸引子。
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此外，至少有一个固定点的物理动力系统，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，总是有多个固定点和吸引子。在倒置碗顶的大理石上，即使碗看起来完美的半球形，和大理石的球形，在显微镜下观察时都是更复杂的表面，它们的形状在接触过程中改变或变形。任何物理表面都可以看到一个由多个山峰、山谷、鞍点、山脊、峡谷和平原组成的崎岖地形。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点，其中一些被归类为吸引子。
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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<sup>n</sup> points, 3×2<sup>n</sup> 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<sup>n</sup> points, 3×2<sup>n</sup> points, etc., for any value of n.
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在一个离散时间系统中，一个吸引子可以采取有限数目的点的形式，这些点按顺序访问。这些点中的每一个都称为周期点。这可以用 logistic 映射来说明，对于 n 的任意值，根据其特定的参数值，可以有一个由2个 sup n / sup 点、32 n / sup 点等组成的吸引子。
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在一个离散时间系统中，一个吸引子可以采取有限数目的点的形式，这些点按顺序访问。这些点中的每一个都称为周期点。这可以用 logistic 映射来说明，对于 n 的任意值，根据其特定的参数值，可以有一个由2个 < sup > n </sup > 点、3 × 2 < sup > n </sup > 点等组成的吸引子。
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Van der Pol phase portrait: an attracting limit cycle</center>]]
 
Van der Pol phase portrait: an attracting limit cycle</center>]]
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范德波尔相图: 一个吸引的极限环 / 中心]
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范德波尔相图: 一个吸引极限环 </center > ]
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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:
 
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:
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在系统通过极限环状态的周期轨迹中，可能存在一个以上的频率。例如，在物理学中，一个频率可以决定一颗行星绕恒星运行的速率，而另一个频率则描述两个天体之间距离的振荡。如果其中两个频率构成一个无理分数(即。它们是不相称的) ，轨迹不再是闭合的，极限环变成了极限环。如果存在不相称的频率，这种吸引子称为环面。例如，这里有一个2-torus:
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在系统通过极限环状态的周期轨迹中，可能存在一个以上的频率。例如，在物理学中，一个频率可能决定行星绕恒星运行的速率，而另一个频率描述两个天体之间距离的振荡。如果其中两个频率构成一个无理分数(即。它们是不相称的) ，轨迹不再是闭合的，极限环变成了极限环。这种吸引子称为环面，如果存在不相称的频率。例如，这里有一个2-torus:
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=== Strange attractor ===<!-- This section is linked from Lorenz attractor -->
 
=== Strange attractor ===<!-- This section is linked from Lorenz attractor -->
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奇怪的吸引子! -- 这一部分是连接从洛伦兹吸引子 --  
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= = = 奇怪的吸引子 = = = < ! -- 这个部分与洛伦兹吸引子相连 -- >
    
[[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]]
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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
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28,10,8 / 3洛伦兹奇怪吸引子图
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关于洛伦兹奇怪吸引子 ρ = 28，σ = 10，β = 8/3的图
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An attractor is called '''strange''' if it has a [[fractal]] structure. This is often the case when the dynamics on it are [[chaos theory|chaotic]], but [[strange nonchaotic attractor]]s 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.<ref>{{cite journal | author = Grebogi Celso, Ott Edward, Yorke James A | year = 1987 | title = Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics | url = | journal = Science | volume = 238 | issue = 4827| pages = 632–638 | doi = 10.1126/science.238.4827.632 }}</ref>
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An attractor is called '''strange''' if it has a [[fractal]] structure. This is often the case when the dynamics on it are [[chaos theory|chaotic]], but [[strange nonchaotic attractor]]s 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.<ref>{{cite journal | author = Grebogi Celso, Ott Edward, Yorke James A | year = 1987 | title = Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics | url = | journal = Science | volume = 238 | issue = 4827| pages = 632–638 | doi = 10.1126/science.238.4827.632 | pmid = 17816542 | bibcode = 1987Sci...238..632G }}</ref>
    
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.
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具有分形结构的吸引子称为奇异吸引子。当系统的动力学是混沌的时候，这种情况经常发生，但是奇异的非混沌吸引子也存在。如果一个奇怪的吸引子是混沌的，表现出对初始条件的敏感依赖性，那么任意两个任意闭合的交替初始点，经过任意数目的迭代，都会导致点相距任意远(受吸引子的限制) ，在任意数目的其他迭代之后，会导致点相距任意近。因此，一个具有混沌吸引子的动力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，邻近的点彼此发散，但永远不会离开吸引子。
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一个吸引子如果具有分形结构就称为奇异吸引子。当系统的动力学是混沌的时候，这种情况经常发生，但是奇异的非混沌吸引子也存在。如果一个奇怪的吸引子是混沌的，表现出对初始条件的敏感依赖性，那么任意两个任意闭合的交替初始点，经过任意数目的迭代，都会导致点相距任意远(受吸引子的限制) ，在任意数目的其他迭代之后，会导致点相距任意近。因此，一个具有混沌吸引子的动力系统是局部不稳定的，但是全局稳定的: 一旦一些序列进入吸引子，邻近的点就会彼此分离，但是永远不会离开吸引子。
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The term '''strange attractor''' was coined by [[David Ruelle]] and [[Floris Takens]] to describe the attractor resulting from a series of [[bifurcation theory|bifurcations]] of a system describing fluid flow.<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> Strange attractors are often [[Differentiable function|differentiable]] in a few directions, but some are [[homeomorphic|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.<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 |last-author-amp=yes |
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The term '''strange attractor''' was coined by [[David Ruelle]] and [[Floris Takens]] to describe the attractor resulting from a series of [[bifurcation theory|bifurcations]] of a system describing fluid flow.<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> Strange attractors are often [[Differentiable function|differentiable]] in a few directions, but some are [[homeomorphic|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.<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|
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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.<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 |last-author-amp=yes |
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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.
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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。奇异吸引子常常在几个方向上是可微的，但有些吸引子就像康托尘埃，因此不可微。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。 随机气候动力学: 随机吸引子和依赖时间的不变量{ cite journal | author1 Chekroun m. d. | author2 Simonnet e. | author3 Ghil m. | author-link3 Michael Ghil | last-author-amp yes |
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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。
    
year = 2011 |
 
year = 2011 |
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year = 2011 |
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2011年
      
title = Stochastic climate dynamics: Random attractors and time-dependent invariant measures |
 
title = Stochastic climate dynamics: Random attractors and time-dependent invariant measures |
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title = Stochastic climate dynamics: Random attractors and time-dependent invariant measures |
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Examples of strange attractors include the double-scroll attractor, Hénon attractor, Rössler attractor, and Lorenz attractor.
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标题随机 climate dynamics: 随机吸引子和依赖时间的不变测度 |
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journal = Physica D |
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奇异吸引子的例子包括双涡卷吸引子、 h é non 吸引子、若斯叻吸引子和 Lorenz 吸引子。
    
journal = Physica D |
 
journal = Physica D |
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物理学杂志 d |
      
volume = 240 |
 
volume = 240 |
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volume = 240 |
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第240卷
      
issue = 21 |
 
issue = 21 |
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issue = 21 |
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第21期
      
pages = 1685–1700 |
 
pages = 1685–1700 |
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pages = 1685–1700 |
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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 10<sup>6</sup> 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).]]
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第1685-1700页
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[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)
    
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>
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doi = 10.1016/j.physd.2011.06.005|citeseerx=10.1.1.156.5891 }}</ref>
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10.1016 / j.physd. 2011.06.005 | citeseerx 10.1.1.156.5891} / ref
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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.
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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。
    
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]].
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Examples of strange attractors include the double-scroll attractor, Hénon attractor, Rössler attractor, and Lorenz attractor.
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奇异吸引子的例子包括双涡卷吸引子、 h 非吸引子、若斯叻吸引子和 Lorenz 吸引子。
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==Attractors characterize the evolution of a system==
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==Effect of parameters on the attractor==
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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.
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吸引子的吸引盆是相空间的一个区域，在这个区域上定义迭代，使得该区域内的任何点(任何初始条件)都渐近迭代到吸引子。对于稳定的线性系统，相空间中的每一点都处于吸引盆中。然而，在非线性系统中，一些点可以直接或渐近地映射到无穷远处，而另一些点可能位于不同的吸引域中，并渐近地映射到不同的吸引子中，其他的初始条件可能位于或直接映射到不吸引点或不吸引周期中。
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[[File:Logistic Map Bifurcation Diagram, Matplotlib.svg|300px|thumb|right|Bifurcation diagram of the [[logistic map]]. The attractor for any value of the parameter ''r'' is shown on the vertical line at that ''r''.]]
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[[File:Logistic Map Bifurcation Diagram, Matplotlib.svg|350px|thumb|right|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 10<sup>6</sup> iterations: frequently encountered values are coloured in blue, less frequently encountered values are yellow. A [[period-doubling bifurcation|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).]]
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Bifurcation diagram of the [[logistic map. The attractor for any value of the parameter r is shown on the vertical line at that r.]]
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[逻辑地图]的分枝图。参数 r 的任意值的吸引子都在 r 的垂直线上显示出来
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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|"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.
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A particular functional form of a dynamic equation can have various types of attractor depending on the particular parameter values used in the function. An example is the well-studied [[logistic map]],  <math>x_{t+1}=rx_t(1-x_t),</math> whose basins of attraction for various values of the parameter ''r'' are shown in the diagram. At some values of the parameter the attractor is a single point, at others it is two points that are visited in turn, at others it is 2<sup>''n''</sup> points or ''k''&nbsp;×&nbsp;2<sup>''n''</sup> points that are visited in turn, for any value of ''n'' depending on the value of the parameter ''r'', and at other values of ''r'' an infinitude of points are visited.
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A particular functional form of a dynamic equation can have various types of attractor depending on the particular parameter values used in the function. An example is the well-studied logistic map,  <math>x_{t+1}=rx_t(1-x_t),</math> whose basins of attraction for various values of the parameter r are shown in the diagram. At some values of the parameter the attractor is a single point, at others it is two points that are visited in turn, at others it is 2<sup>n</sup> points or k&nbsp;×&nbsp;2<sup>n</sup> points that are visited in turn, for any value of n depending on the value of the parameter r, and at other values of r an infinitude of points are visited.
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一个特定的函数形式的动力学方程可以有各种类型的吸引子取决于特定的参数值在函数中使用。一个例子是经过充分研究的 logistic 映射，math x { t + 1} rx t (1-x t) ，/ math，其对参数 r 的各种值的吸引盆地在图中显示。在参数的某些值上，吸引子是一个单点，在其他值上，它是两个点轮流访问，在其他值上，它是2 sup n / sup 点或 k2 sup n / sup 点轮流访问，对于任何值 n 取决于参数 r 的值，在其他值上，r 是一个无穷多的点被访问。
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A single-variable (univariate) linear difference equation of the homogeneous form <math>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.
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一个齐次形式的单变量(单变量)线性差分方程 x _ t = ax _ { t-1} </math > 如果 | a | > 1从除0以外的所有初始点到无穷远，没有吸引子，因此没有吸引盆。但是如果 | a | < 1所有点在数线图上渐近地(或直接在0的情况下)到0; 0是吸引子，而整个数线是吸引盆。
    
==Basins of attraction==
 
==Basins of attraction==
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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 eventually be iterated into the attractor. For a [[stability (mathematics)|stable]] [[linear system]], every point in the phase space is in the basin of attraction. However, in [[nonlinear system]]s, some points may map directly or [[asymptotic behavior|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.<ref>{{cite journal|last1=Strelioff|first1=C.|last2=HUbler|first2=A.|title=Medium-Term Prediction of Chaos|journal=Phys. Rev. Lett.|date=2006|volume=96|issue=4|doi=10.1103/PhysRevLett.96.044101|pmid=16486826|page=044101}}</ref>
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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 eventually 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.
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吸引子的吸引盆是相空间的一个区域，在这个区域上迭代被定义，这样该区域的任何点(任何初始条件)最终将迭代到吸引子。对于稳定的线性系统，相空间中的每一点都处于吸引盆中。然而，在非线性系统中，一些点可以直接或渐近地映射到无穷远处，而另一些点可能位于不同的吸引域中，渐近地映射到不同的吸引子中，其他的初始条件可能位于或直接映射到不吸引点或不吸引周期中。
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===Linear equation or system===
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A single-variable (univariate) linear [[difference equation]] of the [[homogeneous equation|homogeneous form]] <math>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.
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A single-variable (univariate) linear difference equation of the homogeneous form <math>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.
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一个齐次形式的单变量(单变量)线性差分方程 x t ax { t-1} / math 发散到无穷如果 | a | 1从除0以外的所有初始点出发，没有吸引子，因此没有吸引盆。但是如果 | a | 1所有点在数线图上渐近地(或直接在0的情况下)到0; 0是吸引子，而整个数线是吸引盆。
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Likewise, a linear [[matrix difference equation]] in a dynamic [[coordinate vector|vector]] ''X'', of the homogeneous form <math>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.
      
Likewise, a linear matrix difference equation in a dynamic vector X, of the homogeneous form <math>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.
 
Likewise, a linear matrix difference equation in a dynamic vector X, of the homogeneous form <math>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.
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同样，如果动态矢量 x 的最大特征值在绝对值上大于1，则动态矢量 a 的所有元素都发散到无穷远，没有吸引子，也没有吸引盆。但当最大特征值小于1时，所有初始向量都渐近收敛到零向量，即吸引子，位初始向量的整个 n 维空间是吸引盆。
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同样，如果动态矢量 x 的最大特征值在绝对值上大于1，则动态矢量 a 的所有元素都发散到无穷远，不存在吸引子，也不存在吸引盆。但当最大特征值小于1时，所有初始向量都渐近收敛到零向量，零向量是吸引子，位初始向量的整个 n 维空间是吸引盆。
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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 [[asymptotic behavior|asymptotically]] be iterated into the attractor. For a [[stability (mathematics)|stable]] [[linear system]], every point in the phase space is in the basin of attraction. However, in [[nonlinear system]]s, 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.<ref>{{cite journal|last1=Strelioff|first1=C.|last2=Hübler|first2=A.|title=Medium-Term Prediction of Chaos|journal=Phys. Rev. Lett.|date=2006|volume=96|issue=4|doi=10.1103/PhysRevLett.96.044101|pmid=16486826|page=044101}}</ref>
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Similar features apply to linear [[differential equation]]s. The scalar equation <math> 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>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.
      
Similar features apply to linear differential equations. The scalar equation <math> 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>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.
 
Similar features apply to linear differential equations. The scalar equation <math> 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>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.
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类似的特征也适用于线性微分方程。标量方程 dx / dt ax / math 使除0以外的所有 x 的初始值发散到无穷大，如果0收敛到吸引子，如果0收敛到吸引子，使得整个数列为0的吸引盆。如果矩阵 a 的任何特征值是正的，则矩阵系数 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，则零向量是吸引域为整个相空间的吸引子。
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类似的特征也适用于线性微分方程。标量方程 < math > dx/dt = ax </math > 导致除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，使得整个数列沿着吸引盆的方向为0。矩阵系统 < math > dX/dt = AX </math > 如果矩阵 a 的任何特征值是正的，则该矩阵系统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，则零向量是吸引域为整个相空间的吸引子。
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===Linear equation or system===
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===Nonlinear equation or system===
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A single-variable (univariate) linear [[difference equation]] of the [[homogeneous equation|homogeneous form]] <math>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.
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Equations or systems that are [[nonlinear system|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 number|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&mdash;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,<ref>Dence, Thomas, "Cubics, chaos and Newton's method", ''[[Mathematical Gazette]]'' 81, November 1997, 403–408.</ref> for the function <math>f(x)=x^3-2x^2-11x+12</math>, the following initial conditions are in successive basins 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&mdash;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>f(x)=x^3-2x^2-11x+12</math>, the following initial conditions are in successive basins 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&mdash;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>f(x)=x^3-2x^2-11x+12</math>, the following initial conditions are in successive basins of attraction:
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与线性系统相比，非线性方程或系统可以产生更多种类的行为。一个例子是牛顿迭代非线性表达式根的方法。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导致其中一个根，而其他起始点会导致另一个根。表达式根部的吸引盆地通常不是简单的---- 它不是简单地将最靠近一个根部的点全部映射到那里，形成一个由附近点组成的吸引盆地。吸引力的盆地可以是无限的，也可以是任意的小。例如，对于函数数学 f (x) x ^ 3-2x ^ 2-11x + 12 / math，下面的初始条件是在连续的吸引盆中:
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与线性系统相比，非线性方程或系统可以产生更多种类的行为。一个例子是牛顿迭代非线性表达式根的方法。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导致其中一个根，而其他起始点会导致另一个根。表达式根部的吸引盆地通常不是简单的---- 它不是简单地将最靠近一个根部的点全部映射到那里，形成一个由附近点组成的吸引盆地。吸引力的盆地可以是无限的，也可以是任意的小。例如，对于函数 f (x) = x ^ 3-2x ^ 2-11x + 12 </math > ，下面的初始条件是连续的吸引盆地:
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Likewise, a linear [[matrix difference equation]] in a dynamic [[coordinate vector|vector]] ''X'', of the homogeneous form <math>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.
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[[File:newtroot 1 0 0 0 0 m1.png|thumb|Basins of attraction in the complex plane for using Newton's method to solve ''x''<sup>5</sup>&nbsp;−&nbsp;1&nbsp;=&nbsp;0. Points in like-colored regions map to the same root; darker means more iterations are needed to converge.]]
      
Basins of attraction in the complex plane for using Newton's method to solve x<sup>5</sup>&nbsp;−&nbsp;1&nbsp;=&nbsp;0. Points in like-colored regions map to the same root; darker means more iterations are needed to converge.
 
Basins of attraction in the complex plane for using Newton's method to solve x<sup>5</sup>&nbsp;−&nbsp;1&nbsp;=&nbsp;0. Points in like-colored regions map to the same root; darker means more iterations are needed to converge.
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用牛顿法求解 x sup 5 / sup-10的复平面吸引域。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。
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用牛顿法求解 x < sup > 5 </sup >-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。
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Similar features apply to linear [[differential equation]]s. The scalar equation <math> 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>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.
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:2.35287527  converges to 4;
      
2.35287527  converges to 4;
 
2.35287527  converges to 4;
第423行： 第375行：  
2.35287527汇聚为4;
 
2.35287527汇聚为4;
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:2.35284172  converges to −3;
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===Nonlinear equation or system===
    
2.35284172  converges to −3;
 
2.35284172  converges to −3;
第429行： 第381行：  
2.35284172  converges to −3;
 
2.35284172  converges to −3;
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:2.35283735  converges to 4;
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2.35283735  converges to 4;
 
2.35283735  converges to 4;
第435行： 第387行：  
2.35283735汇聚至4;
 
2.35283735汇聚至4;
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:2.352836327  converges to −3;
+
Equations or systems that are [[nonlinear system|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 number|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&mdash;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,<ref>Dence, Thomas, "Cubics, chaos and Newton's method", ''[[Mathematical Gazette]]'' 81, November 1997, 403–408.</ref> for the function <math>f(x)=x^3-2x^2-11x+12</math>, the following initial conditions are in successive basins of attraction:
    
2.352836327  converges to −3;
 
2.352836327  converges to −3;
第441行： 第393行：  
2.352836327  converges to −3;
 
2.352836327  converges to −3;
   −
:2.352836323  converges to 1.
+
 
    
2.352836323  converges to 1.
 
2.352836323  converges to 1.
第447行： 第399行：  
2.352836323汇聚为1。
 
2.352836323汇聚为1。
    +
[[File:newtroot 1 0 0 0 0 m1.png|thumb|Basins of attraction in the complex plane for using Newton's method to solve ''x''<sup>5</sup>&nbsp;−&nbsp;1&nbsp;=&nbsp;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 analysis|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 [[fractal]]s.
      
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.
 
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.
第455行： 第407行：  
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盆; 这些盆地可以如图所示绘制出来。可以看出，组合盆地的吸引力为一个特定的根可以有许多不相连的地区。对于许多复杂的函数，吸引盆地的边界是分形。
 
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盆; 这些盆地可以如图所示绘制出来。可以看出，组合盆地的吸引力为一个特定的根可以有许多不相连的地区。对于许多复杂的函数，吸引盆地的边界是分形。
    +
:2.35287527  converges to 4;
    +
:2.35284172  converges to −3;
   −
== Partial differential equations ==
+
:2.35283735 converges to 4;
−
 
  −
[[Parabolic partial differential equation]]s 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 equation]]s are all known to have global attractors of finite dimension.
      
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.
 
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.
第465行： 第417行：  
抛物型偏微分方程可能具有有限维吸引子。方程的扩散部分阻尼更高的频率，在某些情况下导致一个全局吸引子。Ginzburg-Landau 方程、 Kuramoto-Sivashinsky 方程和二维强迫 Navier-Stokes 方程都具有有限维的全局吸引子。
 
抛物型偏微分方程可能具有有限维吸引子。方程的扩散部分阻尼更高的频率，在某些情况下导致一个全局吸引子。Ginzburg-Landau 方程、 Kuramoto-Sivashinsky 方程和二维强迫 Navier-Stokes 方程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 condition]]s, if it has a global attractor, then this attractor will be of finite dimensions.<ref>[[Geneviève Raugel]], Global Attractors in Partial Differential Equations, ''Handbook of Dynamical Systems'', Elsevier, 2002, pp. 885–982.</ref>
      
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.
 
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.
第475行： 第427行：       −
== Numerical localization (visualization) of attractors: self-excited and hidden attractors ==
+
Newton's method can also be applied to [[complex analysis|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 [[fractal]]s.
 +
 
 +
<!-- This should be uncommented once the <nowiki></nowiki> in hidden attractor is solved. See the talk page for more information.
 +
 
 +
< ! -- 一旦隐藏吸引子中的 < nowiki > </nowiki > 得到解决，这应该被取消评论。更多信息请参见演讲页面。
 +
 
 +
 
   −
[[File:Chua-chaotic-hidden-attractor.jpg|thumb|
+
== Partial differential equations ==
    
[[File:Chua-chaotic-hidden-attractor.jpg|thumb|
 
[[File:Chua-chaotic-hidden-attractor.jpg|thumb|
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[文件: Chua-chaotic-hidden-attractor. jpg | thumb |  
 
[文件: Chua-chaotic-hidden-attractor. jpg | thumb |  
   −
Chaotic [[hidden attractor]] (green domain) in [[Chua's circuit|Chua's system]].
+
[[Parabolic partial differential equation]]s 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 equation]]s are all known to have global attractors of finite dimension.
    
Chaotic hidden attractor (green domain) in Chua's system.
 
Chaotic hidden attractor (green domain) in Chua's system.
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蔡氏系统中的混沌隐藏吸引子(绿域)。
 
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).
+
 
    
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).
 
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(蓝色)的轨迹趋向(红箭头)至无穷大或趋向(黑箭头)至稳定的零平衡点(橙色)。
+
带有初始数据的一个邻近鞍点(蓝色)的轨迹趋向(红色箭头)至无穷大或趋向(黑色箭头)至稳定的零平衡点(橙色)。
   −
]]
+
For the three-dimensional, incompressible Navier–Stokes equation with periodic [[boundary condition]]s, if it has a global attractor, then this attractor will be of finite dimensions.<ref>[[Geneviève Raugel]], Global Attractors in Partial Differential Equations, ''Handbook of Dynamical Systems'', Elsevier, 2002, pp. 885–982.</ref>
    
]]
 
]]
第503行： 第461行：       −
From a computational point of view, attractors can be naturally regarded as ''self-excited attractors'' or
+
<!-- This should be uncommented once the <nowiki>{{Notability}}</nowiki> in [[hidden attractor]] is solved. See the talk page for more information.
    
From a computational point of view, attractors can be naturally regarded as self-excited attractors or
 
From a computational point of view, attractors can be naturally regarded as self-excited attractors or
第509行： 第467行：  
从计算的角度来看，吸引子可以自然地看作自激吸引子或自激吸引子
 
从计算的角度来看，吸引子可以自然地看作自激吸引子或自激吸引子
   −
''[[hidden attractor]]s''.<ref name="2011-PLA-Hidden-Chua-attractor">{{cite journal |author1=Leonov G.A. |author2=Vagaitsev V.I. |author3=Kuznetsov N.V. |
+
== Numerical localization (visualization) of attractors: self-excited and hidden attractors ==
 +
 
 +
hidden attractors. 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.
   −
hidden attractors.<ref name="2011-PLA-Hidden-Chua-attractor">{{cite journal |author1=Leonov G.A. |author2=Vagaitsev V.I. |author3=Kuznetsov N.V. |
+
隐藏吸引子。自激吸引子可以用标准的计算程序进行数值局部化，在一个瞬态序列之后，从不稳定平衡点的小邻域的不稳定流形上的一个点出发的轨迹到达一个吸引子，如 Van der Pol、 Belousov-Zhabotinsky、 Lorenz 等许多其他动力系统中的经典吸引子。相反，一个隐藏吸引子的吸引盆不包含平衡邻域，因此隐藏吸引子不能被标准的计算程序局部化。
   −
2011-pla-hidden-chua-attractor { cite journal | author1 Leonov g.a.2 Vagaitsev v.i.3 Kuznetsov n.v. |  
+
[[File:Chua-chaotic-hidden-attractor.jpg|thumb|
   −
year = 2011 |
+
-->
   −
year = 2011 |
+
-->
   −
2011年
+
Chaotic [[hidden attractor]] (green domain) in [[Chua's circuit|Chua's system]].
   −
title = Localization of hidden Chua's attractors |
+
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).
   −
title = Localization of hidden Chua's attractors |
+
]]
   −
标题本地化的隐藏蔡氏吸引子 |
     −
journal = Physics Letters A |
     −
journal = Physics Letters A |
+
From a computational point of view, attractors can be naturally regarded as ''self-excited attractors'' or
   −
物理学快报 a |  
+
''[[hidden attractor]]s''.<ref name="2011-PLA-Hidden-Chua-attractor">{{cite journal |author1=Leonov G.A. |author2=Vagaitsev V.I. |author3=Kuznetsov N.V. |
   −
volume = 375 |
+
year = 2011 |
   −
volume = 375 |
+
title = Localization of hidden Chua's attractors |
   −
第375卷
+
journal = Physics Letters A |
   −
issue = 23 |
+
volume = 375 |
    
issue = 23 |
 
issue = 23 |
−
  −
第23期
      
pages = 2230–2233 |
 
pages = 2230–2233 |
−
  −
pages = 2230–2233 |
  −
  −
第2230-2233页
      
url = http://www.math.spbu.ru/user/nk/PDF/2011-PhysLetA-Hidden-Attractor-Chua.pdf |
 
url = http://www.math.spbu.ru/user/nk/PDF/2011-PhysLetA-Hidden-Attractor-Chua.pdf |
−
  −
url = http://www.math.spbu.ru/user/nk/PDF/2011-PhysLetA-Hidden-Attractor-Chua.pdf |
  −
  −
网址 http://www.math.spbu.ru/user/nk/pdf/2011-physleta-hidden-attractor-chua.pdf  |
  −
  −
doi = 10.1016/j.physleta.2011.04.037}}
      
doi = 10.1016/j.physleta.2011.04.037}}
 
doi = 10.1016/j.physleta.2011.04.037}}
−
  −
10.1016 / j.physleta. 2011.04.037}
      
</ref><ref>{{cite journal
 
</ref><ref>{{cite journal
−
  −
</ref><ref>{{cite journal
  −
  −
/ ref { cite journal
      
  |author1=Bragin V.O. |author2=Vagaitsev V.I. |author3=Kuznetsov N.V. |author4=Leonov G.A. | year = 2011
 
  |author1=Bragin V.O. |author2=Vagaitsev V.I. |author3=Kuznetsov N.V. |author4=Leonov G.A. | year = 2011
−
  −
|author1=Bragin V.O. |author2=Vagaitsev V.I. |author3=Kuznetsov N.V. |author4=Leonov G.A. | year = 2011
  −
  −
1 Bragin v.o.2 Vagaitsev v.i.3 Kuznetsov n.v. | author4 Leonov g.a.2011年
  −
  −
| title = Algorithms for Finding Hidden Oscillations in Nonlinear Systems. The Aizerman and Kalman Conjectures and Chua's Circuits
      
  | title = Algorithms for Finding Hidden Oscillations in Nonlinear Systems. The Aizerman and Kalman Conjectures and Chua's Circuits
 
  | title = Algorithms for Finding Hidden Oscillations in Nonlinear Systems. The Aizerman and Kalman Conjectures and Chua's Circuits
−
  −
发现非线性系统中隐藏振荡的算法。Aizerman 猜想和 Kalman 猜想与蔡氏电路
      
  | journal = Journal of Computer and Systems Sciences International
 
  | journal = Journal of Computer and Systems Sciences International
−
  −
| journal = Journal of Computer and Systems Sciences International
  −
  −
国际计算机与系统科学杂志
      
  | volume = 50
 
  | volume = 50
−
  −
| volume = 50
  −
  −
第50卷
  −
  −
| number = 5
      
  | number = 5
 
  | number = 5
−
  −
5号
      
  | pages = 511–543
 
  | pages = 511–543
   −
| pages = 511–543
+
  | last = Ruelle
   −
第511-543页
+
| last = Ruelle
    
  | url = http://www.math.spbu.ru/user/nk/PDF/2011-TiSU-Hidden-oscillations-attractors-Aizerman-Kalman-conjectures.pdf
 
  | url = http://www.math.spbu.ru/user/nk/PDF/2011-TiSU-Hidden-oscillations-attractors-Aizerman-Kalman-conjectures.pdf
   −
| url = http://www.math.spbu.ru/user/nk/PDF/2011-TiSU-Hidden-oscillations-attractors-Aizerman-Kalman-conjectures.pdf
+
  | first = David
   −
Http://www.math.spbu.ru/user/nk/pdf/2011-tisu-hidden-oscillations-attractors-aizerman-kalman-conjectures.pdf
+
第一名: David
    
  | doi = 10.1134/S106423071104006X}}
 
  | doi = 10.1134/S106423071104006X}}
   −
| doi = 10.1134/S106423071104006X}}
+
  | authorlink = David Ruelle
   −
10.1134 / S106423071104006X }
+
| authorlink = David Ruelle
    
</ref><ref name="2012-Physica-D-Hidden-attractor-Chua-circuit-smooth">{{cite journal |author1=Leonov G.A. |author2=Vagaitsev V.I. |author3=Kuznetsov N.V. |
 
</ref><ref name="2012-Physica-D-Hidden-attractor-Chua-circuit-smooth">{{cite journal |author1=Leonov G.A. |author2=Vagaitsev V.I. |author3=Kuznetsov N.V. |
   −
</ref><ref name="2012-Physica-D-Hidden-attractor-Chua-circuit-smooth">{{cite journal |author1=Leonov G.A. |author2=Vagaitsev V.I. |author3=Kuznetsov N.V. |
+
  | title = What is...a Strange Attractor?
   −
2012-phyica-d hidden-attrator-chua-circuit-smooth { cite journal | author1 Leonov g.a.2 Vagaitsev v.i.3 Kuznetsov n.v. |
+
| title = 什么是... 奇异吸引子？
    
year = 2012 |
 
year = 2012 |
   −
year = 2012 |
+
  | journal = Notices of the American Mathematical Society
   −
2012年
+
| journal = 美国数学学会公告
    
title = Hidden attractor in smooth Chua systems |
 
title = Hidden attractor in smooth Chua systems |
   −
title = Hidden attractor in smooth Chua systems |
+
  |date=August 2006
   −
光滑蔡氏系统中的隐藏吸引子 |  
+
| date = August 2006
    
journal = Physica D |
 
journal = Physica D |
   −
journal = Physica D |
+
  | volume = 53
   −
物理学杂志 d |
+
53
    
volume = 241 |
 
volume = 241 |
   −
volume = 241 |
+
  | issue = 7
   −
第241卷
+
第七期
    
issue = 18 |
 
issue = 18 |
   −
issue = 18 |
+
  | pages = 764–765
   −
第18期
+
| 页数 = 764-765
    
pages = 1482–1486 |
 
pages = 1482–1486 |
   −
pages = 1482–1486 |
+
  | url = http://www.ams.org/notices/200607/what-is-ruelle.pdf
   −
第1482-1486页
+
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 |
 
url = http://www.math.spbu.ru/user/nk/PDF/2012-Physica-D-Hidden-attractor-Chua-circuit-smooth.pdf |
   −
url = http://www.math.spbu.ru/user/nk/PDF/2012-Physica-D-Hidden-attractor-Chua-circuit-smooth.pdf |
+
  | accessdate = 16 January 2008 }}
−
 
  −
网址 http://www.math.spbu.ru/user/nk/pdf/2012-physica-d-hidden-attractor-chua-circuit-smooth.pdf  |
     −
doi = 10.1016/j.physd.2012.05.016}}
+
16 January 2008}}
    
doi = 10.1016/j.physd.2012.05.016}}
 
doi = 10.1016/j.physd.2012.05.016}}
−
  −
10.1016 / j.physd. 2012.05.016}
      
</ref><ref name="2013-IJBC-Hidden-attractors">{{cite journal |author1=Leonov G.A. |author2=Kuznetsov N.V. |
 
</ref><ref name="2013-IJBC-Hidden-attractors">{{cite journal |author1=Leonov G.A. |author2=Kuznetsov N.V. |
−
  −
</ref><ref name="2013-IJBC-Hidden-attractors">{{cite journal |author1=Leonov G.A. |author2=Kuznetsov N.V. |
  −
  −
/ ref name"2013-ijbc-hidden-attrtors"{ cite journal | author1 Leonov g.a.2 Kuznetsov n.v. |
      
year = 2013 |
 
year = 2013 |
−
  −
year = 2013 |
  −
  −
2013年
  −
  −
title = Hidden attractors in dynamical systems. From hidden oscillations in Hilbert–Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits |
      
title = Hidden attractors in dynamical systems. From hidden oscillations in Hilbert–Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits |
 
title = Hidden attractors in dynamical systems. From hidden oscillations in Hilbert–Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits |
−
  −
动力系统中的隐含吸引子。从 Hilbert-Kolmogorov 的隐藏振荡、 Aizerman 和 Kalman 问题到 Chua 电路的隐藏混沌吸引子 |
      
journal = International Journal of Bifurcation and Chaos |
 
journal = International Journal of Bifurcation and Chaos |
−
  −
journal = International Journal of Bifurcation and Chaos |
  −
  −
国际分歧与混沌杂志 |
      
volume = 23 |
 
volume = 23 |
−
  −
volume = 23 |
  −
  −
第23卷
      
issue = 1 |
 
issue = 1 |
−
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issue = 1 |
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第一期
      
pages = art. no. 1330002|
 
pages = art. no. 1330002|
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pages = art. no. 1330002|
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书页艺术。没有。1330002|
      
doi = 10.1142/S0218127413300024|
 
doi = 10.1142/S0218127413300024|
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doi = 10.1142/S0218127413300024|
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10.1142 / S0218127413300024 |
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doi-access = free }}
      
doi-access = free }}
 
doi-access = free }}
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免费开放
      
</ref> 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 oscillator|Van der Pol]], [[Belousov–Zhabotinsky reaction|Belousov–Zhabotinsky]], [[Lorenz attractor|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.
 
</ref> 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 oscillator|Van der Pol]], [[Belousov–Zhabotinsky reaction|Belousov–Zhabotinsky]], [[Lorenz attractor|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.
   −
</ref> 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局部化，在一个瞬态序列之后，从不稳定平衡点附近的不稳定流形上的一个点出发的轨迹到达一个吸引子，如 Van der Pol、 Belousov-Zhabotinsky、 Lorenz 等许多动力学系统中的经典吸引子。相反，一个隐藏吸引子的吸引盆不包含平衡邻域，因此隐藏吸引子不能被标准的计算程序局部化。
  −
 
  −
 
      
== See also ==
 
== See also ==
第744行： 第620行：     
* [[Steady state]]
 
* [[Steady state]]
−
  −
* [[Wada basin]]
  −
  −
* [[Hidden oscillation]]
  −
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* [[Rossler attractor]]
  −
  −
* [[Stable distribution]]
  −
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  −
== References ==
  −
  −
{{Reflist}}
  −
  −
  −
  −
== Further reading ==
  −
  −
* {{Scholarpedia|title=Attractor|curator=[[John Milnor]]|urlname=Attractor}}
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  −
* {{cite journal | author=David Ruelle | author-link=David Ruelle |author2=Floris Takens |author2-link=Floris Takens | title= On the nature of turbulence | journal=Communications in Mathematical Physics | year=1971 | volume=20 | pages=167–192 | doi= 10.1007/BF01646553 | issue=3}}
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* {{cite journal | author=D. Ruelle| title= Small random perturbations of dynamical systems and the definition of attractors | journal=Communications in Mathematical Physics | year=1981 | volume=82 | pages=137–151| doi= 10.1007/BF01206949}}
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* {{cite journal | author=John Milnor | author-link=John Milnor | title= On the concept of attractor | journal=Communications in Mathematical Physics | year=1985 | volume=99 | pages=177–195| doi= 10.1007/BF01212280 | issue=2}}
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* {{cite book | author=David Ruelle | title=Elements of Differentiable Dynamics and Bifurcation Theory | publisher=Academic Press | year=1989 | isbn=978-0-12-601710-6}}
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* {{cite journal
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  | last = Ruelle
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  | last = Ruelle
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最后的 Ruelle
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  | first = David
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  | first = David
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首先是大卫
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  | authorlink = David Ruelle
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  | authorlink = David Ruelle
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作者: David Ruelle
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  | title = What is...a Strange Attractor?
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  | title = What is...a Strange Attractor?
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| 题目什么是... 奇异吸引子？
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  | journal = [[Notices of the American Mathematical Society]]
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  | journal = Notices of the American Mathematical Society
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美国数学学会公告
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  |date=August 2006
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  |date=August 2006
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2006年8月
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  | volume = 53
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  | volume = 53
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第53卷
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  | issue = 7
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  | issue = 7
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第七期
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  | pages = 764–765
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  | pages = 764–765
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764-765页
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  | url = http://www.ams.org/notices/200607/what-is-ruelle.pdf
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  | url = http://www.ams.org/notices/200607/what-is-ruelle.pdf
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Http://www.ams.org/notices/200607/what-is-ruelle.pdf
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  | accessdate = 16 January 2008 }}
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  | accessdate = 16 January 2008 }}
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2008年1月16日}
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* {{cite journal | doi=10.1016/0167-2789(84)90282-3 | author1=Celso Grebogi | author2-link=Edward Ott | author2=Edward Ott | author3=Pelikan | author4=Yorke | title = Strange attractors that are not chaotic | journal = Physica D | year =  1984 | volume = 13 | issue=1–2 | pages = 261–268| author1-link=Celso Grebogi }}
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* {{cite journal | doi=10.1016/j.physd.2011.06.005 | author=Chekroun, M. D. |author2=E. Simonnet |author3=M. Ghil |author-link3=Michael Ghil| title=Stochastic climate dynamics: Random attractors and time-dependent invariant measures | journal = Physica D | year =  2011 | volume = 240 | issue = 21 | pages = 1685–1700 | citeseerx=10.1.1.156.5891 }}
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* [[Edward Lorenz|Edward N. Lorenz]] (1996) ''The Essence of Chaos'' {{ISBN|0-295-97514-8}}
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* [[James Gleick]] (1988) ''Chaos: Making a New Science'' {{ISBN|0-14-009250-1}}
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== External links ==
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* [http://www.scholarpedia.org/article/Basin_of_attraction Basin of attraction on Scholarpedia]
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* [http://slide.nethium.pl/album_en.net?gNwADMfFmY A gallery of trigonometric strange attractors]
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* [http://www.chuacircuits.com/sim.php Double scroll attractor] Chua's circuit simulation
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* [https://ccrma.stanford.edu/~stilti/images/chaotic_attractors/poly.html A gallery of polynomial strange attractors]
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* [http://www.chaoscope.org Chaoscope, a 3D Strange Attractor rendering freeware]
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* [http://ronrecord.com/PhD/intro.html Research abstract] and [ftp://ftp2.sco.com/pub/skunkware/src/x11/misc/mathrec-1.1c.tar.gz software laboratory] <!--[needs shorter summary] for exploring new algorithms to determining the attractor basin boundaries of iterated endomorphisms.-->
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* [http://wokos.nethium.pl/attractors_en.net Online strange attractors generator]
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* [https://web.archive.org/web/20131112192849/http://1618.pl/home/math_viz/attractor/attractor.html Interactive trigonometric attractors generator]
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* [https://web.archive.org/web/20131220102737/http://www.bentamari.com/attractors Economic attractor]
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[[Category:Limit sets]]
      
Category:Limit sets
 
Category:Limit sets
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