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{{short description|Concept in dynamical systems}}

{{short description|Concept in dynamical systems}}

 

{{other uses}}

{{other uses}}

 

{{redirect|Strange attractor|other uses|Strange Attractor (disambiguation)}}

{{redirect|Strange attractor|other uses|Strange Attractor (disambiguation)}}

 

{{Use dmy dates|date=May 2013}}

{{Use dmy dates|date=May 2013}}

 

{{more footnotes|date=March 2013}}

{{more footnotes|date=March 2013}}

 

[[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>.]]

[[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>.]]

[[奇怪的吸引子]]

 

[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.

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.

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

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

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

在有限维系统中，演化变量可用代数表示为 n 维向量。<font color="#ff8000"> 吸引子</font>是 n 维空间中的一个区域。在物理系统中，n 维可以是，例如，一个或多个物理实体的两个或三个位置坐标; 在经济系统中，它们可以是单独的变量，如通货膨胀率和失业率。

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

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

如果演化变量是二维或三维的，则动态过程的吸引子可以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。

如果演化变量是二维或三维的，则动态过程的<font color="#ff8000"> 吸引子</font>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<font color="#ff8000"> 吸引子</font>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<font color="#ff8000"> 奇怪吸引子Strange attractor</font>(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<font color="#ff8000"> 混沌动力学系统Chaotic dynamical systems</font>的吸引子是混沌理论的重要成果之一。

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

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

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)。

动力系统在吸引子中的轨迹除了保持在<font color="#ff8000"> 吸引子</font>上的时间向前外，不必满足任何特殊的约束条件。轨迹可能是周期性的，也可能是混沌的。如果一组点是周期性的或混沌的，但附近的流远离该集合，则该集合不是吸引子，而是称为<font color="#ff8000"> 排斥点（或斥点）Repeller (or repellor)</font>

== Motivation of attractors ==

== Motivation of attractors 吸引子的动力机制==

A [[dynamical system]] is generally described by one or more [[differential equations|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 [[Integral|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 equations|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 [[Integral|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.

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.

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。

<font color="#ff8000"> 动力系统</font>通常由一个或多个<font color="#ff8000"> 微分方程</font>或<font color="#ff8000"> 差分方程</font>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<font color="#ff8000"> 迭代Iteration</font>(通常借助于计算机)对方程进行积分。

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

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

物理世界中的动力系统往往产生于耗散系统: 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。

物理世界中的动力系统往往产生于<font color="#ff8000"> 耗散系统Dissipative system</font>: 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<font color="#ff8000">初始瞬态Initial transients</font>，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<font color="#ff8000"> 吸引子</font>。

 

 

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 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.

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。

<font color="#ff8000"> 不变集</font>和<font color="#ff8000"> 极限集</font>类似于吸引子的概念。<font color="#ff8000"> 不变集</font>是在动力学下向自身演化的集合。<font color="#ff8000"> 吸引子</font>可能包含不变集。<font color="#ff8000"> 极限集</font>是一组点，这些点存在一些初始状态，这些初始状态随着时间的推移到无穷远最终将任意接近极限集(即到集合的每个点）。<font color="#ff8000"> 吸引子</font>是<font color="#ff8000"> 极限集</font>，但不是所有的<font color="#ff8000"> 极限集</font>都是<font color="#ff8000"> 吸引子</font>: 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。

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

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

例如，阻尼摆有两个不变点: 最小高度点和最大高度点。这一点也是一个极限设置，因为轨迹收敛到它; 这一点不是一个极限设置。由于空气阻力的耗散，这个点也是一个吸引子。如果没有耗散，就不会有吸引力。亚里士多德认为物体只有在被推动的情况下才会移动，这是耗散吸引子的早期表述。

例如，<font color="#ff8000"> 阻尼摆Damping ratio|damped</font>有两个不变点: 最小高度点{{math|x₀}}和最大高度点{{math|x₁}}。点{{math|x₀}}也是一个极限集,因为轨迹向它收敛；点 {{math|x₁}}不是一个极限集。由于空气阻力的耗散，点{{math|x₀}}也是吸引子。如果没有耗散，{{math|x₀}}就不会是<font color="#ff8000"> 吸引子</font>。亚里士多德认为物体只有在被推动时才会移动，这是<font color="#ff8000"> 耗散吸引子</font>的早期表述。

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

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.

== Mathematical definition ==

== Mathematical definition数学定义 ==

Let ''t'' represent time and let ''f''(''t'', •) be a function which specifies the dynamics of the system.  That is, if ''a'' is a point in an ''n''-dimensional phase space, representing the initial state of the system, then ''f''(0, ''a'') = ''a'' and, for a positive value of ''t'', ''f''(''t'', ''a'') is the result of the evolution of this state after ''t'' units of time.  For example, if the system describes the evolution of a free particle in one dimension then the phase space is the plane '''R'''² with coordinates (''x'',''v''), where ''x'' is the position of the particle, ''v'' is its velocity, ''a''&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'''² 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 R² 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 R² 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

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，那么相空间是平面 r < sup > 2 </sup > ，坐标为(x，v) ，其中 x 是粒子的位置，v 是粒子的速度，a = (x，v)

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，以及对于 t 的正值，f (t，a)是该状态在 t 个时间单位之后演化的结果。例如，如果系统描述了自由粒子在一维空间中的演化，那么相空间是坐标为(x，v)的平面 '''R'''² ，，其中 x 是粒子的位置，v 是粒子的速度，a&nbsp;=&nbsp;(x,v),由以下给出

[[File:Julia immediate basin 1 3.png|right|thumb|Attracting period-3 cycle and its immediate basin of attraction for a certain parametrization of ''f''(''z'')&nbsp;=&nbsp;''z''²&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.]]

[[File:Julia immediate basin 1 3.png|right|thumb|Attracting period-3 cycle and its immediate basin of attraction for a certain parametrization of ''f''(''z'')&nbsp;=&nbsp;''z''²&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;z²&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;z²&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.

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。

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

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:

* ''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.

* There exists a [[Neighbourhood (mathematics)|neighborhood]] of ''A'', called the '''basin of attraction''' for ''A'' and denoted ''B''(''A''), which consists of all points ''b'' that "enter ''A'' in the limit ''t''&nbsp;→&nbsp;∞". More formally, ''B''(''A'') is the set of all points ''b'' in the phase space with the following property:

* There exists a [[Neighbourhood (mathematics)|neighborhood]] of ''A'', called the '''basin of attraction''' for ''A'' and denoted ''B''(''A''), which consists of all points ''b'' that "enter ''A'' in the limit ''t''&nbsp;→&nbsp;∞". More formally, ''B''(''A'') is the set of all points ''b'' in the phase space with the following property:

:: 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''.

  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.

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。

●对于“A”的任何开邻域“N”，存在一个正常数“T”，使得对所有实数“T”>“T”，有''f''(''t'',''b'') ∈ ''N''，。

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

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

 

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 space|metric]] on the phase space, but the resulting notion usually depends only on the topology of the phase space. In the case of '''R'''^''n'', the Euclidean norm is typically used.

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

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

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。

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Rⁿ的情况下，通常使用<font color="#ff8000"> 欧氏范数Euclidean norm</font>。

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.  

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.  

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。

<font color="#ff8000"> 吸引子</font>的许多其他定义出现在文献中。例如，一些作者要求<font color="#ff8000"> 吸引子</font>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。