[[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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[[File:Attractor Poisson Saturne.jpg|right|333px|thumb|奇异吸引子<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>的可视化]]
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[[strange attractor.]]
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在'''有限维系统 finite-dimensional systems'''中,演化变量可用代数表示为 n 维向量。吸引子是n维空间中的一个区域。在物理系统中,n维可以是一个或多个物理实体的两个或三个位置坐标;在经济系统中,它们可以是单独的变量,如'''通货膨胀率 inflation rate'''和'''失业率 unemployment rate'''。
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 finite-dimensional systems, the evolving variable may be represented [[algebra]]ically as an ''n''-dimensional [[Coordinate vector|vector]]. The attractor is a region in [[space (mathematics)|''n''-dimensional space]]. In [[Physics|physical systems]], the ''n'' dimensions may be, for example, two or three positional coordinates for each of one or more physical entities; in [[Economics|economic systems]], they may be separate variables such as the [[inflation rate]] and the [[unemployment rate]].
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有些吸引子是混沌的(参见奇异吸引子),在这种情况下,吸引子的任意两个不同点的演化都会触发指数发散轨迹。此时即使系统中有一点噪声,预测也会因此而变得复杂。<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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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<font color="#ff8000">有限维系统finite-dimensional systems</font>中,演化变量可用代数表示为 n 维向量。吸引子是 n 维空间中的一个区域。在物理系统中,n 维可以是一个或多个物理实体的两个或三个位置坐标; 在经济系统中,它们可以是单独的变量,如<font color="#ff8000">通货膨胀率 inflation rate</font>和<font color="#ff8000">失业率unemployment rate</font>。
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If the evolving variable is two- or three-dimensional, the attractor of the dynamic process can be represented [[Geometry|geometrically]] in two or three dimensions, (as for example in the three-dimensional case depicted to the right). An attractor can be a [[point (geometry)|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 [[Attractor#Strange attractor|strange attractor]] below). If the variable is a [[scalar (mathematics)|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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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.
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 function|periodic]] or [[Chaos theory|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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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 [[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.
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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.
Dynamical systems in the physical world tend to arise from [[dissipative system|dissipative systems]]: if it were not for some driving force, the motion would cease. (Dissipation may come from [[friction|internal friction]], [[thermodynamics|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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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.
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.
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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.
For example, the [[damping ratio|damped]] [[pendulum]] has two invariant points: the point {{math|x<sub>0</sub>}} of minimum height and the point {{math|x<sub>1</sub>}} of maximum height. The point {{math|x<sub>0</sub>}} is also a limit set, as trajectories converge to it; the point {{math|x<sub>1</sub>}} is not a limit set. Because of the dissipation due to air resistance, the point {{math|x<sub>0</sub>}} is also an attractor. If there was no dissipation, {{math|x<sub>0</sub>}} 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.
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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.
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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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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有些吸引子是混沌的(参见#奇异吸引子),在这种情况下,吸引子的任意两个不同点的演化都会触发指数发散轨迹。此时即使系统中有一点噪声,预测也会因此而变得复杂。<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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== Mathematical definition数学定义 ==
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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'' = (''x'',''v''), and the evolution is given by
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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 = (x,v), and the evolution is given by
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设 t 表示时间,设 f (t,•)为指定系统动力学的函数。也就是说,如果 a 是 n 维相空间中一个表示系统初始状态的点,那么 f (0,a) = a。对于 t 的正值,f (t,a)是该状态在 t 个时间单位之后演化的结果。例如,如果系统描述了自由粒子在一维空间中的演化,那么相空间是坐标为(x,v)的平面 '''R'''<sup>2</sup> ,其中 x 是粒子的位置,v 是粒子的速度,a = (x,v),由以下给出
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[[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'') = ''z''<sup>2</sup> + ''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) = z<sup>2</sup> + 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.
An '''attractor''' is a [[subset]] ''A'' of the [[phase space]] characterized by the following three conditions:
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在文献中有吸引子的其他定义出现。例如,一些作者要求吸引子具有正测度(防止一个点成为吸引子),另一些作者则弱化了B(A)作为一个邻域的要求。<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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An attractor is a subset A of the phase space characterized by the following three conditions:
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==吸引子的类型==
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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 topologically wild sets'''——虽然在当时是已知的,但却被认为是脆弱的异常事物。斯蒂芬·斯梅尔 Stephen Smale能够证明他的马蹄映射是稳定的,它的吸引子具有'''康托尔集 Cantor set'''结构。
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吸引子是<font color="#ff8000">相空间 phase space </font>的<font color="#ff8000">子集 subset </font>A——这需要具有以下三个条件:
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* ''A'' is ''forward invariant'' under ''f'': if ''a'' is an element of ''A'' then so is ''f''(''t'',''a''), for all ''t'' > 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'' → ∞". More formally, ''B''(''A'') is the set of all points ''b'' in the phase space with the following property:
* There is no proper (non-empty) subset of ''A'' having the first two properties.
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*“A”中不存在具有前两个属性的正确(非空)子集。
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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 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'''<sup>''n''</sup>, the Euclidean norm is typically used.
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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.
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由于吸引域包含一个含有 a 的开集合,所以每一个足够接近 a 的点都会被 a 吸引。吸引子的定义使用了相空间上的一个度量,但得到的结果通常只取决于相空间的拓扑结构。在R<sup>n</sup>的情况下,我们通常会使用<font color="#ff8000">欧氏范数Euclidean norm</font>。
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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.
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在文献中有吸引子的其他定义出现。例如,一些作者要求吸引子具有正测度(防止一个点成为吸引子) ,另一些作者则弱化了B(A)作为一个邻域的要求。<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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== Types of attractors 吸引子的类型==
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Attractors are portions or [[subset]]s of the [[Configuration space (physics)|phase space]] of a [[dynamical system]]. Until the 1960s, attractors were thought of as being [[Geometric primitive|simple geometric subsets]] of the phase space, like [[Point (geometry)|points]], [[Line (mathematics)|lines]], [[Surface (topology)|surface]]s, and simple regions of [[three-dimensional space]]. More complex attractors that cannot be categorized as simple geometric subsets, such as [[topology|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 [[structural stability|robust]] and that its attractor had the structure of a [[Cantor set]].
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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.
Two simple attractors are a [[Fixed point (mathematics)|fixed point]] and the [[limit cycle]]. Attractors can take on many other geometric shapes (phase space subsets). But when these sets (or the motions within them) cannot be easily described as simple combinations (e.g. [[intersection (set theory)|intersection]] and [[union (set theory)|union]]) of [[Geometric primitive|fundamental geometric objects]] (e.g. [[Line (mathematics)|lines]], [[Surface (topology)|surface]]s, [[sphere]]s, [[toroid]]s, [[manifold]]s), then the attractor is called a ''[[attractor#Strange attractor|strange attractor]]''.
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Two simple attractors are a fixed point and the limit cycle. Attractors can take on many other geometric shapes (phase space subsets). But when these sets (or the motions within them) cannot be easily described as simple combinations (e.g. intersection and union) of fundamental geometric objects (e.g. lines, surfaces, spheres, toroids, manifolds), then the attractor is called a strange attractor.
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两个简单的吸引子是一个<font color="#ff8000">不动点fixed point </font>和一个<font color="#ff8000">极限环 limit cycle </font>。吸引子可以呈现出许多几何形状(相空间子集)。但当这些集合(或其中的运动)不能简单地描述为基本几何对象(例如,直线,曲面,球体,环面,流形)的简单组合(例如,交集和并集)时,这个吸引子就被称为“[[奇异吸引子]]”。
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=== Fixed point驻点 ===
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[[File:Critical orbit 3d.png|right|thumb|Weakly attracting fixed point for a complex number evolving according to a [[complex quadratic polynomial]]. The phase space is the horizontal complex plane; the vertical axis measures the frequency with which points in the complex plane are visited. The point in the complex plane directly below the peak frequency is the fixed point attractor.]]
A [[Fixed point (mathematics)|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 [[damping ratio|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 [[Stability theory#Stability of fixed points|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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此外,由于物理世界动力学的现实性——包括非线性动力学的粘滞,摩擦,表面粗糙度,变形(弹性和塑性),甚至'''量子力学 quantum mechanics'''——至少拥有一个固定点的物理动力系统总是有多个固定点和吸引子。<ref name="Contact of Nominally Flat Surfaces">{{cite journal|last=Greenwood|first=J. A.|author2=J. B. P. Williamson|title=Contact of Nominally Flat Surfaces|journal=Proceedings of the Royal Society|date=6 December 1966|volume=295|issue=1442|pages=300–319|doi=10.1098/rspa.1966.0242}}</ref>回到倒置碗顶上的大理石这一例子,即使碗看起来是完美的半球形,大理石是规范的球形,在显微镜下观察时它们的表面实际上都十分复杂,它们的形状在接触过程中改变。任何物理表面都可以被视作一个由多个山峰、山谷、鞍点、山脊、峡谷和平原组成的崎岖地形。<ref name="NISTIR 89-4088">{{cite book|last=Vorberger|first=T. V.|title=Surface Finish Metrology Tutorial|year=1990|publisher=U.S. Department of Commerce, National Institute of Standards (NIST)|page=5|url=https://www.nist.gov/calibrations/upload/89-4088.pdf}}</ref>在这个表面地形中有许多点(以及在这个微观地形上滚动的同样粗糙的大理石的动力系统)被认为是静止的或不动的,其中一些被归类为吸引子。
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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).
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 (engineering)|deformation]] (both [[Elastic deformation|elastic]] and [[plastic]]ity), and even [[quantum mechanics]].<ref name="Contact of Nominally Flat Surfaces">{{cite journal|last=Greenwood|first=J. A.|author2=J. B. P. Williamson|title=Contact of Nominally Flat Surfaces|journal=Proceedings of the Royal Society|date=6 December 1966|volume=295|issue=1442|pages=300–319|doi=10.1098/rspa.1966.0242}}</ref> In the case of a marble on top of an inverted bowl, even if the bowl seems perfectly [[Sphere#Hemisphere|hemispherical]], and the marble's [[sphere|spherical]] shape, are both much more complex surfaces when examined under a microscope, and their [[Contact mechanics#History|shapes change]] or [[deformation (mechanics)|deform]] during contact. Any physical surface can be seen to have a rough terrain of multiple peaks, valleys, saddle points, ridges, ravines, and plains.<ref name="NISTIR 89-4088">{{cite book|last=Vorberger|first=T. V.|title=Surface Finish Metrology Tutorial|year=1990|publisher=U.S. Department of Commerce, National Institute of Standards (NIST)|page=5|url=https://www.nist.gov/calibrations/upload/89-4088.pdf}}</ref> 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 [[Critical point (mathematics)|stationary]] or fixed points, some of which are categorized as attractors.
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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.
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此外,由于物理世界动力学的现实性——包括非线性动力学的粘滞,摩擦,表面粗糙度,变形(弹性和塑性) ,甚至<font color="#ff8000">量子力学quantum mechanics</font>——至少拥有一个固定点的物理动力系统总是有多个固定点和吸引子<ref name="Contact of Nominally Flat Surfaces">{{cite journal|last=Greenwood|first=J. A.|author2=J. B. P. Williamson|title=Contact of Nominally Flat Surfaces|journal=Proceedings of the Royal Society|date=6 December 1966|volume=295|issue=1442|pages=300–319|doi=10.1098/rspa.1966.0242}}</ref>。回到倒置碗顶上的大理石这一例子,即使碗看起来是完美的半球形,大理石是规范的球形,在显微镜下观察时它们的表面实际上都十分复杂,它们的形状在接触过程中改变。任何物理表面都可以被视作一个由多个山峰、山谷、鞍点、山脊、峡谷和平原组成的崎岖地形<ref name="NISTIR 89-4088">{{cite book|last=Vorberger|first=T. V.|title=Surface Finish Metrology Tutorial|year=1990|publisher=U.S. Department of Commerce, National Institute of Standards (NIST)|page=5|url=https://www.nist.gov/calibrations/upload/89-4088.pdf}}</ref>。在这个表面地形中有许多点(以及在这个微观地形上滚动的同样粗糙的大理石的动力系统)被认为是静止的或不动的,其中一些被归类为吸引子。
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===Finite number of points有限点数===
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In a [[Discrete time and continuous time|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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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.
A [[limit cycle]] is a periodic orbit of a continuous dynamical system that is [[isolated point|isolated]]. Examples include the swings of a [[pendulum clock]], and the heartbeat while resting. (The limit cycle of an ideal pendulum is not an example of a limit cycle attractor because its orbits are not isolated: in the phase space of the ideal pendulum, near any point of a periodic orbit there is another point that belongs to a different periodic orbit, so the former orbit is not attracting).
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A limit cycle is a periodic orbit of a continuous dynamical system that is isolated. Examples include the swings of a pendulum clock, and the heartbeat while resting. (The limit cycle of an ideal pendulum is not an example of a limit cycle attractor because its orbits are not isolated: in the phase space of the ideal pendulum, near any point of a periodic orbit there is another point that belongs to a different periodic orbit, so the former orbit is not attracting).
[[File:VanDerPolPhaseSpace.png|center|250px|thumb|<center>[[Van der Pol oscillator|Van der Pol]] [[phase portrait]]: an attracting limit cycle</center>]]
[[文件:VanDerPolPhaseSpace.png|center| 250px |拇指|<center>[[Van der Pol振荡器| Van der Pol]][[相位肖像]]:吸引极限环</center>]]
[[文件:VanDerPolPhaseSpace.png|center| 250px |拇指|<center>[[Van der Pol振荡器| Van der Pol]][[相位肖像]]:吸引极限环</center>]]
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 number|irrational fraction]] (i.e. they are [[commensurability (mathematics)|incommensurate]]), the trajectory is no longer closed, and the limit cycle becomes a limit [[torus]]. This kind of attractor is called an {{math|''N''<sub>''t''</sub>}} -torus if there are {{math|N<sub>t</sub>}} incommensurate frequencies. For example, here is a 2-torus:
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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:
A time series corresponding to this attractor is a [[quasiperiodic]] series: A discretely sampled sum of {{math|N<sub>t</sub>}} periodic functions (not necessarily [[sine]] waves) with incommensurate frequencies. Such a time series does not have a strict periodicity, but its [[power spectrum]] still consists only of sharp lines.
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A time series corresponding to this attractor is a quasiperiodic series: A discretely sampled sum of periodic functions (not necessarily sine waves) with incommensurate frequencies. Such a time series does not have a strict periodicity, but its power spectrum still consists only of sharp lines.
=== Strange attractor 奇异吸引子===<!-- This section is linked from [[Lorenz attractor]] 本节链接自[[洛伦兹吸引子]]-->
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[[File:Lorenz attractor yb.svg|thumb|200px|right|A plot of Lorenz's strange attractor for values ''ρ'' = 28, ''σ'' = 10, ''β'' = 8/3]]
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===奇异吸引子===
[[文件:洛伦兹吸引子yb.svg公司|thumb | 200px | right | 洛伦兹奇异吸引子的图, ''ρ'' = 28, ''σ'' = 10, ''β'' = 8/3]]
[[文件:洛伦兹吸引子yb.svg公司|thumb | 200px | right | 洛伦兹奇异吸引子的图, ''ρ'' = 28, ''σ'' = 10, ''β'' = 8/3]]
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关于洛伦兹奇异吸引子 ρ = 28,σ = 10,β = 8/3的图
关于洛伦兹奇异吸引子 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 | pmid = 17816542 | bibcode = 1987Sci...238..632G }}</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 chaotic, but strange nonchaotic attractors also exist. If a strange attractor is chaotic, exhibiting sensitive dependence on initial conditions, then any two arbitrarily close alternative initial points on the attractor, after any of various numbers of iterations, will lead to points that are arbitrarily far apart (subject to the confines of the attractor), and after any of various other numbers of iterations will lead to points that are arbitrarily close together. Thus a dynamic system with a chaotic attractor is locally unstable yet globally stable: once some sequences have entered the attractor, nearby points diverge from one another but never depart from the attractor.
如果吸引子具有分形结构,则称为“奇异”。这种情况通常发生在在当它的动力学系统符合混沌理论时,但是奇异的非混沌吸引子也存在。如果一个奇异吸引子是混沌的,表现出对初始条件的敏感依赖性,那么在吸引子上两个任意接近的备选初始点经过多次迭代后,都会指向任意相距很远的点(受吸引子的限制),而在经历其他次数的迭代之后,会指向任意接近的点。因此,具有混沌吸引子的动态系统是局部不稳定但全局稳定的:一旦一些序列进入吸引子,附近的点就会发散,但不会离开。<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>
如果吸引子具有分形结构,则称为“奇异”。这种情况通常发生在在当它的动力学系统符合混沌理论时,但是奇异的非混沌吸引子也存在。如果一个奇异吸引子是混沌的,表现出对初始条件的敏感依赖性,那么在吸引子上两个任意接近的备选初始点经过多次迭代后,都会指向任意相距很远的点(受吸引子的限制),而在经历其他次数的迭代之后,会指向任意接近的点。因此,具有混沌吸引子的动态系统是局部不稳定但全局稳定的:一旦一些序列进入吸引子,附近的点就会发散,但不会离开。<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>
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术语“奇异吸引子”由大卫·吕埃勒 David Ruelle和弗洛里斯·塔肯斯 Floris Takens提出,用来描述吸引子——产生于刻画流体的系统的一系列分叉。<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>奇异吸引子通常在几个方向上可微,但有些吸引子与康托尘类似,因此不可微。在噪声条件下,人们也能发现奇异吸引子,这可以用来支持Sinai-Ruelle-Bowen型的不变随机概率测度。<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 |year = 2011 |title = Stochastic climate dynamics: Random attractors and time-dependent invariant measures |Examples of strange attractors include the double-scroll attractor, Hénon attractor, Rössler attractor, and Lorenz attractor. |journal = Physica D |volume = 240 |issue = 21 |pages = 1685–1700 |doi = 10.1016/j.physd.2011.06.005|citeseerx=10.1.1.156.5891 }}</ref>
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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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year = 2011 |
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title = Stochastic climate dynamics: Random attractors and time-dependent invariant measures |
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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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术语“奇异吸引子”由大卫·吕埃勒David Ruelle和弗洛里斯·塔肯斯Floris Takens提出,用来描述吸引子——产生于刻画流体的系统的一系列分叉。<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>奇异吸引子通常在几个方向上可微,但有些吸引子与康托尘类似,因此不可微。在噪声条件下,人们也能发现奇异吸引子,这可以用来支持Sinai-Ruelle-Bowen型的不变随机概率测度。<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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year = 2011 |
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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.
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).]]
doi = 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.
Examples of strange attractors include the [[Double scroll attractor|double-scroll attractor]], [[Hénon map|Hénon attractor]], [[Rössler attractor]], and [[Lorenz attractor]].
==Attractors characterize the evolution of a system吸引子表征系统的演化==
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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.
[[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).]]
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.
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.
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.
齐次形式的单变量(单变量)线性差分方程<math>x_t=ax{t-1}</math>从除0以外的所有初始点| a|>1发散到无穷大;没有吸引子,因此没有吸引池。但是如果| a |<1,则数线图上的所有点渐进地(或在0的情况下直接)趋向0;0是吸引子,整个数线是吸引域。
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==Basins of attraction吸引池==
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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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同样地,动态向量X中的线性矩阵差分方程,如果a的最大特征值绝对值大于1,则动态向量X中的所有元素<math>X_t=AX_{t-1}</math> 都将发散到无穷大;不存在吸引子和吸引池。但如果最大特征值小于1,则所有初始向量将渐近收敛于零向量,即零为吸引子;潜在初始向量的整个n维空间就是<font color="#ff8000">吸引池 basin of attraction </font>。
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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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==吸引池==
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同样地,动态向量X中的线性矩阵差分方程,如果a的最大特征值绝对值大于1,则动态向量X中的所有元素<math>X_t=AX_{t-1}</math> 都将发散到无穷大;不存在吸引子和吸引池。但如果最大特征值小于1,则所有初始向量将渐近收敛于零向量,即零为吸引子;潜在初始向量的整个n维空间就是'''吸引池 basin of attraction'''。
吸引子的吸引域是相空间的区域,迭代在这个区域上得到定义,使得该区域中的任何点(任何初始条件)都将渐近地迭代到吸引子中。对于一个稳定的线性系统,相空间中的每一点都在吸引域中。然而,在非线性系统中,有些点可能直接或渐近地映射到无穷大,而另一些点可能位于不同的吸引域中并渐近映射到不同的吸引子;其他初始条件则可能位于或直接映射到非吸引点或循环中。<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>
吸引子的吸引域是相空间的区域,迭代在这个区域上得到定义,使得该区域中的任何点(任何初始条件)都将渐近地迭代到吸引子中。对于一个稳定的线性系统,相空间中的每一点都在吸引域中。然而,在非线性系统中,有些点可能直接或渐近地映射到无穷大,而另一些点可能位于不同的吸引域中并渐近映射到不同的吸引子;其他初始条件则可能位于或直接映射到非吸引点或循环中。<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>
吸引子是动力系统的相空间的一部分或子集。直到20世纪60年代,吸引子被认为是相空间的简单几何子集——像点、线、面和简单的三维空间。更复杂的吸引子,不能被归类为简单的几何子集,如拓扑野生集 topologically wild sets——虽然在当时是已知的,但却被认为是脆弱的异常事物。斯蒂芬·斯梅尔 Stephen Smale能够证明他的马蹄映射是稳定的,它的吸引子具有康托尔集 Cantor set结构。
术语“奇异吸引子”由大卫·吕埃勒 David Ruelle和弗洛里斯·塔肯斯 Floris Takens提出,用来描述吸引子——产生于刻画流体的系统的一系列分叉。[7]奇异吸引子通常在几个方向上可微,但有些吸引子与康托尘类似,因此不可微。在噪声条件下,人们也能发现奇异吸引子,这可以用来支持Sinai-Ruelle-Bowen型的不变随机概率测度。[8]
[[文件:逻辑图分岔图,Matplotlib.svg|350px |拇指|右|分岔图逻辑图。参数“r”所有的吸引子显示在区间[math]\displaystyle{ 0\lt x\lt 1 }[/math]的纵坐标上。点的颜色表示在10次6次迭代过程中访问点[math]\displaystyle{ (r,x) }[/math]的频率:经常遇到的值用蓝色表示,不太常见的值用黄色表示。在[math]\displaystyle{ r\approx3.0 }[/math]附近出现分岔,在[math]\displaystyle{ r\approx3.5 }[/math]附近出现第二个分岔(导致四个吸引子值)。当[math]\displaystyle{ r\gt 3.6\lt math\gt 时,行为变得越来越复杂,中间穿插着简单行为区域(白色条纹)。]]
The parameters of a dynamic equation evolve as the equation is iterated, and the specific values may depend on the starting parameters. An example is the well-studied [[logistic map]], \lt math\gt x_{n+1}=rx_n(1-x_n) }[/math], whose basins of attraction for various values of the parameter r are shown in the figure. If [math]\displaystyle{ r=2.6 }[/math], all starting x values of [math]\displaystyle{ x\lt 0 }[/math] will rapidly lead to function values that go to negative infinity; starting x values of [math]\displaystyle{ x\gt 0 }[/math] will go to infinity. But for [math]\displaystyle{ 0\lt x\lt 1 }[/math] the x values rapidly converge to [math]\displaystyle{ x\approx0.615 }[/math], i.e. at this value of r, a single value of x is an attractor for the function's behaviour. For other values of r, more than one value of x may be visited: if r is 3.2, starting values of [math]\displaystyle{ 0\lt x\lt 1 }[/math] will lead to function values that alternate between [math]\displaystyle{ x\approx0.513 }[/math] and [math]\displaystyle{ x\approx0.799 }[/math]. At some values of r, the attractor is a single point (a "fixed point"), at other values of r two values of x are visited in turn (a period-doubling bifurcation); at yet other values of r, any given number of values of x are visited in turn; finally, for some values of r, an infinitude of points are visited. Thus one and the same dynamic equation can have various types of attractors, depending on its starting parameters.
Similar features apply to linear differential equations. The scalar equation [math]\displaystyle{ dx/dt =ax }[/math] causes all initial values of x except zero to diverge to infinity if a > 0 but to converge to an attractor at the value 0 if a < 0, making the entire number line the basin of attraction for 0. And the matrix system [math]\displaystyle{ dX/dt=AX }[/math] gives divergence from all initial points except the vector of zeroes if any eigenvalue of the matrix A is positive; but if all the eigenvalues are negative the vector of zeroes is an attractor whose basin of attraction is the entire phase space.
类似的特征也适用于线性微分方程。标量方程[math]\displaystyle{ dx/dt =ax }[/math] 使得除0以外的所有 x 的初始值在 a > 0时发散到无穷大,但在 a < 0时收敛到吸引子,使整条数线成为0的吸引池。如果矩阵 A 的任何特征值是正的,则该矩阵系统[math]\displaystyle{ dX/dt=AX }[/math]从除零向量以外的所有初始点发散; 但如果所有特征值都是负的,则零向量就是吸引池为整个相空间的吸引子。
Linear equation or system线性方程或系统
A single-variable (univariate) linear difference equation of the homogeneous form[math]\displaystyle{ x_t=ax_{t-1} }[/math] diverges to infinity if |a| > 1 from all initial points except 0; there is no attractor and therefore no basin of attraction. But if |a| < 1 all points on the number line map asymptotically (or directly in the case of 0) to 0; 0 is the attractor, and the entire number line is the basin of attraction.
Equations or systems that are nonlinear can give rise to a richer variety of behavior than can linear systems. One example is Newton's method of iterating to a root of a nonlinear expression. If the expression has more than one real root, some starting points for the iterative algorithm will lead to one of the roots asymptotically, and other starting points will lead to another. The basins of attraction for the expression's roots are generally not simple—it is not simply that the points nearest one root all map there, giving a basin of attraction consisting of nearby points. The basins of attraction can be infinite in number and arbitrarily small. For example, for the function [math]\displaystyle{ f(x)=x^3-2x^2-11x+12 }[/math], the following initial conditions are in successive basins of attraction:
Likewise, a linear matrix difference equation in a dynamic vectorX, of the homogeneous form [math]\displaystyle{ X_t=AX_{t-1} }[/math] in terms of square matrixA will have all elements of the dynamic vector diverge to infinity if the largest eigenvalue of A is greater than 1 in absolute value; there is no attractor and no basin of attraction. But if the largest eigenvalue is less than 1 in magnitude, all initial vectors will asymptotically converge to the zero vector, which is the attractor; the entire n-dimensional space of potential initial vectors is the basin of attraction.
Basins of attraction in the complex plane for using Newton's method to solve x5 − 1 = 0. Points in like-colored regions map to the same root; darker means more iterations are needed to converge.
Similar features apply to linear differential equations. The scalar equation [math]\displaystyle{ dx/dt =ax }[/math] causes all initial values of x except zero to diverge to infinity if a > 0 but to converge to an attractor at the value 0 if a < 0, making the entire number line the basin of attraction for 0. And the matrix system [math]\displaystyle{ dX/dt=AX }[/math] gives divergence from all initial points except the vector of zeroes if any eigenvalue of the matrix A is positive; but if all the eigenvalues are negative the vector of zeroes is an attractor whose basin of attraction is the entire phase space.
Equations or systems that are nonlinear can give rise to a richer variety of behavior than can linear systems. One example is Newton's method of iterating to a root of a nonlinear expression. If the expression has more than one real root, some starting points for the iterative algorithm will lead to one of the roots asymptotically, and other starting points will lead to another. The basins of attraction for the expression's roots are generally not simple—it is not simply that the points nearest one root all map there, giving a basin of attraction consisting of nearby points. The basins of attraction can be infinite in number and arbitrarily small. For example,[10] for the function [math]\displaystyle{ f(x)=x^3-2x^2-11x+12 }[/math], the following initial conditions are in successive basins of attraction:
与线性系统相比,非线性方程或系统可以产生更多行为。一个例子是非线性表达式根的牛顿迭代法。如果表达式有多个实根,则迭代算法的某些起始点会渐近地靠近其中一个根,而其他起始点会得出另一个根。表达式根的吸引池通常并不简单,最接近某一个根的点都被映射到那里,从而形成由附近点组成的吸引区。吸引池在数值上可以是无限的,可以任意小。例如[11]对于函数[math]\displaystyle{ f(x)=x^3-2x^2-11x+12 }[/math],以下初始条件在连续的吸引池中
2.352836327 converges to −3;
2.352836327 converges to −3;
2.352836323 converges to 1.
2.352836323汇聚为1。
Basins of attraction in the complex plane for using Newton's method to solve x5 − 1 = 0. Points in like-colored regions map to the same root; darker means more iterations are needed to converge.
Newton's method can also be applied to complex functions to find their roots. Each root has a basin of attraction in the complex plane; these basins can be mapped as in the image shown. As can be seen, the combined basin of attraction for a particular root can have many disconnected regions. For many complex functions, the boundaries of the basins of attraction are fractals.
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.
For the three-dimensional, incompressible Navier–Stokes equation with periodic boundary conditions, if it has a global attractor, then this attractor will be of finite dimensions.
Newton's method can also be applied to complex functions to find their roots. Each root has a basin of attraction in the complex plane; these basins can be mapped as in the image shown. As can be seen, the combined basin of attraction for a particular root can have many disconnected regions. For many complex functions, the boundaries of the basins of attraction are fractals.
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).
从计算的角度来看,吸引子可以分为“自激吸引子”或隐藏吸引子[12][13][14][15] Self-excited attractors can be localized numerically by standard computational procedures, in which after a transient sequence, a trajectory starting from a point on an unstable manifold in a small neighborhood of an unstable equilibrium reaches an attractor, such as the classical attractors in the Van der Pol, Belousov–Zhabotinsky, Lorenz, and many other dynamical systems. In contrast, the basin of attraction of a hidden attractor does not contain neighborhoods of equilibria, so the hidden attractor cannot be localized by standard computational procedures.
自激吸引子可以用标准的计算程序进行数值定域,在一个瞬态序列之后,从不稳定平衡小邻域中不稳定流形上的点出发的轨迹将到达一个吸引子,例如范德波尔振荡器中的经典吸引子Van der Pol oscillator,别洛乌索夫·扎伯廷斯基反应 Belousov–Zhabotinsky reaction,洛伦兹吸引子 Lorenz attractor,以及其他许多动力系统。相比之下,隐藏吸引子的吸引池不包含平衡邻域,因此隐藏吸引子不能用标准的计算程序进行局部化。
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↑Carvalho, A.; Langa, J.A.; Robinson, J. (2012). Attractors for infinite-dimensional non-autonomous dynamical systems. 182. Springer.
↑Kantz, H.; Schreiber, T. (2004). Nonlinear time series analysis. Cambridge university press.
↑John Milnor (1985). "On the concept of attractor". Communications in Mathematical Physics. 99 (2): 177–195. doi:10.1007/BF01212280.
↑Greenwood, J. A.; J. B. P. Williamson (6 December 1966). "Contact of Nominally Flat Surfaces". Proceedings of the Royal Society. 295 (1442): 300–319. doi:10.1098/rspa.1966.0242.
↑Chekroun M. D.; Simonnet E. & Ghil M. (2011). "Stochastic climate dynamics: Random attractors and time-dependent invariant measures". Physica D. 240 (21): 1685–1700. CiteSeerX10.1.1.156.5891. doi:10.1016/j.physd.2011.06.005. {{cite journal}}: Text "Examples of strange attractors include the double-scroll attractor, Hénon attractor, Rössler attractor, and Lorenz attractor." ignored (help)
↑Leonov G.A.; Vagaitsev V.I.; Kuznetsov N.V. (August 2006
journal = Physica D). [http://www.ams.org/notices/200607/what-is-ruelle.pdfHttp://www.ams.org/notices/200607/what-is-ruelle.pdf
url = http://www.math.spbu.ru/user/nk/PDF/2012-Physica-D-Hidden-attractor-Chua-circuit-smooth.pdf "什么是... 奇异吸引子? year = 2012"] (PDF). 美国数学学会公告 title = Hidden attractor in smooth Chua systems. 53
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