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第45行： − 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>+ 第63行：
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{{更多脚注{日期=2013年3月}}
{{更多脚注{日期=2013年3月}}
[[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>.]]
[[泊松文件：吸引子土星.jpg|右| 333px |拇指|视觉表示[[#奇怪吸引器|奇怪吸引器]]<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>.]]
[[泊松文件：吸引子土星.jpg|右| 333px |拇指|视觉表示[[#奇异吸引子|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>.]]
[[strange attractor.]]
[[strange attractor.]]
[[奇怪的吸引子]]
[[奇异吸引子]]
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.
在动力系统的数学领域中，<font color="#ff8000"> 吸引子Attractor</font>是系统在各种初始条件下演化趋向于的一组数值。即使稍微受到干扰，与吸引子值足够接近的系统值仍然保持足够接近<font color="#ff8000"> 吸引子</font>。
在动力系统的数学领域中，<font color="#ff8000"> 吸引子Attractor</font>是系统在各种初始条件下演化趋向于的一组数值。即使稍微受到干扰，与吸引子的值足够接近的系统值仍然保持足够接近<font color="#ff8000"> 吸引子</font>。
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<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。
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"> 奇异吸引子Strange attractor</font>(见下面的奇异吸引子)。如果变量是标量，那么吸引子就是实数线的子集。描述<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).
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，但其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>
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<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>。
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<font color="#ff8000"> 吸引子</font>，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.
<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。
<font color="#ff8000"> 不变集</font>和<font color="#ff8000"> 极限集</font>是类似于吸引子的概念。<font color="#ff8000"> 不变集</font>是在动力学作用下向自身演化的集合。<font color="#ff8000"> 不变集</font>可能包含于<font color="#ff8000"> 吸引子</font>。<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时最终将任意接近极限集(即收敛到集合的每个点）。<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。