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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.

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

两个简单的<font color="#ff8000"> 吸引子</font>是一个<font color="#ff8000"> 不动点</font>和一个<font color="#ff8000"> 极限环</font>。<font color="#ff8000"> 吸引子</font>可以呈现许多其他几何形状（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[[几何本原|基本几何对象]]（例如，[直线（数学）|直线]]，[[曲面（拓扑）|曲面]]s，[[球体]]s，[[环面]]s，[[环面]]s，[[流形]]s的简单组合（例如，[交集（集合论）|交集]]和[[并集理论）|并集]]，则这个吸引子被称为“[[奇怪吸引子]]”。

两个简单的<font color="#ff8000"> 吸引子</font>是一个<font color="#ff8000"> 不动点</font>和一个<font color="#ff8000"> 极限环</font>。<font color="#ff8000"> 吸引子</font>可以呈现许多其他几何形状（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[[几何本原|基本几何对象]]（例如，[直线（数学）|直线]]，[[曲面（拓扑）|曲面]]s，[[球体]]s，[[环面]]s，[[环面]]s，[[流形]]s的简单组合（例如，[交集（集合论）|交集]]和[[并集理论）|并集]]，则这个吸引子被称为“[[奇异吸引子]]”。

=== Fixed point驻点 ===

=== Fixed point不动点 ===

[[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.]]

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

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

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

函数或变换的不动点是通过函数或变换映射到自身的点。如果我们把动力系统的演化看作是一系列的转变，那么在每一个转变下，可能会有一个点是固定的，也可能没有。动力系统的最终状态对应于该系统演化函数的吸引不动点，例如阻尼摆的中心底部位置，玻璃杯中晃动水的水平线和平坦线，或碗的底部中心含有滚动的大理石。但是动态系统的不动点不一定是系统的吸引子。例如，如果装有滚动大理石的碗被倒置，大理石平衡在碗的顶部，碗的中心底部(现在是顶部)是一个固定的状态，但不是一个吸引子。这等价于稳定平衡点和不稳定平衡点之差。如果一个大理石在一个倒碗(山)的顶部，这个点在碗(山)的顶部是一个不动点(平衡) ，但不是一个吸引子(稳定的平衡)。

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

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

此外，至少有一个固定点的物理动力系统，由于物理世界动力学的现实性，包括非线性动力学的粘滞，摩擦，表面粗糙度，变形(弹性和塑性) ，甚至量子力学，总是有多个固定点和吸引子。在倒置碗顶的大理石上，即使碗看起来完美的半球形，和大理石的球形，在显微镜下观察时都是更复杂的表面，它们的形状在接触过程中改变或变形。任何物理表面都可以看到一个由多个山峰、山谷、鞍点、山脊、峡谷和平原组成的崎岖地形。在这个表面地形中有许多点(以及在这个微观地形上滚动的同样粗糙的大理石的动力系统)被认为是静止的或不动的点，其中一些被归类为吸引子。

此外，至少有一个不动点的物理动力系统，由于物理世界动力学的现实性，包括非线性动力学的粘滞，摩擦，表面粗糙度，变形(弹性和塑性) ，甚至量子力学，总是有多个不动点和吸引子。在倒置碗顶的大理石上，即使碗看起来完美的半球形，和大理石的球形，在显微镜下观察时都是更复杂的表面，它们的形状在接触过程中改变或变形。任何物理表面都可以看到一个由多个山峰、山谷、鞍点、山脊、峡谷和平原组成的崎岖地形。在这个表面地形中有许多点(以及在这个微观地形上滚动的同样粗糙的大理石的动力系统)被认为是静止的或不动的点，其中一些被归类为吸引子。

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 2ⁿ points, 3×2ⁿ points, etc., for any value of n.

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

在一个[[离散和连续时间|离散时间]]系统中，<font color="#ff8000"> 吸引子</font>可以以有限数量的点的形式依次访问。每个点都称为[[周期点]]。[[逻辑图]]说明了这一点，根据其特定参数值，对于任何“n”值，可以有由2^''n''点、3×2^''n''点等组成的<font color="#ff8000"> 吸引子</font>。

在一个[[离散和连续时间|离散时间]]系统中，<font color="#ff8000"> 吸引子</font>可以用有限数量的点的形式依次访问。每个点都称为[[周期点]]。[[逻辑图]]说明了这一点，根据其特定参数值，对于任何“n”值，可以有由2^''n''点、3×2^''n''点等组成的<font color="#ff8000"> 吸引子</font>。

=== Limit cycle 极限环===

=== Limit cycle 极限环===

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

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

关于洛伦兹奇怪吸引子 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

An attractor is called '''strange''' if it has a [[fractal]] structure. This is often the case when the dynamics on it are [[chaos theory|chaotic]], but [[strange nonchaotic attractor]]s also exist.  If a strange attractor is chaotic, exhibiting [[sensitive dependence on initial conditions]], then any two arbitrarily close alternative initial points on the attractor, after any of various numbers of iterations, will lead to points that are arbitrarily far apart (subject to the confines of the attractor), and after any of various other numbers of iterations will lead to points that are arbitrarily close together. Thus a dynamic system with a chaotic attractor is locally unstable yet globally stable: once some sequences have entered the attractor, nearby points diverge from one another but never depart from the attractor.<ref>{{cite journal | author = Grebogi Celso, Ott Edward, Yorke James A | year = 1987 | title = Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics | url = | journal = Science | volume = 238 | issue = 4827| pages = 632–638 | doi = 10.1126/science.238.4827.632 | pmid = 17816542 | bibcode = 1987Sci...238..632G }}</ref>

An attractor is called '''strange''' if it has a [[fractal]] structure. This is often the case when the dynamics on it are [[chaos theory|chaotic]], but [[strange nonchaotic attractor]]s also exist.  If a strange attractor is chaotic, exhibiting [[sensitive dependence on initial conditions]], then any two arbitrarily close alternative initial points on the attractor, after any of various numbers of iterations, will lead to points that are arbitrarily far apart (subject to the confines of the attractor), and after any of various other numbers of iterations will lead to points that are arbitrarily close together. Thus a dynamic system with a chaotic attractor is locally unstable yet globally stable: once some sequences have entered the attractor, nearby points diverge from one another but never depart from the attractor.<ref>{{cite journal | author = Grebogi Celso, Ott Edward, Yorke James A | year = 1987 | title = Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics | url = | journal = Science | volume = 238 | issue = 4827| pages = 632–638 | doi = 10.1126/science.238.4827.632 | pmid = 17816542 | bibcode = 1987Sci...238..632G }}</ref>

An attractor is called strange if it has a fractal structure. This is often the case when the dynamics on it are chaotic, but strange nonchaotic attractors also exist.  If a strange attractor is chaotic, exhibiting sensitive dependence on initial conditions, then any two arbitrarily close alternative initial points on the attractor, after any of various numbers of iterations, will lead to points that are arbitrarily far apart (subject to the confines of the attractor), and after any of various other numbers of iterations will lead to points that are arbitrarily close together. Thus a dynamic system with a chaotic attractor is locally unstable yet globally stable: once some sequences have entered the attractor, nearby points diverge from one another but never depart from the attractor.

An attractor is called strange if it has a fractal structure. This is often the case when the dynamics on it are chaotic, but strange nonchaotic attractors also exist.  If a strange attractor is chaotic, exhibiting sensitive dependence on initial conditions, then any two arbitrarily close alternative initial points on the attractor, after any of various numbers of iterations, will lead to points that are arbitrarily far apart (subject to the confines of the attractor), and after any of various other numbers of iterations will lead to points that are arbitrarily close together. Thus a dynamic system with a chaotic attractor is locally unstable yet globally stable: once some sequences have entered the attractor, nearby points diverge from one another but never depart from the attractor.

如果吸引子具有[[分形]]结构，则称为“奇异”。当它的动力学是[[混沌理论|混沌]]时，通常会出现这种情况，但是[[奇异的非混沌吸引子]]也存在。如果一个<font color="#ff8000"> 奇异吸引子</font>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，经过任意多次迭代后，都会导致任意相距很远的点（受吸引子的限制），而在其他次数的迭代之后，都会导致任意接近的点。因此，具有混沌吸引子的动态系统是局部不稳定的但全局稳定的：一旦一些序列进入<font color="#ff8000"> 吸引子</font>，附近的点就会彼此发散，但不会离开<font color="#ff8000"> 吸引子</font>。

如果吸引子具有[[分形]]结构，则称为“奇异”。当它的动力学是[[混沌理论|混沌]]时，通常会出现这种情况，但是[[奇异的非混沌吸引子]]也存在。如果一个<font color="#ff8000"> 奇异吸引子</font>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，经过任意多次迭代后，会导致任意相距很远的点（受吸引子的限制），而在其他次数的迭代之后，会导致任意接近的点。因此，具有混沌吸引子的动态系统是局部不稳定的但全局稳定的：一旦一些序列进入<font color="#ff8000"> 吸引子</font>，附近的点就会彼此发散，但不会离开<font color="#ff8000"> 吸引子</font>。