1,568
个编辑
更改
跳到导航
跳到搜索
小第87行：
第87行： − 一些吸引子被认为是混沌的(见 # 奇异吸引子) ，在这种情况下，吸引子的任意两个不同点的演化导致了指数发散轨迹，当系统中甚至存在最小的噪声时，这使预测变得复杂。+ 第97行：
第97行： − + 第121行：
第121行： − + 第141行：
第141行： − + 第159行：
第159行： − 吸引子是动力系统的相空间的一部分或子集。直到20世纪60年代，吸引子都被认为是相空间的简单几何子集，像点、线、面和简单的三维空间区域。更复杂的吸引子，不能被归类为简单的几何子集，如拓扑野生集，在当时是已知的，但被认为是脆弱的异常。斯蒂芬 · 斯梅尔能够证明他的马蹄地图是健壮的，它的吸引子具有康托集的结构。+ 第565行：
第565行： − 第七期+ 第571行：
第571行： − +
Moved page from wikipedia:en:Attractor (history)
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.
有些吸引子是混沌的，在这种情况下，吸引子的任意两个不同点的演化都会导致指数发散轨迹，即使系统中存在最小的噪声，预测也会变得复杂。
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
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
设 t 表示时间，设 f (t，•)是指定系统动力学的函数。也就是说，如果 a 是 n 维相空间中的一个点，表示系统的初始状态，那么 f (0，a) = a，对于 t 的正值，f (t，a)是该状态在 t 个时间单位之后演化的结果。例如，如果系统描述了自由粒子在一维空间中的演化，那么相空间就是平面 r < sup > 2 </sup > 和坐标(x，v) ，其中 x 是粒子的位置，v 是粒子的速度，a = (x，v)
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)
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:
吸引子是相空间 a 的一个子集，它具有以下三个拥有属性:
吸引子是相空间 a 的子集，它具有以下三个拥有属性:
* ''A'' is ''forward invariant'' under ''f'': if ''a'' is an element of ''A'' then so is ''f''(''t'',''a''), for all ''t'' > 0.
* ''A'' is ''forward invariant'' under ''f'': if ''a'' is an element of ''A'' then so is ''f''(''t'',''a''), for all ''t'' > 0.
Since the basin of attraction contains an open set containing A, every point that is sufficiently close to A is attracted to A. The definition of an attractor uses a metric on the phase space, but the resulting notion usually depends only on the topology of the phase space. In the case of R<sup>n</sup>, the Euclidean norm is typically used.
Since the basin of attraction contains an open set containing A, every point that is sufficiently close to A is attracted to A. The definition of an attractor uses a metric on the phase space, but the resulting notion usually depends only on the topology of the phase space. In the case of R<sup>n</sup>, the Euclidean norm is typically used.
由于吸引盆包含一个含有 a 的开集，所以每一个离 a 足够近的点都会被 a 吸引。吸引子的定义使用了相空间上的一个度量，但得到的结果通常只依赖于相空间的拓扑结构。在 r < sup > n </sup > 的情况下，通常使用欧氏范数。
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。
Attractors are portions or subsets of the phase space of a dynamical system. Until the 1960s, attractors were thought of as being simple geometric subsets of the phase space, like points, lines, surfaces, and simple regions of three-dimensional space. More complex attractors that cannot be categorized as simple geometric subsets, such as topologically wild sets, were known of at the time but were thought to be fragile anomalies. Stephen Smale was able to show that his horseshoe map was robust and that its attractor had the structure of a Cantor set.
Attractors are portions or subsets of the phase space of a dynamical system. Until the 1960s, attractors were thought of as being simple geometric subsets of the phase space, like points, lines, surfaces, and simple regions of three-dimensional space. More complex attractors that cannot be categorized as simple geometric subsets, such as topologically wild sets, were known of at the time but were thought to be fragile anomalies. Stephen Smale was able to show that his horseshoe map was robust and that its attractor had the structure of a Cantor set.
吸引子是动力系统的相空间的一部分或子集。直到20世纪60年代，吸引子被认为是相空间的简单几何子集，像点、线、面和简单的三维空间。更复杂的吸引子，不能被归类为简单的几何子集，如拓扑野生集，在当时是已知的，但被认为是脆弱的异常。斯蒂芬 · 斯梅尔能够证明他的马蹄地图是健壮的，它的吸引子具有康托集的结构。
| issue = 7
| issue = 7
第7期
issue = 18 |
issue = 18 |
| pages = 764–765
| pages = 764–765
| 页数 = 764-765
| 页 = 764-765
pages = 1482–1486 |
pages = 1482–1486 |