更改

== Types of attractors ==

== Types of attractors 吸引子的类型==

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]].

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]].

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。

<font color="#ff8000"> 吸引子</font>是动力系统的<font color="#ff8000"> 相空间</font>的一部分或<font color="#ff8000"> 子集</font>。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<font color="#ff8000"> 拓扑野生集Topologically wild sets,</font>，在当时是已知的，但被认为是脆弱的异常。斯蒂芬 · 斯梅尔Stephen Smale能够证明他的马蹄地图是健壮的，它的吸引子具有<font color="#ff8000"> 康托集Cantor set</font>的结构。

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.

=== Fixed point ===

=== Fixed point驻点 ===

[[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.]]

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.]]

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.]]

根据[复二次多项式]演化的复数的弱吸引不动点。相空间为水平复平面，垂直轴测量复平面上点的频率。在复平面上的点直接低于峰值频率是不动点吸引子。]

根据[[复二次多项式演化的复数的弱吸引不动点。相空间是水平复平面；纵轴测量访问复平面中的点的频率。复平面中峰值频率正下方的点是不动点吸引子。]]

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).

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).

Examples of strange attractors include the [[Double scroll attractor|double-scroll attractor]], [[Hénon map|Hénon attractor]], [[Rössler attractor]], and [[Lorenz attractor]].

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==

==Attractors characterize the evolution of a system==