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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''² + ''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.]]

[[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''² + ''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.]]

[[资料图：茱莉亚立即盆地1 3.png |右|拇指|吸引周期-3旋回及其对“f”（“z”）参数化的直接吸引盆地。三个最暗的点是3个循环的点，它们按顺序相互连接，从吸引盆地中的任何点迭代会导致（通常是渐进的）收敛到这三个点的序列。]]

[[资料图：茱莉亚立即盆地1 3.png |右|拇指|吸引周期-3旋回及其对“f”（“z”）参数化的直接吸引域。三个最暗的点是3循环的点，它们按顺序相互连接，从吸引域中的任何点迭代会导致（通常是渐进的）收敛到这三个点的序列。]]

Attracting period-3 cycle and its immediate basin of attraction for a certain parametrization of f(z) = z² + 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) = z² + 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.

f(z) = z² + c的某一特定参数的吸引3-周期循环及其直接吸引域。最暗的三个点是3-周期循环的点，它们依次相向，从吸引域中的任何一点迭代都会导致（通常是渐近的）收敛到这三个点的序列。

f(z) = z² + c的某一特定参数的吸引3-周期循环及其直接吸引池。最暗的三个点是3-周期循环的点，它们依次相向，从吸引域中的任何一点迭代都会导致（通常是渐近的）收敛到这三个点的序列。

  <math> f(t,(x,v))=(x+tv,v).\ </math>

  <math> f(t,(x,v))=(x+tv,v).\ </math>

F (t，(x，v) = (x + tv，v)

F (t，(x，v)） = (x + tv，v)

* ''A'' is ''forward invariant'' under ''f'': if ''a'' is an element of ''A'' then so is ''f''(''t'',''a''), for all&nbsp;''t''&nbsp;>&nbsp;0.

* ''A'' is ''forward invariant'' under ''f'': if ''a'' is an element of ''A'' then so is ''f''(''t'',''a''), for all&nbsp;''t''&nbsp;>&nbsp;0.

*“A”是“f”下的“前向不变”：如果“A”是“A”的元素，则“f”（“t”，“A”）也是，对于所有“t”>0。

*“A”是“f”下的“前向不变”：如果“A”是“A”的元素，则对于所有“t”>0，“f”（“t”，“A”）也是。

* 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''&nbsp;→&nbsp;∞". More formally, ''B''(''A'') is the set of all points ''b'' in the phase space with the following property:

* 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''&nbsp;→&nbsp;∞". More formally, ''B''(''A'') is the set of all points ''b'' in the phase space with the following property:

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 Rⁿ, 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 Rⁿ, the Euclidean norm is typically used.

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Rⁿ的情况下，通常使用<font color="#ff8000"> 欧氏范数Euclidean norm</font>。

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依赖于相空间的拓扑结构。在Rⁿ的情况下，通常使用<font color="#ff8000"> 欧氏范数Euclidean norm</font>。

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.  

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.  

<font color="#ff8000"> 吸引子</font>的许多其他定义出现在文献中。例如，一些作者要求<font color="#ff8000"> 吸引子</font>具有正测度(防止一个点成为吸引子) ，另一些作者放松了 B(A)是一个邻域的要求。

在文献中有<font color="#ff8000"> 吸引子</font>的许多其他定义出现。例如，一些作者要求<font color="#ff8000"> 吸引子</font>具有正测度(防止一个点成为吸引子) ，另一些作者放松了 B(A)是一个邻域的要求。

 

 

== Types of attractors 吸引子的类型==

== Types of attractors 吸引子的类型==