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第316行：
第316行： − + 第322行：
第322行： − 吸引子的吸引盆是相空间的一个区域，在这个区域上定义迭代，使得该区域内的任何点(任何初始条件)都渐近迭代到吸引子。对于稳定的线性系统，相空间中的每一点都处于吸引盆中。然而，在非线性系统中，一些点可以直接或渐近地映射到无穷远处，而另一些点可能位于不同的吸引域中，并渐近地映射到不同的吸引子中，其他的初始条件可能位于或直接映射到不吸引点或不吸引周期中。+ − + − + − 一个齐次形式的单变量(单变量)线性差分方程 x _ t = ax _ { t-1} </math > 如果 | a | > 1从除0以外的所有初始点到无穷远，没有吸引子，因此没有吸引盆。但是如果 | a | < 1所有点在数线图上渐近地(或直接在0的情况下)到0; 0是吸引子，而整个数线是吸引盆。+
→Attractors characterize the evolution of a system
奇异吸引子的例子包括[[双滚动吸引子|双滚动吸引子]]、[[Hénon-map | Hénon吸引子]]、[[Rössler吸引子]]和[[Lorenz吸引子]]。
奇异吸引子的例子包括[[双滚动吸引子|双滚动吸引子]]、[[Hénon-map | Hénon吸引子]]、[[Rössler吸引子]]和[[Lorenz吸引子]]。
==Attractors characterize the evolution of a system==
==Attractors characterize the evolution of a system吸引子表征系统的演化==
An attractor's basin of attraction is the region of the phase space, over which iterations are defined, such that any point (any initial condition) in that region will asymptotically be iterated into the attractor. For a stable linear system, every point in the phase space is in the basin of attraction. However, in nonlinear systems, some points may map directly or asymptotically to infinity, while other points may lie in a different basin of attraction and map asymptotically into a different attractor; other initial conditions may be in or map directly into a non-attracting point or cycle.
An attractor's basin of attraction is the region of the phase space, over which iterations are defined, such that any point (any initial condition) in that region will asymptotically be iterated into the attractor. For a stable linear system, every point in the phase space is in the basin of attraction. However, in nonlinear systems, some points may map directly or asymptotically to infinity, while other points may lie in a different basin of attraction and map asymptotically into a different attractor; other initial conditions may be in or map directly into a non-attracting point or cycle.
吸引子的吸引域是相空间的区域，在这个区域上定义了迭代，使得该区域中的任何点（任何初始条件）都将渐近地迭代到吸引子中。对于一个稳定的线性系统，相空间中的每一点都在吸引域中。然而，在非线性系统中，有些点可能直接或渐近地映射到无穷大，而另一些点可能位于不同的吸引域中并渐近映射到不同的吸引子；其他初始条件可能位于或直接映射到非吸引点或循环中。
[[File:Logistic Map Bifurcation Diagram, Matplotlib.svg|350px|thumb|right|Bifurcation diagram of the [[logistic map]]. The attractor(s) for any value of the parameter ''r'' are shown on the ordinate in the domain <math>0<x<1</math>. The colour of a point indicates how often the point <math>(r, x)</math> is visited over the course of 10<sup>6</sup> iterations: frequently encountered values are coloured in blue, less frequently encountered values are yellow. A [[period-doubling bifurcation|bifurcation]] appears around <math>r\approx3.0</math>, a second bifurcation (leading to four attractor values) around <math>r\approx3.5</math>. The behaviour is increasingly complicated for <math>r>3.6</math>, interspersed with regions of simpler behaviour (white stripes).]]
[[File:Logistic Map Bifurcation Diagram, Matplotlib.svg|350px|thumb|right|Bifurcation diagram of the [[logistic map]]. The attractor(s) for any value of the parameter ''r'' are shown on the ordinate in the domain <math>0<x<1</math>. The colour of a point indicates how often the point <math>(r, x)</math> is visited over the course of 10<sup>6</sup> iterations: frequently encountered values are coloured in blue, less frequently encountered values are yellow. A [[period-doubling bifurcation|bifurcation]] appears around <math>r\approx3.0</math>, a second bifurcation (leading to four attractor values) around <math>r\approx3.5</math>. The behaviour is increasingly complicated for <math>r>3.6</math>, interspersed with regions of simpler behaviour (white stripes).]]
[[文件：逻辑图分岔图，Matplotlib.svg|350px |拇指|右|分岔图[[逻辑图]]。参数“r”的任何值的吸引子显示在域<math>0<x<1</math>的纵坐标上。点的颜色表示在10次<sup>6次迭代过程中访问点<math>（r，x）</math>的频率：经常遇到的值用蓝色表示，不太常见的值用黄色表示。在<math>r\approx3.0</math>附近出现[[倍周期分岔|分岔]]，在<math>r\approx3.5</math>附近出现第二个分岔（导致四个吸引子值）。当<math>r>3.6<math>时，行为变得越来越复杂，中间穿插着行为更简单的区域（白色条纹）。]]
The parameters of a dynamic equation evolve as the equation is iterated, and the specific values may depend on the starting parameters. An example is the well-studied [[logistic map]], <math>x_{n+1}=rx_n(1-x_n)</math>, whose basins of attraction for various values of the parameter ''r'' are shown in the figure. If <math>r=2.6</math>, all starting ''x'' values of <math>x<0</math> will rapidly lead to function values that go to negative infinity; starting ''x'' values of <math>x>0</math> will go to infinity. But for <math>0<x<1</math> the ''x'' values rapidly converge to <math>x\approx0.615</math>, i.e. at this value of ''r'', a single value of ''x'' is an attractor for the function's behaviour. For other values of ''r'', more than one value of x may be visited: if ''r'' is 3.2, starting values of <math>0<x<1</math> will lead to function values that alternate between <math>x\approx0.513</math> and <math>x\approx0.799</math>. At some values of ''r'', the attractor is a single point (a [[#Fixed_point|"fixed point"]]), at other values of ''r'' two values of ''x'' are visited in turn (a [[period-doubling bifurcation]]); at yet other values of r, any given number of values of ''x'' are visited in turn; finally, for some values of ''r'', an infinitude of points are visited. Thus one and the same dynamic equation can have various types of attractors, depending on its starting parameters.
The parameters of a dynamic equation evolve as the equation is iterated, and the specific values may depend on the starting parameters. An example is the well-studied [[logistic map]], <math>x_{n+1}=rx_n(1-x_n)</math>, whose basins of attraction for various values of the parameter ''r'' are shown in the figure. If <math>r=2.6</math>, all starting ''x'' values of <math>x<0</math> will rapidly lead to function values that go to negative infinity; starting ''x'' values of <math>x>0</math> will go to infinity. But for <math>0<x<1</math> the ''x'' values rapidly converge to <math>x\approx0.615</math>, i.e. at this value of ''r'', a single value of ''x'' is an attractor for the function's behaviour. For other values of ''r'', more than one value of x may be visited: if ''r'' is 3.2, starting values of <math>0<x<1</math> will lead to function values that alternate between <math>x\approx0.513</math> and <math>x\approx0.799</math>. At some values of ''r'', the attractor is a single point (a [[#Fixed_point|"fixed point"]]), at other values of ''r'' two values of ''x'' are visited in turn (a [[period-doubling bifurcation]]); at yet other values of r, any given number of values of ''x'' are visited in turn; finally, for some values of ''r'', an infinitude of points are visited. Thus one and the same dynamic equation can have various types of attractors, depending on its starting parameters.
动力学方程的参数随着方程的迭代而变化，具体值可能取决于初始参数。一个例子是经过充分研究的[[logistic map]]，<math>x{n+1}=rx\u n（1-xün）</math>，其对参数“r”的各种值的吸引范围如图所示。如果<math>r=2.6</math>，所有开始的<math>x<0</math>的“x”值将迅速导致函数值变为负无穷大；<math>x>0开始的“x”值将变为无穷大。但对于<math>0<x<1</math>“x”值迅速收敛到<math>x\approx0.615</math>，即在“r”值处，单个值“x”是函数行为的吸引子。对于“r”的其他值，可以访问x的多个值：如果“r”为3.2，<math>0<x<1</math>的起始值将导致函数值在<math>x\approx0.513</math>和<math>x\approx0.799</math>之间交替。在“r”的某些值处，吸引子是一个单点（a[[#不动点|“不动点”]]），在“r”的其他值处，依次访问“x”的两个值（a[[倍周期分岔]]）；在r的其他值处，依次访问任意给定数量的“x”值；最后，对于“r”的某些值，访问无穷多个点。因此，同一个动力学方程可以有不同类型的吸引子，这取决于它的起始参数。
A single-variable (univariate) linear difference equation of the homogeneous form <math>x_t=ax_{t-1}</math> diverges to infinity if |a| > 1 from all initial points except 0; there is no attractor and therefore no basin of attraction. But if |a| < 1 all points on the number line map asymptotically (or directly in the case of 0) to 0; 0 is the attractor, and the entire number line is the basin of attraction.
A single-variable (univariate) linear difference equation of the homogeneous form <math>x_t=ax_{t-1}</math> diverges to infinity if |a| > 1 from all initial points except 0; there is no attractor and therefore no basin of attraction. But if |a| < 1 all points on the number line map asymptotically (or directly in the case of 0) to 0; 0 is the attractor, and the entire number line is the basin of attraction.
齐次形式的单变量（单变量）线性差分方程<math>x_t=ax{t-1}</math>从除0以外的所有初始点| A>1发散到无穷大；没有吸引子，因此没有吸引池。但是如果| a |<1，则数线图上的所有点渐进地（或在0的情况下直接）到0；0是吸引子，整个数线是吸引域。
==Basins of attraction==
==Basins of attraction==