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== 迭代函数 ==

== 迭代函数 ==

给定一个从集合<math>X</math>到自身的映射<math>f</math>,

给定一个从集合<math>X</math>到自身的映射<math>f</math>,

where <math>f_n</math> is the ''n''th [[iterated function|iterate]] of ''f''. The smallest positive [[integer]] ''n'' satisfying the above is called the ''prime period'' or ''least period'' of the point ''x''. If every point in ''X'' is a periodic point with the same period ''n'', then ''f'' is called ''periodic'' with period ''n'' (this is not to be confused with the notion of a [[periodic function]]).

where <math>f_n</math> is the ''n''th [[iterated function|iterate]] of ''f''. The smallest positive [[integer]] ''n'' satisfying the above is called the ''prime period'' or ''least period'' of the point ''x''. If every point in ''X'' is a periodic point with the same period ''n'', then ''f'' is called ''periodic'' with period ''n'' (this is not to be confused with the notion of a [[periodic function]]).

其中<math>f_n</math>为<math>f</math>的第<math>n</math>次迭代。满足上述条件的最小正整数<math>n</math>称为点<math>x</math>的素数周期或最小周期。如果X中的每一个点都是周期为n的周期点，那么 f被称为周期点，周期为n(这不能和周期函数的概念混淆)。

其中<math>f_n</math>为<math>f</math>的第<math>n</math>次迭代。满足上述条件的最小正整数<math>n</math>称为点<math>x</math>的素数周期prime period或最小周期。如果<math>X</math>中的每一个点都是周期为<math>n</math>的周期点，那么<math>f</math>有周期性，周期为<math>n</math>(这不能和周期函数的概念混淆)。

If there exist distinct ''n'' and ''m'' such that <math>f_n(x) = f_m(x)</math>

If there exist distinct ''n'' and ''m'' such that <math>f_n(x) = f_m(x)</math>

如果存在不同的n和m使:<math>f_n(x) = f_m(x)</math>

如果存在不同的<math>n</math>和<math>m</math>使:<math>f_n(x) = f_m(x)</math>

then ''x'' is called a '''preperiodic point'''. All periodic points are preperiodic.

then ''x'' is called a '''preperiodic point'''. All periodic points are preperiodic.

If ''f'' is a [[diffeomorphism]] of a [[differentiable manifold]], so that the [[derivative]] <math>f_n^\prime</math> is defined, then one says that a periodic point is ''hyperbolic'' if

If ''f'' is a [[diffeomorphism]] of a [[differentiable manifold]], so that the [[derivative]] <math>f_n^\prime</math> is defined, then one says that a periodic point is ''hyperbolic'' if

:<math>|f_n^\prime|\ne 1,</math>

:<math>|f_n^\prime|\ne 1,</math>

如果f是微分流形的微分同胚，则定义了导数<math>f_n^\prime</math>，如果:<math>|f_n^\prime|\ne 1,</math>，那么f是双曲周期点，

如果<math>x</math>是微分流形的微分同胚，则定义了导数<math>f_n^\prime</math>，如果:<math>|f_n^\prime|\ne 1,</math>，那么<math>f</math>是双曲周期点，

that it is ''[[Attractor|attractive]]'' if :<math>|f_n^\prime|< 1,</math>

that it is ''[[Attractor|attractive]]'' if :<math>|f_n^\prime|< 1,</math>

如果:<math>|f_n^\prime|< 1,</math>，则称周期点f为吸引子，

如果:<math>|f_n^\prime|< 1,</math>，则称周期点<math>f</math>为吸引子，

and it is ''repelling'' if:<math>|f_n^\prime|> 1.</math>

and it is ''repelling'' if:<math>|f_n^\prime|> 1.</math>

如果:<math>|f_n^\prime|> 1.</math>，则称周期点f为排斥子。

如果:<math>|f_n^\prime|> 1.</math>，则称周期点<math>f</math>为排斥子。

If the [[dimension]] of the [[stable manifold]] of a periodic point or fixed point is zero, the point is called a ''source''; if the dimension of its [[unstable manifold]] is zero, it is called a ''sink''; and if both the stable and unstable manifold have nonzero dimension, it is called a ''saddle'' or [[saddle point]].

If the [[dimension]] of the [[stable manifold]] of a periodic point or fixed point is zero, the point is called a ''source''; if the dimension of its [[unstable manifold]] is zero, it is called a ''sink''; and if both the stable and unstable manifold have nonzero dimension, it is called a ''saddle'' or [[saddle point]].

=== Examples ===

===示例 ===

A period-one point is called a [[fixed point (mathematics)|fixed point]].

A period-one point is called a [[fixed point (mathematics)|fixed point]].

参数r的各种值呈现周期性。对于介于0到1之间的r，0是唯一的周期点，周期为1（给出了吸引所有轨道的序列0,0,0，... ）。对于介于1到3之间的r，值0仍然是周期性的，但不是吸引点，而该值是周期1的吸引周期点。当r大于3但小于1 + 时，存在一对周期2的点，它们共同构成一个吸引序列，非吸引周期1点为0。当参数r的值上升到4时，会出现周期为正的一组周期点；对于 r 的某些值，这些重复序列中的一个被吸引，而对于其他值，则没有一个被吸引(几乎所有的轨道都是混乱的)。

参数r的各种值呈现周期性。对于介于0到1之间的r，0是唯一的周期点，周期为1（给出了吸引所有轨道的序列0,0,0，... ）。对于介于1到3之间的r，值0仍然是周期性的，但不是吸引点，而该值是周期1的吸引周期点。当r大于3但小于1 + 时，存在一对周期2的点，它们共同构成一个吸引序列，非吸引周期1点为0。当参数r的值上升到4时，会出现周期为正的一组周期点；对于 r 的某些值，这些重复序列中的一个被吸引，而对于其他值，则没有一个被吸引(几乎所有的轨道都是混乱的)。

== Dynamical system ==

== Dynamical system ==