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添加43字节 、 2021年1月22日 (五) 17:38
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== 迭代函数 ==
 
== 迭代函数 ==
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Given a [[mapping (mathematics)|mapping]] ''f'' from a [[set (mathematics)|set]] ''X'' into itself,
      
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>,
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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]]).
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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)。
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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)。
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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>
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如果存在不同的n和m使:<math>f_n(x) = f_m(x)</math>
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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.
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那么x称为前周期点。所有周期点都是预周期点。
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那么<math>x</math>称为前周期点。所有周期点都是前周期点。
    
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
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:<math>|f_n^\prime|\ne 1,</math>
 
:<math>|f_n^\prime|\ne 1,</math>
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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,
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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>
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如果:<math>|f_n^\prime|< 1,</math>,则称周期点f为吸引子,
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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>
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如果:<math>|f_n^\prime|> 1.</math>,则称周期点f为排斥子。
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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]].
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=== Examples ===
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===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]].
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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)。
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== Dynamical system ==
 
== Dynamical system ==