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此词条暂由彩云小译翻译，翻译字数共488，未经人工整理和审校，带来阅读不便，请见谅。

此词条暂由彩云小译翻译，翻译字数共488，未经人工整理和审校，带来阅读不便，请见谅。

In [[mathematics]], in the study of [[iterated function]]s and [[dynamical system]]s, a '''periodic point''' of a [[function (mathematics)|function]] is a point which the system returns to after a certain number of function iterations or a certain amount of time.

In [[mathematics]], in the study of [[iterated function]]s and [[dynamical system]]s, a '''periodic point''' of a [[function (mathematics)|function]] is a point which the system returns to after a certain number of function iterations or a certain amount of time.

If there exist distinct ''n'' and ''m'' such that

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

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

 

:<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.

那么x称为前周期点。所有周期点都是预周期点。

那么x称为前周期点。所有周期点都是预周期点。

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

:<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f_n，如果f_n^prime不等于1，那么f是双曲周期点，

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，

 

 

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

如果f_n^prime < 1，则称周期点f为吸引子，

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

:<math>|f_n^\prime|> 1.</math>

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

如果f_n^prime > 1，则称周期点f为排斥子。

如果:<math>|f_n^\prime|> 1.</math>，则称周期点f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]].

The [[logistic map]]  

The [[logistic map]]  

后勤地图

后勤地图

:<math>x_{t+1}=rx_t(1-x_t), \qquad 0 \leq x_t \leq 1, \qquad 0 \leq r \leq 4</math>

:<math>x_{t+1}=rx_t(1-x_t), \qquad 0 \leq x_t \leq 1, \qquad 0 \leq r \leq 4</math>

exhibits periodicity for various values of the parameter ''r''. For ''r'' between 0 and 1, 0 is the sole periodic point, with period 1 (giving the sequence 0, 0, 0, ..., which [[attractor|attracts]] all orbits). For ''r'' between 1 and 3, the value 0 is still periodic but is not attracting, while the value {{nowrap|(''r'' −&thinsp;1) /&thinsp;''r''}} is an attracting periodic point of period 1. With ''r'' greater than 3 but less than 1 + {{radic|6}}, there are a pair of period-2 points which together form an attracting sequence, as well as the non-attracting period-1 points 0 and {{nowrap|(''r'' −&thinsp;1) /&thinsp;''r''}}. As the value of parameter ''r'' rises toward 4, there arise groups of periodic points with any positive integer for the period; for some values of ''r'' one of these repeating sequences is attracting while for others none of them are (with almost all orbits being chaotic).

exhibits periodicity for various values of the parameter ''r''. For ''r'' between 0 and 1, 0 is the sole periodic point, with period 1 (giving the sequence 0, 0, 0, ..., which [[attractor|attracts]] all orbits). For ''r'' between 1 and 3, the value 0 is still periodic but is not attracting, while the value {{nowrap|(''r'' −&thinsp;1) /&thinsp;''r''}} is an attracting periodic point of period 1. With ''r'' greater than 3 but less than 1 + {{radic|6}}, there are a pair of period-2 points which together form an attracting sequence, as well as the non-attracting period-1 points 0 and {{nowrap|(''r'' −&thinsp;1) /&thinsp;''r''}}. As the value of parameter ''r'' rises toward 4, there arise groups of periodic points with any positive integer for the period; for some values of ''r'' one of these repeating sequences is attracting while for others none of them are (with almost all orbits being chaotic).

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)。

给定一个实整体动力系统(R，X，Φ) ，其中X为相空间，Φ为演化函数，

给定一个实整体动力系统(R，X，Φ) ，其中X为相空间，Φ为演化函数，

<math>\Phi: \mathbb{R} \times X \to X</math>

<math>\Phi: \mathbb{R} \times X \to X</math>

a point ''x'' in ''X'' is called ''periodic'' with ''period'' ''t'' if there exists a ''t'' >&thinsp;0 so that

a point ''x'' in ''X'' is called ''periodic'' with ''period'' ''t'' if there exists a ''t'' >&thinsp;0 so that

如果存在 t > & thinsp; 0，则X中的点x称为周期为t的周期。因此

如果存在 t > & thinsp; 0，则X中的点x称为周期为t的周期。因此

:<math>\Phi(t, x) = x\,</math>

<math>\Phi(t, x) = x\,</math>

The smallest positive ''t'' with this property is called ''prime period'' of the point ''x''.

The smallest positive ''t'' with this property is called ''prime period'' of the point ''x''.

具有此性质的最小正t称为点x的素数周期。

具有此性质的最小正t称为点x的素数周期。

* Given a periodic point ''x'' with period ''p'', then <math>\Phi(t,x) = \Phi(t+p,x)</math> for all ''t'' in '''R'''

* Given a periodic point ''x'' with period ''p'', then <math>\Phi(t,x) = \Phi(t+p,x)</math> for all ''t'' in '''R'''

* 给定一个周期为“p”的周期点“x”，则对于“R”中所有“t”的Phi(t,x) = \Phi(t+p,x)

* 给定一个周期为“p”的周期点“x”，则对于“R”中所有“t”的Phi(t,x) = \Phi(t+p,x)

* Given a periodic point ''x'' then all points on the [[orbit (dynamics)|orbit]] <math>\gamma_x</math> through ''x'' are periodic with the same prime period.

* Given a periodic point ''x'' then all points on the [[orbit (dynamics)|orbit]] <math>\gamma_x</math> through ''x'' are periodic with the same prime period.

* 给定周期点“x”，则在轨道gamma_x上的所有点都具有相同的素数周期

* 给定周期点“x”，则在轨道 <math>\gamma_x</math>上的所有点都具有相同的素数周期