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第3行： − In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time.+ − − 在数学中，在迭代函数和动力系统的研究中，函数的周期点是系统在一定次数的函数迭代或一定时间后返回的点。 第13行：
第11行： − Given a mapping f from a set X into itself,+ − − 给定一个从集合 x 到自身的映射 f, − <math>f: X \to X,</math>+ − − x 到 x，数学 − a point x in X is called periodic point if there exists an n so that+ − − X 中的点 x 称为周期点，如果存在一个 n − <math>\ f_n(x) = x</math>+ − − [ math ] f _ n (x) = x − where <math>f_n</math> is the nth 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 f 的第 n 次迭代。满足上述条件的最小正整数 n 称为点 x 的素周期或最小周期。如果 x 中的每一个点都是周期点，周期 n 相同，那么 f 被称为周期点，周期 n (这不能和周期函数的概念混淆)。 第47行：
第35行： − 如果存在不同的 n 和 m+ − <math>f_n(x) = f_m(x)</math>+ − − [ math > f _ n (x) = f _ m (x)] − 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。 第66行：
第50行： − − 如果 f 是微分流形的微分同胚，因此定义了导数 f _ n ^ prime </math > ，那么周期点是双曲的，如果 − − − <math>|f_n^\prime|\ne 1,</math>+ − − 数学 | f _ n ^ prime | ne 1，</math > − − − − that it is attractive if − − 这是有吸引力的，如果 − − − <math>|f_n^\prime|< 1,</math>+ − − [ math > | f _ n ^ prime | < 1，</math > − − − − and it is repelling if − − 而且它会排斥如果 − − − <math>|f_n^\prime|> 1.</math>+ − − [ math > | f _ n ^ prime | > 1 − − − 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.+ − − 如果周期点或不动点的稳定流形维数为零，则称其为源; 如果不稳定流形维数为零，则称其为汇; 如果稳定流形和不稳定流形都有非零维数，则称其为鞍点或鞍点。 第122行：
第76行： − − A period-one point is called a fixed point. 第141行：
第93行： − < math > x _ { t + 1} = rx _ t (1-x _ t) ，qquad 0 leq x _ t leq 1，qquad 0 leq r leq 4 </math > + 第147行：
第99行： − 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 attracts all orbits). For r between 1 and 3, the value 0 is still periodic but is not attracting, while the value is an attracting periodic point of period 1. With r greater than 3 but less than 1 + , 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 . 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0到1之间的 r，0是唯一的周期点，周期为1(给出序列0,0,0，... ，吸引所有轨道)。对于1到3之间的 r，值0仍然是周期性的，但不是吸引点，而值是周期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，周期内出现一组周期点，其中任意一个正整数，对于 r 的某些值，这些重复序列中的一个是吸引的，而对于其他的序列，它们都不是吸引的(几乎所有的轨道都是混沌的)。+ 第157行：
第108行： − Given a real global dynamical system (R, X, Φ) with X the phase space and Φ the evolution function,+ − 给定一个具有 x 相空间和 φ 演化函数的实整体动力系统(r，x，φ) , 第165行：
第115行： − 数学: 数学{ r }乘 x 到 x+ 第171行：
第121行： − + − <math>\Phi(t, x) = x\,</math>+ − − 第183行：
第131行： − 这个性质的最小正 t 称为点 x 的素周期。+ 第190行：
第138行： + + 第209行：
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无编辑摘要
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.
在数学中，特别是在迭代函数和动力系统的研究领域中，函数的周期点是系统在一定次数的函数迭代或一定时间后返回的点。
Given a [[mapping (mathematics)|mapping]] ''f'' from a [[set (mathematics)|set]] ''X'' into itself,
Given a [[mapping (mathematics)|mapping]] ''f'' from a [[set (mathematics)|set]] ''X'' into itself,
给定一个从集合X到自身的映射f,
:<math>f: X \to X,</math>
:<math>f: X \to X,</math>
f：X→X，
a point ''x'' in ''X'' is called periodic point if there exists an ''n'' so that
a point ''x'' in ''X'' is called periodic point if there exists an ''n'' so that
X中的点x称为周期点，如果存在一个n使
:<math>\ f_n(x) = x</math>
:<math>\ f_n(x) = x</math>
f_n(x) = x
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f_n为f的第n次迭代。满足上述条件的最小正整数n称为点x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)。
If there exist distinct n and m such that
If there exist distinct n and m such that
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>
f_n(x) = f_m(x)
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。
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，
that it is ''[[Attractor|attractive]]'' if
that it is ''[[Attractor|attractive]]'' if
:<math>|f_n^\prime|< 1,</math>
:<math>|f_n^\prime|< 1,</math>
如果f_n^prime < 1，则称周期点f为吸引子，
and it is ''repelling'' if
and it is ''repelling'' if
:<math>|f_n^\prime|> 1.</math>
:<math>|f_n^\prime|> 1.</math>
如果f_n^prime > 1，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]].
如果周期点或不动点的稳定流形的维数为零，则称其为源点；如果不稳定流形的维数为零，则称其为汇点；如果稳定流形和不稳定流形都有非零维数，则称其为鞍点。
A period-one point is called a [[fixed point (mathematics)|fixed point]].
A period-one point is called a [[fixed point (mathematics)|fixed point]].
一个周期——一个点叫做不动点。
一个周期——一个点叫做不动点。
<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>
x_{t+1}=rx_t(1-x_t)，qquad 0 leq x _ t leq 1，qquad 0 leq r leq 4
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'' − 1) / ''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'' − 1) / ''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'' − 1) / ''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'' − 1) / ''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)。
Given a [[real global dynamical system]] ('''R''', ''X'', Φ) with ''X'' the [[Phase space (dynamical system)|phase space]] and Φ the [[evolution function]],
Given a [[real global dynamical system]] ('''R''', ''X'', Φ) with ''X'' the [[Phase space (dynamical system)|phase space]] and Φ the [[evolution function]],
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>
<math>\Phi: \mathbb{R} \times X \to X</math>
<math>\Phi: \mathbb{R} \times X \to X</math>
Phi:实数集{R}乘 X → X
a point ''x'' in ''X'' is called ''periodic'' with ''period'' ''t'' if there exists a ''t'' > 0 so that
a point ''x'' in ''X'' is called ''periodic'' with ''period'' ''t'' if there exists a ''t'' > 0 so that
a point x in X is called periodic with period t if there exists a t > 0 so that
a point x in X is called periodic with period t if there exists a t > 0 so that
如果存在 t > & thinsp; 0，则 x 中的点 x 称为周期 t
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>
Phi (t，x) = x，
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''.
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。
* 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)
* 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
* [[Periodic points of complex quadratic mappings]]
* [[Periodic points of complex quadratic mappings]]
* [[限制周期]]
* [[限量套]]
* [[稳定歧管|稳定集]]
* [[Sharkovsky定理]]
* [[固定点]]
* [[复杂二次映射的周期点]]