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第1行： − #REDIRECT [[Periodic point]]+ − REDIRECT Periodic point+ − 重定向周期点+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + − +
Moved page from wikipedia:en:Periodic point (history)
此词条暂由彩云小译翻译，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，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 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。
== Iterated functions ==
Given a [[mapping (mathematics)|mapping]] ''f'' from a [[set (mathematics)|set]] ''X'' into itself,
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>
<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
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</math>
[ 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 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)。
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
:<math>f_n(x) = f_m(x)</math>
<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.
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
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>
<math>|f_n^\prime|\ne 1,</math>
数学 | f _ n ^ prime | ne 1，</math >
that it is ''[[Attractor|attractive]]'' if
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>
[ math > | f _ n ^ prime | < 1，</math >
and it is ''repelling'' if
and it is repelling if
而且它会排斥如果
:<math>|f_n^\prime|> 1.</math>
<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]].
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。
=== Examples ===
A period-one point is called a [[fixed point (mathematics)|fixed point]].
A period-one point is called a fixed point.
一个周期——一个点叫做不动点。
The [[logistic map]]
The logistic map
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>
< 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'' − 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 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)。
== Dynamical system ==
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 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) ,
:<math>\Phi: \mathbb{R} \times X \to X</math>
<math>\Phi: \mathbb{R} \times X \to X</math>
数学: 数学{ r }乘 x 到 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
如果存在 t > & thinsp; 0，则 x 中的点 x 称为周期 t
:<math>\Phi(t, x) = x\,</math>
<math>\Phi(t, x) = x\,</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.
这个性质的最小正 t 称为点 x 的素周期。
=== Properties ===
* 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'' then all points on the [[orbit (dynamics)|orbit]] <math>\gamma_x</math> through ''x'' are periodic with the same prime period.
==See also==
* [[Limit cycle]]
* [[Limit set]]
* [[Stable manifold|Stable set]]
* [[Sharkovsky's theorem]]
* [[Stationary point]]
* [[Periodic points of complex quadratic mappings]]
{{PlanetMath attribution|id=4516|title=hyperbolic fixed point}}
[[Category:Limit sets]]
Category:Limit sets
类别: 极限集
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<noinclude>
<small>This page was moved from [[wikipedia:en:Periodic orbit]]. Its edit history can be viewed at [[周期点/edithistory]]</small></noinclude>
<small>This page was moved from [[wikipedia:en:Periodic point]]. Its edit history can be viewed at [[周期点/edithistory]]</small></noinclude>
[[Category:待整理页面]]
[[Category:待整理页面]]