“周期点”的版本间的差异

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(Moved page from wikipedia:en:Periodic orbit (history))
 
(Moved page from wikipedia:en:Periodic point (history))
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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。
  
#REDIRECT [[Periodic point]]
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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.
  
REDIRECT Periodic point
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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.
  
重定向周期点
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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。
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== Iterated functions ==
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Given a [[mapping (mathematics)|mapping]] ''f'' from a [[set (mathematics)|set]] ''X'' into itself,
 +
 
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Given a mapping f from a set X into itself,
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给定一个从集合 x 到自身的映射 f,
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:<math>f: X \to X,</math>
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<math>f: X \to X,</math>
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x 到 x,数学
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a point ''x'' in ''X'' is called periodic point if there exists an ''n'' so that
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a point x in X is called periodic point if there exists an n so that
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X 中的点 x 称为周期点,如果存在一个 n
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:<math>\ f_n(x) = x</math>
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<math>\ f_n(x) = x</math>
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[ math ] f _ n (x) = x
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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]]).
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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).
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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)。
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If there exist distinct ''n'' and ''m'' such that
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If there exist distinct n and m such that
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如果存在不同的 n 和 m
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:<math>f_n(x) = f_m(x)</math>
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<math>f_n(x) = f_m(x)</math>
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[ math > f _ n (x) = f _ m (x)]
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then ''x'' is called a '''preperiodic point'''. All periodic points are preperiodic.
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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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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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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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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
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 +
 
 +
 
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:<math>|f_n^\prime|\ne 1,</math>
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<math>|f_n^\prime|\ne 1,</math>
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数学 | f _ n ^ prime | ne 1,</math >
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that it is ''[[Attractor|attractive]]'' if
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that it is attractive if
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这是有吸引力的,如果
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:<math>|f_n^\prime|< 1,</math>
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<math>|f_n^\prime|< 1,</math>
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[ math > | f _ n ^ prime | < 1,</math >
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and it is ''repelling'' if
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and it is repelling if
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而且它会排斥如果
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:<math>|f_n^\prime|> 1.</math>
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<math>|f_n^\prime|> 1.</math>
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[ math > | f _ n ^ prime | > 1
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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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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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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。
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=== Examples ===
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A period-one point is called a [[fixed point (mathematics)|fixed point]].
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A period-one point is called a fixed point.
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一个周期——一个点叫做不动点。
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The [[logistic map]]
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The logistic map
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后勤地图
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:<math>x_{t+1}=rx_t(1-x_t), \qquad 0 \leq x_t \leq 1, \qquad 0 \leq r \leq 4</math>
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<math>x_{t+1}=rx_t(1-x_t), \qquad 0 \leq x_t \leq 1, \qquad 0 \leq r \leq 4</math>
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< math > x _ { t + 1} = rx _ t (1-x _ t) ,qquad 0 leq x _ t leq 1,qquad 0 leq r leq 4 </math >
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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).
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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).
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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)。
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== Dynamical system ==
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Given a [[real global dynamical system]] ('''R''', ''X'', Φ) with ''X'' the [[Phase space (dynamical system)|phase space]] and Φ the [[evolution function]],
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Given a real global dynamical system (R, X, Φ) with X the phase space and Φ the evolution function,
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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) ,
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:<math>\Phi: \mathbb{R} \times X \to X</math>
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<math>\Phi: \mathbb{R} \times X \to X</math>
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数学: 数学{ r }乘 x 到 x
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a point ''x'' in ''X'' is called ''periodic'' with ''period'' ''t'' if there exists a ''t'' >&thinsp;0 so that
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a point x in X is called periodic with period t if there exists a t >&thinsp;0 so that
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如果存在 t > & thinsp; 0,则 x 中的点 x 称为周期 t
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:<math>\Phi(t, x) = x\,</math>
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<math>\Phi(t, x) = x\,</math>
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Phi (t,x) = x,</math >
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The smallest positive ''t'' with this property is called ''prime period'' of the point ''x''.
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The smallest positive t with this property is called prime period of the point x.
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这个性质的最小正 t 称为点 x 的素周期。
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=== Properties ===
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* Given a periodic point ''x'' with period ''p'', then <math>\Phi(t,x) = \Phi(t+p,x)</math> for all ''t'' in '''R'''
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* 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.
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==See also==
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* [[Limit cycle]]
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* [[Limit set]]
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* [[Stable manifold|Stable set]]
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* [[Sharkovsky's theorem]]
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* [[Stationary point]]
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* [[Periodic points of complex quadratic mappings]]
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{{PlanetMath attribution|id=4516|title=hyperbolic fixed point}}
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[[Category:Limit sets]]
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Category:Limit sets
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类别: 极限集
  
 
<noinclude>
 
<noinclude>
  
<small>This page was moved from [[wikipedia:en:Periodic orbit]]. Its edit history can be viewed at [[周期点/edithistory]]</small></noinclude>
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<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]]

2020年10月25日 (日) 16:03的版本

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 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.

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 f from a 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]\displaystyle{ f: X \to X, }[/math]

[math]\displaystyle{ 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]\displaystyle{ \ f_n(x) = x }[/math]

[math]\displaystyle{ \ f_n(x) = x }[/math]

[ math ] f _ n (x) = x

where [math]\displaystyle{ 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).

where [math]\displaystyle{ 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]\displaystyle{ f_n(x) = f_m(x) }[/math]

[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ |f_n^\prime|\ne 1, }[/math]

[math]\displaystyle{ |f_n^\prime|\ne 1, }[/math]

数学 | f _ n ^ prime | ne 1,</math >


that it is attractive if

that it is attractive if

这是有吸引力的,如果


[math]\displaystyle{ |f_n^\prime|\lt 1, }[/math]

[math]\displaystyle{ |f_n^\prime|\lt 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]\displaystyle{ |f_n^\prime|\gt 1. }[/math]

[math]\displaystyle{ |f_n^\prime|\gt 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.

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]\displaystyle{ x_{t+1}=rx_t(1-x_t), \qquad 0 \leq x_t \leq 1, \qquad 0 \leq r \leq 4 }[/math]

[math]\displaystyle{ 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 attracts all orbits). For r between 1 and 3, the value 0 is still periodic but is not attracting, while the value (r − 1) / r is an attracting periodic point of period 1. With r greater than 3 but less than 1 + 模板:Radic, 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 (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 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]\displaystyle{ \Phi: \mathbb{R} \times X \to X }[/math]

[math]\displaystyle{ \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]\displaystyle{ \Phi(t, x) = x\, }[/math]

[math]\displaystyle{ \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]\displaystyle{ \Phi(t,x) = \Phi(t+p,x) }[/math] for all t in R
  • Given a periodic point x then all points on the orbit [math]\displaystyle{ \gamma_x }[/math] through x are periodic with the same prime period.


See also


模板:PlanetMath attribution

Category:Limit sets

类别: 极限集


This page was moved from wikipedia:en:Periodic point. Its edit history can be viewed at 周期点/edithistory