“周期点”的版本间的差异
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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. | 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。 | |
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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, | |
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:<math>f: X \to X,</math> | :<math>f: X \to X,</math> | ||
| − | + | f：X→X， | |
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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 | |
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:<math>\ f_n(x) = x</math> | :<math>\ f_n(x) = x</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]]). | 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)。 | |
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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) | |
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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。 | |
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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 | 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> | ||
| − | + | 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， | |
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that it is ''[[Attractor|attractive]]'' if | that it is ''[[Attractor|attractive]]'' if | ||
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:<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， | |
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and it is ''repelling'' if | and it is ''repelling'' if | ||
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:<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。 | |
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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]]. | 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。 | |
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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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一个周期——一个点叫做不动点。 | 一个周期——一个点叫做不动点。 | ||
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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> | <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 | |
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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'' − 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). | ||
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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)。 | |
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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]], | 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， | |
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:<math>\Phi: \mathbb{R} \times X \to X</math> | :<math>\Phi: \mathbb{R} \times X \to X</math> | ||
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<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 | ||
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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; | + | 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， | |
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| − | Phi (t，x) = 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''. | ||
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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。 | |
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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''' | * 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 | ||
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* [[Periodic points of complex quadratic mappings]] | * [[Periodic points of complex quadratic mappings]] | ||
| + | * [[限制周期]] | ||
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| + | * [[限量套]] | ||
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| + | * [[稳定歧管|稳定集]] | ||
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| + | * [[Sharkovsky定理]] | ||
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| + | * [[固定点]] | ||
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| + | * [[复杂二次映射的周期点]] | ||
2021年1月3日 (日) 23:21的版本
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 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，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。
Iterated functions
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]
f：X→X，
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]\displaystyle{ \ f_n(x) = x }[/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).
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]\displaystyle{ 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.
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
- [math]\displaystyle{ |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 attractive if
- [math]\displaystyle{ |f_n^\prime|\lt 1, }[/math]
如果f_n^prime < 1，则称周期点f为吸引子，
and it is repelling if
- [math]\displaystyle{ |f_n^\prime|\gt 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.
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。
Examples
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]
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 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).
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)。
Dynamical system
Given a real global dynamical system (R, X, Φ) with X the 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]\displaystyle{ \Phi: \mathbb{R} \times X \to X }[/math]
[math]\displaystyle{ \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
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]\displaystyle{ \Phi(t, x) = x\, }[/math]
Phi (t，x) = 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。
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
- 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 [math]\displaystyle{ \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
See also
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