周期点

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舒寒讨论 | 贡献2021年1月3日 (日) 23:21的版本
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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。

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


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