“周期点”的版本间的差异

2021年1月3日 (日) 23:24的版本

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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.

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]

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]

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

如果存在不同的n和m使

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

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

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]

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]

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

@@ 第14行： / 第14行： @@
 :<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
@@ 第22行： / 第20行： @@
 :<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]]).
@@ 第32行： / 第28行： @@
 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>
-f_n(x) = f_m(x)
 then ''x'' is called a '''preperiodic point'''. All periodic points are preperiodic.
@@ 第148行： / 第140行： @@
 * [[Limit cycle]]
+* [[限制周期]]
 * [[Limit set]]
+* [[限量套]]
 * [[Stable manifold|Stable set]]
+* [[稳定歧管|稳定集]]
 * [[Sharkovsky's theorem]]
+* [[Sharkovsky定理]]
 * [[Stationary point]]
+* [[固定点]]
 * [[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]]
 {{PlanetMath attribution|id=4516|title=hyperbolic fixed point}}