周期点

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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.

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,

给定一个从集合 x 到自身的映射 f,

[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

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 ] 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

如果存在不同的 n 和 m

[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]

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 > | 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 > | 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.

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]

< 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,

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]

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]

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.

周期点

目录

Iterated functions

Examples

Dynamical system

Properties

See also

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