周期点

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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。这里的迭代次数叫做周期。周期为1的周期点被称为不动点。

迭代函数

给定一个从集合[math]\displaystyle{ X }[/math]到自身的映射[math]\displaystyle{ f }[/math],

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

a point x in X is called periodic point if there exists an n so that

[math]\displaystyle{ X }[/math]中的点[math]\displaystyle{ x }[/math]称为周期点，如果存在一个[math]\displaystyle{ n }[/math]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[math]\displaystyle{ f_n }[/math]为[math]\displaystyle{ f }[/math]的第[math]\displaystyle{ n }[/math]次迭代。满足上述条件的最小正整数[math]\displaystyle{ n }[/math]称为点[math]\displaystyle{ x }[/math]的素数周期prime period或最小周期。如果[math]\displaystyle{ X }[/math]中的每一个点都是周期为[math]\displaystyle{ n }[/math]的周期点，那么[math]\displaystyle{ f }[/math]有周期性，周期为[math]\displaystyle{ n }[/math](这不能和周期函数的概念混淆)。

If there exist distinct n and m such that [math]\displaystyle{ f_n(x) = f_m(x) }[/math]

如果存在不同的[math]\displaystyle{ n }[/math]和[math]\displaystyle{ m }[/math]使:[math]\displaystyle{ f_n(x) = f_m(x) }[/math]

then x is called a preperiodic point. All periodic points are preperiodic.

那么[math]\displaystyle{ x }[/math]称为前周期点。所有周期点都是前周期点。

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

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

如果[math]\displaystyle{ x }[/math]是微分流形的微分同胚，则定义了导数[math]\displaystyle{ f_n^\prime }[/math]，如果:[math]\displaystyle{ |f_n^\prime|\ne 1, }[/math]，那么[math]\displaystyle{ f }[/math]是双曲周期点，

that it is attractive if :[math]\displaystyle{ |f_n^\prime|\lt 1, }[/math]

如果:[math]\displaystyle{ |f_n^\prime|\lt 1, }[/math]，则称周期点[math]\displaystyle{ f }[/math]为吸引子，

and it is repelling if:[math]\displaystyle{ |f_n^\prime|\gt 1. }[/math]

如果:[math]\displaystyle{ |f_n^\prime|\gt 1. }[/math]，则称周期点[math]\displaystyle{ f }[/math]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。

示例

A period-one point is called a fixed point.

周期为1的点也叫做不动点。

逻辑斯谛克映射函数表达式：

[math]\displaystyle{ 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).

参数[math]\displaystyle{ r }[/math]随着取值的不同，呈现周期性。对于介于0到1之间的[math]\displaystyle{ r }[/math]，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[math]\displaystyle{ r }[/math]，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[math]\displaystyle{ r }[/math]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[math]\displaystyle{ r }[/math]的值上升到4时，会出现周期为正的一组周期点；对于[math]\displaystyle{ r }[/math]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)。

动力系统

Given a real global dynamical system (R, X, Φ) with X the phase space and Φ the evolution function,

给定一个连续时间的动力系统[math]\displaystyle{ (R，X，Φ) }[/math]，其中[math]\displaystyle{ X }[/math]为相空间，[math]\displaystyle{ Φ }[/math]为状态转移函数，

[math]\displaystyle{ \Phi: \mathbb{R} \times X \to X }[/math]

a point x in X is called periodic with period t if there exists a t > 0 so that

如果存在 [math]\displaystyle{ t \gt =0 }[/math]，则[math]\displaystyle{ X }[/math]中的点[math]\displaystyle{ x }[/math]称为周期为[math]\displaystyle{ t }[/math]的周期。因此

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

The smallest positive t with this property is called prime period of the point x.

具有此性质的最小正[math]\displaystyle{ t }[/math]称为点[math]\displaystyle{ x }[/math]的素数周期。

性质

• 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[math]\displaystyle{ R }[/math]中所有[math]\displaystyle{ x }[/math]的[math]\displaystyle{ \Phi(t,x) = \Phi(t+p,x) }[/math]

• 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 [math]\displaystyle{ \gamma_x }[/math]上的所有点都具有相同的素数周期

请参见

• 极限环Limit cycle

• 极限集合Limit set

• 稳定集 Stable set

• Sharkovsky定理Sharkovsky's theorem

• 驻点 Stationary point

• 复杂二次映射的周期点Periodic points of complex quadratic mappings