# 周期点

## 迭代函数

$\displaystyle{ f: X \to X, }$

$\displaystyle{ X }$中的点$\displaystyle{ x }$称为周期点，如果存在一个$\displaystyle{ n }$使

$\displaystyle{ \ f_n(x) = x }$

### 示例

$\displaystyle{ x_{t+1}=rx_t(1-x_t), \qquad 0 \leq x_t \leq 1, \qquad 0 \leq r \leq 4 }$

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).

• 对于介于0到1之间的$\displaystyle{ r }$，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$\displaystyle{ r }$，值0仍然是周期性的，但不是吸引子，而该值是周期1的周期吸引子;
• $\displaystyle{ r }$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$\displaystyle{ r }$的值上升到4时，会出现周期为正的一组周期点；
• 对于$\displaystyle{ r }$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)。

## 动力系统

$\displaystyle{ \Phi: \mathbb{R} \times X \to X }$

$\displaystyle{ \Phi(t, x) = x\, }$

### 性质

• 给定一个周期为$\displaystyle{ “p” }$的周期点$\displaystyle{ “x” }$，则对于$\displaystyle{ t∈R }$$\displaystyle{ \Phi(t,x) = \Phi(t+p,x) }$

• 给定周期点“x”，则在轨道 $\displaystyle{ \gamma_x }$上的所有点都具有相同的素数周期prime period