periodic point

周期点

Dynamical Systems · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

A point $x$ for which $T^rx=x$ for some integer $r\ge1$; its least period is the smallest such positive integer, and the orbit is called periodic.

存在正整数次迭代后返回自身的点；其中最小次数是最小周期，经过该点的轨道为周期轨道。

Example

Under circle rotation by $1/3$, every point has least period $3$.

Usage and distinctions

A periodic point is one point; its periodic orbit (周期轨道) is the set of points visited through iteration. A least-period-r orbit has r distinct periodic points. A fixed point has period one; distinguish a return time from the least period.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Gerald Teschl — Ordinary Differential Equations and Dynamical Systems§10.2 Fixed and periodic points, periodic-point definition · Checked 2026-10-14

Translation sources

1. 陈绥阳、褚蕾蕾 — 动力系统基础及其方法（科学出版社预览）§2.2，定义2.2.1，p.13：周期点、周期轨道、周期 · Checked 2026-10-14

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