Poincaré return map

首次返回映射

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

Other names / variants: Poincaré map · Poincaré 映射

Translation evidence: Literature usage

Meaning in context

For a flow and a transverse section, the map taking a point to its next positive-time intersection with that section, on the domain where the first return exists.

流的一条横截面上，把有首次正时间返回的点送至其下一次交点的映射。

Example

For unit-speed motion around a circle, a section at one point has return time 11 and identity return map.

Usage and distinctions

A return time must exist; a return map is not generally the time-one map and depends on the section.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Gerald Teschl — Ordinary Differential Equations and Dynamical Systems§6.4 The Poincaré map · 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）§2.1，注2.1.1，p.12：首次返回映射（Poincaré 映射） · Checked 2026-10-14
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