---
tag: WWZTW11664
kind: concept
name: Invariant set
msc: [34C45, 37C10, 37B25]
sources:
  - url: https://en.wikipedia.org/wiki/Invariant_(mathematics)
---

## Summary

An invariant set of a differential equation is a set of states that no solution can leave or enter.
A solution that starts in the set stays in it for all times.
Equilibria, [[periodic orbits|0AA97XA70K]], limit sets and stable manifolds are invariant sets.
A positively invariant set only needs to be kept for positive times.
Such sets are used to trap solutions and to prove stability and existence of periodic orbits.

## Explanations

### L1 human

Picture the state of a system moving along its [[trajectory|07TZ2Q4C1M]] in the phase plane.
A set is invariant if every trajectory that touches it lies entirely inside it.
For the rotation $x' = -y$, $y' = x$, every disc $x^2 + y^2 \le r^2$ is invariant, since each trajectory is a circle around the origin.
The half-plane $x \ge 0$ is not invariant, because the circles leave it.
A union of whole trajectories is always invariant.