---
tag: FHNDX22WMK
kind: concept
name: Periodic solution
msc: [34C25]
sources:
  - url: https://en.wikipedia.org/wiki/Periodic_function
---

## Summary

A periodic solution of a differential equation is a solution defined for all times that repeats itself after a fixed time $T > 0$.
For an autonomous system, a periodic solution that is not constant traces a [[periodic orbit|0AA97XA70K]] in phase space.
For an equation whose right-hand side is $T$-periodic in time, such as a periodically forced oscillator, one looks for solutions with the same period $T$.
Periodic solutions model rhythms such as oscillating circuits, heartbeats and planetary motion.

## Explanations

### L1 human

A periodic solution is a motion that comes back to exactly the same state after a fixed time and then repeats forever.
Its graph is a [[periodic function|0ATF6Q0633]] of time.
The function $x(t) = \cos t$ solves $x'' + x = 0$ and repeats after the time $2\pi$.
A pendulum that swings back and forth without friction moves along a periodic solution.
A damped pendulum does not, because its swings get smaller and smaller.