---
tag: WQF5ASXZTH
kind: concept
name: Arnold tongue
msc: ["37E45", "37E10", "37G15"]
sources:
  - url: https://en.wikipedia.org/wiki/Arnold_tongue
same_as:
  wikidata: Q2040587
  wikipedia: Arnold tongue
---

## Summary

An Arnold tongue is a region in the parameter plane of the [[circle map|N2SVK69S53]] where the [[rotation number|DP4WWQBJJF]] takes one fixed rational value $p/q$.
Each tongue starts at the single point $\Omega = p/q$ when there is no nonlinearity, and widens as the nonlinearity $K$ grows.
Inside a tongue the system is mode locked: it has periodic orbits of period $q$.
Arnold tongues describe synchronization of oscillators with a periodic forcing.

## Explanations

### L1 human

Drive an oscillator, such as a swing or a pendulum clock, with a periodic push.
For some combinations of push strength and frequency, the oscillator locks into a rhythm, for example two swings for every three pushes.
Drawing these locked regions in the plane of frequency and strength gives thin wedge shapes that grow wider as the push gets stronger.
They look like tongues hanging down to the axis of zero strength, hence the name.