---
tag: N2SVK69S53
kind: concept
name: Circle map
msc: ["37E10", "37E45"]
sources:
  - url: https://en.wikipedia.org/wiki/Arnold_tongue
---

## Summary

The circle map, or standard circle map, is the two-parameter family $\theta \mapsto \theta + \Omega - \frac{K}{2\pi} \sin(2\pi\theta)$ of maps of the [[circle|060W5CBZP6]] $\R/\Z$.
It models a rotation by $\Omega$ disturbed by a nonlinear kick of strength $K$.
It is the basic model of mode locking: its [[rotation number|DP4WWQBJJF]] locks to rational values on the [[Arnold tongues|WQF5ASXZTH]].
For $0 \le K < 1$ it is a family of circle [[diffeomorphisms|0KZZ70ZHXP]], and at $K = 1$ it has a critical point.

## Explanations

### L1 human

Picture a point moving around a circle in steps.
Without the kick ($K = 0$), each step is the same turn by $\Omega$.
The kick speeds the point up in some places and slows it down in others.
A pendulum clock pushed by a periodic force, or a heart driven by a pacemaker, can be modelled this way.
The question is whether the motion locks to the rhythm of the drive.