Arnold Tongue

Consider the Circle Map. If $K$ is Nonzero, then the motion is periodic in some Finite region surrounding each rational $\Omega$. This execution of periodic motion in response to an irrational forcing is known as Mode Locking. If a plot is made of $K$ versus $\Omega$ with the regions of periodic Mode-Locked parameter space plotted around rational $\Omega$ values (the Winding Numbers), then the regions are seen to widen upward from 0 at $K = 0$ to some Finite width at $K = 1$. The region surrounding each Rational Number is known as an Arnold Tongue.

At $K = 0$, the Arnold tongues are an isolated set of Measure zero. At $K = 1$, they form a general Cantor Set of dimension $d \approx 0.8700$. In general, an Arnold tongue is defined as a resonance zone emanating out from Rational Numbers in a two-dimensional parameter space of variables.

See also Circle Map