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Consider the Circle Map. If is Nonzero, then the motion is periodic in some Finite region
surrounding each rational
. This execution of periodic motion in response to an irrational forcing is known as
Mode Locking. If a plot is made of
versus
with the regions of periodic Mode-Locked parameter space plotted around rational
values (the Winding Numbers),
then the regions are seen to widen upward from 0 at
to some Finite width at
. The region surrounding
each Rational Number is known as an Arnold Tongue.
At , the Arnold tongues are an isolated set of Measure zero. At
, they form a general Cantor
Set of dimension
. 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