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 map 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 
(Rasband 1990, p. 131). In general, an Arnold tongue is defined as a resonance
zone emanating out from rational numbers in a
two-dimensional parameter space of variables.
