Arnold Tongue

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.

REFERENCES:

Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, pp. 130-131, 1990.