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Date: 30-11-2021
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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 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.
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