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 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.