## Circle Map

A 1-D Map which maps a Circle onto itself

 (1)

where is computed mod 1. Note that the circle map has two parameters: and . can be interpreted as an externally applied frequency, and as a strength of nonlinearity. The 1-D Jacobian is
 (2)

so the circle map is not Area-Preserving. It is related to the Standard Map
 (3) (4)

for and computed mod 1. Writing as
 (5)

gives the circle map with = and . The unperturbed circle map has the form
 (6)

If is Rational, then it is known as the map Winding Number, defined by
 (7)

and implies a periodic trajectory, since will return to the same point (at most) every Orbits. If is Irrational, then the motion is quasiperiodic. If is Nonzero, then the motion may be 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 vs. with the regions of periodic Mode-Locked parameter space plotted around Rational values (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 Cantor Set of Dimension . For , the tongues overlap, and the circle map becomes noninvertible. The circle map has a Feigenbaum Constant

 (8)