Circle Map

The circle map is a one-dimensional map which maps a circle onto itself

where theta_(n+1) is computed mod 1 and K is a constant. Note that the circle map has two parameters: Omega and K. Omega can be interpreted as an externally applied frequency, and K as a strength of nonlinearity. The circle map exhibits very unexpected behavior as a function of parameters, as illustrated above.

It is related to the standard map

for I and theta computed mod 1. Writing theta_(n+1) as

gives the circle map with I_n=Omega and K=-K.

The one-dimensional Jacobian of the circle map is

so the circle map is not area-preserving.

The unperturbed circle map has the form

If Omega is rational, then it is known as the map map winding number, defined by

and implies a periodic trajectory, since theta_n will return to the same point (at most) every q map orbits. If Omega is irrational, then the motion is quasiperiodic. If K is nonzero, then the motion may be periodic in some finite region surrounding each rational Omega. This execution of periodic motion in response to an irrational forcing is known as mode locking.

If a plot is made of K vs. Omega with the regions of periodic mode-locked parameter space plotted around rational Omega values (map winding numbers), then the regions are seen to widen upward from 0 at K=0 to some finite width at K=1. The region surrounding each rational number is known as an Arnold tongue. At K=0, the Arnold tongues are an isolated set of measure zero. At K=1, they form a Cantor set of dimension d approx 0.08700. For K>1, the tongues overlap, and the circle map becomes noninvertible.

Let Omega_n be the parameter value of the circle map for a cycle with map winding number W_n=F_n/F_(n+1) passing with an angle theta=0, where F_n is a Fibonacci number. Then the parameter values Omega_n accumulate at the rate

(Feigenbaum et al. 1982).

See also

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References

Referenced on Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Circle Map." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CircleMap.html

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