Circle Map

A 1-D Map which maps a Circle onto itself

$$\theta_{n+1} = \theta_n+\Omega-{K\over 2\pi}\sin(2\pi\theta_n),$$ (1)

where $\theta_{n+1}$ is computed mod 1. 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 1-D Jacobian is

$${\partial\theta_{n+1}\over\partial\theta_n} = 1-K\cos(2\pi\theta_n),$$ (2)

so the circle map is not Area-Preserving. It is related to the Standard Map

$$I_{n+1} = I_n+{K\over2\pi}\sin(2\pi\theta_n)$$ (3)

$$\theta_{n+1} = \theta_n+I_{n+1},$$ (4)

for $I$ and $\theta$ computed mod 1. Writing $\theta_{n+1}$ as

$$\theta_{n+1} = \theta_n+I_n+{K\over 2\pi} \sin(2\pi\theta_n)$$ (5)

gives the circle map with $I_n$ = $\Omega$ and $K = -K$. The unperturbed circle map has the form

$$\theta_{n+1} =\theta_n+\Omega.$$ (6)

If $\Omega$ is Rational, then it is known as the map Winding Number, defined by

$$\Omega = W \equiv {p\over q},$$ (7)

and implies a periodic trajectory, since $\theta_n$ will return to the same point (at most) every $q$ 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 (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. The circle map has a Feigenbaum Constant

$$\delta\equiv\lim_{n\to\infty} {\theta_n-\theta_{n-1}\over\theta_{n+1}-\theta_n} = 2.833.$$ (8)

See also Arnold Tongue, Devil's Staircase, Mode Locking, Winding Number (Map)