Circle Map - Mode Locking

For small to intermediate values of K (that is, in the range of K = 0 to about K = 1), and certain values of Ω, the map exhibits a phenomenon called mode locking or phase locking. In a phase-locked region, the values advance essentially as a rational multiple of n, although they may do so chaotically on the small scale.

The limiting behavior in the mode-locked regions is given by the rotation number

which is also sometimes referred to as the map winding number.

The phase-locked regions, or Arnold tongues, are illustrated in black in the figure above. Each such V-shaped region touches down to a rational value in the limit of . The values of (K,Ω) in one of these regions will all result in a motion such that the rotation number . For example, all values of (K,Ω) in the large V-shaped region in the bottom-center of the figure correspond to a rotation number of . One reason the term "locking" is used is that the individual values can be perturbed by rather large random disturbances (up to the width of the tongue, for a given value of K), without disturbing the limiting rotation number. That is, the sequence stays "locked on" to the signal, despite the addition of significant noise to the series . This ability to "lock on" in the presence of noise is central to the utility of phase-locked loop electronic circuit.

There is a mode-locked region for every rational number . It is sometimes said that the circle map maps the rationals, a set of measure zero at K = 0, to a set of non-zero measure for . The largest tongues, ordered by size, occur at the Farey fractions. Fixing K and taking a cross-section through this image, so that ω is plotted as a function of Ω gives the "Devil's staircase", a shape that is generically similar to the Cantor function.

The circle map also exhibits subharmonic routes to chaos, that is, period doubling of the form 3,6,12,24,....