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In Mathematics , the circle map is a Chaotic Map showing a number of interesting chaotic behaviors. It was first proposed by Andrey Kolmogorov as a simplified model for driven mechanical rotors (specifically, a free-spinning wheel weakly coupled by a spring to a motor). The circle map equations also describe a simplified model of the Phase-locked Loop in Electronics . The circle map exhibits certain regions of its parameters where it is locked to the driving frequency (phase-locking or mode-locking in the language of electronic circuits); these are referred to as '''Arnold tongues''', after Vladimir Arnold . Among other applications, the circle map has been used to study the dynamical behaviour of a beating Heart . DEFINITION The circle map is given by iterating the map : It has two parameters, the coupling strength ''K'' and the driving phase Ω. As a model for phase-locked loops, Ω may be interpreted as a driving frequency. 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 : 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 winding number . For example, all values of (''K'',Ω) in the large ''V''-shaped region in the bottom-center of the figure correspond to a winding 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 winding 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 Fraction s. 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,.... REFERENCES
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