Circle map

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Circle map showing mode-locked regions or Arnold tongues in black. Ω varies from 0 to 1 along the x-axis, and K varies from 0 at the bottom to 4π at the top.

Winding number as a function of Ω with K held constant at K = 1

Winding number, with black corresponding to 0, green to 1/2 and red to 1. Ω varies from 0 to 1 along the x-axis, and K varies from 0 at the bottom to 2π at the top.

Bifurcation diagram for Ω held fixed at 1/3, and K running from 0 at bottom to 4π at top. Black regions correspond to Arnold tongues.

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.

1 Definition
2 Mode locking
3 References
4 External links

[edit] Definition

The circle map is given by iterating the map

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

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.

[edit] 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 $θ n$ 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

$\omega=\lim_{n\to\infty} \frac{\theta_n}{n}.$

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 $Ω = p / q$ in the limit of $K\to 0$ . The values of (K,Ω) in one of these regions will all result in a motion such that the winding number $ω = p / q$ . 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 $ω = 1 / 2$ . One reason the term "locking" is used is that the individual values $θ n$ 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 $θ n$ . 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 $p / q$ . 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 $K\neq 0$ . 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,....

[edit] References

Eric W. Weisstein, Circle Map at MathWorld.
Robert Gilmore and Marc Lefranc, The Topology of Chaos, Alice in Stretch and Squeezeland, (2002) Wiley Interscience ISBN 0-471-40816-6 (Provides a brief review of basic facts in section 2.12).
Leon Glass, Micheal R. Guevara, Alvin Shrier, Rafael Perez, "Bifurcation and Chaos in a Periodically Stimulated Cardiac Oscillator", Physica 7D (1983) pp 89-101. Bound as Order in Chaos, Proceedings of the International Conference on Order and Chaos held at the Center for Nonlinear Studies, Los Alamos, New Mexico 87545,USA 24-28 May 1982, Eds. David Campbell, Harvey Rose; North-Holland Amsterdam ISBN 0-444-86727-9. (Performs a detailed analysis of heart cardiac rhythms in the context of the circle map.)
Mark McGuinness and Young Hong, Arnold tongues in human cardiorespiratory system, Chaos, March 2004, 14, 1.