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.

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

Rotation number, with black corresponding to 0, green to 1/2 and red to 1. This article is about the rotation number, which is sometimes called the map winding number or simply winding number. 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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, chaos theory describes the behavior of certain dynamical systems – that is systems whose state evolves with time – that may exhibit dynamics that 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). Andrey Nikolaevich Kolmogorov (Андрей Николаевич Колмогоров ( April 25, 1903 - October 20, 1987) was a Soviet The circle map equations also describe a simplified model of the phase-locked loop in electronics. A phase-locked loop or phase lock loop (PLL is a Control system that generates a signal that has a fixed relation to the phase of a "reference" Electronics refers to the flow of charge (moving Electrons through Nonmetal conductors (mainly Semiconductors, whereas electrical 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. Vladimir Igorevich Arnol'd or Arnold (Влади́мир И́горевич Арно́льд born June 12, 1937 in Odessa, Ukrainian SSR Among other applications, the circle map has been used to study the dynamical behaviour of a beating heart. The heart is a muscular organ in all Vertebrates responsible for pumping Blood through the Blood vessels by repeated rhythmic

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. For K = 0 and Ω irrational, the map reduces to an irrational rotation. In Mathematics, an irrational rotation is a map r: \rightarrow given by r(x = x + \theta \mod 1

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. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions

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. This article is about the rotation number, which is sometimes called the map winding number or simply 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 Ω = p / q 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 ω = p / q. This article is about the rotation number, which is sometimes called the map winding number or simply 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 rotation 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 rotation 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 . In Mathematics, a null set is a set that is negligible in some sense. The largest tongues, ordered by size, occur at the Farey fractions. In Mathematics, the Farey sequence of order n is the Sequence of completely reduced fractions between 0 and 1 which when In lowest terms 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. In Mathematics, a singular function is any function &fnof( x) defined on the interval ''b'' that has the following properties In Mathematics, the Cantor function, named after Georg Cantor, is an example of a function that is continuous, but not Absolutely continuous

The circle map also exhibits subharmonic routes to chaos, that is, period doubling of the form 3,6,12,24,. In Mathematics, Sharkovskii 's theorem is a result about Discrete dynamical systems One of the implications of the theorem is that if a continuous discrete . . .

References

• Eric W. Weisstein, Circle Map at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
• Philip L. Boyland : Bifurcations of circle maps: Arnol'd tongues, bistability and rotation intervals Comm. Math. Phys. Volume 106, Number 3 (1986), 353-381.
• 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. The heart is a muscular organ in all Vertebrates responsible for pumping Blood through the Blood vessels by repeated rhythmic )
• Mark McGuinness and Young Hong, Arnold tongues in human cardiorespiratory system, Chaos, March 2004, 14, 1.

External links

• Circle map with interactive Java applet

See also

• Process equation