In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly in
dynamical systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...
, Arnold tongues (named after
Vladimir Arnold
Vladimir Igorevich Arnold (alternative spelling Arnol'd, russian: link=no, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. While he is best known for the Kolmogorov– ...
)
[ Section 12 in page 78 has a figure showing Arnold tongues.] are a pictorial phenomenon that occur when visualizing how the
rotation number
In mathematics, the rotation number is an invariant of homeomorphisms of the circle.
History
It was first defined by Henri Poincaré in 1885, in relation to the precession of the perihelion of a planetary orbit. Poincaré later proved a theore ...
of a dynamical system, or other related
invariant property thereof, changes according to two or more of its parameters. The regions of constant rotation number have been observed, for some dynamical systems, to form
geometric shapes that resemble tongues, in which case they are called Arnold tongues.
Arnold tongues are observed in a large variety of natural phenomena that involve oscillating quantities, such as concentration of enzymes and substrates in biological processes and
cardiac electric waves. Sometimes the frequency of oscillation depends on, or is constrained (i.e., ''phase-locked'' or ''mode-locked'', in some contexts) based on some quantity, and it is often of interest to study this relation. For instance, the outset of a
tumor
A neoplasm () is a type of abnormal and excessive growth of tissue. The process that occurs to form or produce a neoplasm is called neoplasia. The growth of a neoplasm is uncoordinated with that of the normal surrounding tissue, and persists ...
triggers in the area a series of substance (mainly proteins) oscillations that interact with each other; simulations show that these interactions cause Arnold tongues to appear, that is, the frequency of some oscillations constrain the others, and this can be used to control tumor growth.
Other examples where Arnold tongues can be found include the
inharmonicity
In music, inharmonicity is the degree to which the frequencies of overtones (also known as partials or partial tones) depart from whole multiples of the fundamental frequency ( harmonic series).
Acoustically, a note perceived to have a singl ...
of musical instruments,
orbital resonance
In celestial mechanics, orbital resonance occurs when orbiting bodies exert regular, periodic gravitational influence on each other, usually because their orbital periods are related by a ratio of small integers. Most commonly, this relationsh ...
and
tidal locking
Tidal locking between a pair of co-orbiting astronomical bodies occurs when one of the objects reaches a state where there is no longer any net change in its rotation rate over the course of a complete orbit. In the case where a tidally locked b ...
of orbiting moons,
mode-locking
Mode locking is a technique in optics by which a laser can be made to produce pulses of light of extremely short duration, on the order of picoseconds (10−12 s) or femtoseconds (10−15 s). A laser operated in this way is sometimes r ...
in
fiber optics
An optical fiber, or optical fibre in Commonwealth English, is a flexible, transparent fiber made by drawing glass (silica) or plastic to a diameter slightly thicker than that of a human hair. Optical fibers are used most often as a means to ...
and
phase-locked loop
A phase-locked loop or phase lock loop (PLL) is a control system that generates an output signal whose phase is related to the phase of an input signal. There are several different types; the simplest is an electronic circuit consisting of a ...
s and other
electronic oscillator
An electronic oscillator is an electronic circuit that produces a periodic, oscillation, oscillating electronic signal, often a sine wave or a square wave or a triangle wave. Oscillation, Oscillators convert direct current (DC) from a power supp ...
s, as well as in
cardiac rhythms,
heart arrhythmia
Arrhythmias, also known as cardiac arrhythmias, heart arrhythmias, or dysrhythmias, are irregularities in the heartbeat, including when it is too fast or too slow. A resting heart rate that is too fast – above 100 beats per minute in adult ...
s and
cell cycle
The cell cycle, or cell-division cycle, is the series of events that take place in a cell that cause it to divide into two daughter cells. These events include the duplication of its DNA (DNA replication) and some of its organelles, and subs ...
.
One of the simplest physical models that exhibits mode-locking consists of two rotating disks connected by a weak spring. One disk is allowed to spin freely, and the other is driven by a motor. Mode locking occurs when the freely-spinning disk turns at a frequency that is a
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
multiple of that of the driven rotator.
The simplest mathematical model that exhibits mode-locking is the circle map, which attempts to capture the motion of the spinning disks at discrete time intervals.
Standard circle map
Arnold tongues appear most frequently when studying the interaction between
oscillators, particularly in the case where one oscillator ''drives'' another. That is, one oscillator depends on the other but not other way around, so they do not mutually influence each other as happens in
Kuramoto model
The Kuramoto model (or Kuramoto–Daido model), first proposed by , is a mathematical model used to describing synchronization. More specifically, it is a model for the behavior of a large set of coupled oscillators. Its formulation was motivated b ...
s, for example. This is a particular case of
driven oscillators, with a driving force that has a periodic behaviour. As a practical example,
heart cells (the external oscillator) produce periodic electric signals to stimulate heart contractions (the driven oscillator); here, it could be useful to determine the relation between the frequency of the oscillators, possibly to design better
artificial pacemaker
An artificial cardiac pacemaker (or artificial pacemaker, so as not to be confused with the natural cardiac pacemaker) or pacemaker is a medical device that generates electrical impulses delivered by electrodes to the chambers of the heart eith ...
s. The family of circle maps serves as a useful mathematical model for this biological phenomenon, as well as many others.
The family of circle maps are functions (or
endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
s) of the circle to itself. It is mathematically simpler to consider a point in the circle as being a point
in the real line that should be interpreted
modulo , representing the angle at which the point is located in the circle. When the modulo is taken with a value other than
, the result still represents an angle, but must be normalized so that the whole range