In
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 ...
, Arnold tongues (named after
Vladimir Arnold)
[ Section 12 in page 78 has a figure showing Arnold tongues.] are a pictorial phenomenon that occur when visualizing how the
rotation number 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 shape
A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type.
A plane shape or plane figure is constrained to lie ...
s 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 of musical instruments,
orbital resonance and
tidal locking of orbiting moons,
mode-locking 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 ...
and
phase-locked loops and other
electronic oscillators, as well as in
cardiac rhythm
The cardiac conduction system (CCS) (also called the electrical conduction system of the heart) transmits the signals generated by the sinoatrial node – the heart's pacemaker, to cause the heart muscle to contract, and pump blood through the b ...
s,
heart arrhythmias 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 sub ...
.
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 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 models, 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 pacemakers. 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
endomorphisms) 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
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation).
Given two positive numbers and , modulo (often abbreviated as ) is t ...
, 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