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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 x in the real line that should be interpreted modulo 2 \pi, representing the angle at which the point is located in the circle. When the modulo is taken with a value other than 2 \pi, the result still represents an angle, but must be normalized so that the whole range , 2 \pi/math> can be represented. With this in mind, the family of circle maps is given by: :\theta_ = g(\theta_i) + \Omega where \Omega is the oscillator's "natural" frequency and g is a periodic function that yields the influence caused by the external oscillator. Note that if g(\theta) = \theta for all \theta the particle simply walks around the circle at \Omega units at a time; in particular, if \Omega is irrational the map reduces to an
irrational rotation In the mathematical theory of dynamical systems, an irrational rotation is a map : T_\theta : ,1\rightarrow ,1\quad T_\theta(x) \triangleq x + \theta \mod 1 where ''θ'' is an irrational number. Under the identification of a circle wit ...
. The particular circle map originally studied by Arnold, and which continues to prove useful even nowadays, is: :\theta_ = \theta_i + \Omega + \frac \sin(2 \pi \theta_i) where K is called coupling strength, and \theta_i should be interpreted modulo 1. This map displays very diverse behavior depending on the parameters K and \Omega; if we fix \Omega = 1/3 and vary K, the
bifurcation diagram In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the syst ...
around this paragraph is obtained, where we can observe
periodic orbits In mathematics, specifically in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. It can be understood as the subset of phase space covered by the trajectory of the dynami ...
, period-doubling bifurcations as well as possible
chaotic behavior Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have c ...
.


Deriving the circle map

Another way to view the circle map is as follows. Consider a function y(t) that decreases linearly with slope a. Once it reaches zero, its value is reset to a certain oscillating value, described by a function z(t) = c + b \sin(2 \pi t). We are now interested in the sequence of times \ at which y(t) reaches zero. This model tells us that at time t_ it is valid that y(t_) = c + b \sin(2 \pi t_). From this point, y will then decrease linearly until t_n, where the function y is zero, thus yielding: : \begin 0 &= y(t_) - a \cdot (t_ - t_) \\ .5em 0 &= \left c + b \sin(2 \pi t_) \right- a t_n + a t_ \\ .5em t_n &= \frac \left c + b \sin(2 \pi t_) \right+ t_ \\ .5em t_n &= t_ + \frac + \frac \sin(2 \pi t_) \end and by choosing \Omega = c/a and K = 2 \pi b/a we obtain the circle map discussed previously: : t_n = t_ + \Omega + \frac \sin(2 \pi t_). argues that this simple model is applicable to some biological systems, such as regulation of substance concentration in cells or blood, with y(t) above representing the concentration of a certain substance. In this model, a phase-locking of N:M would mean that y(t) is reset ''exactly'' N times every M periods of the sinusoidal z(t). The rotation number, in turn, would be the quotient N/M.


Properties

Consider the general family of circle endomorphisms: : \theta_ = g(\theta_i) + \Omega where, for the standard circle map, we have that g(\theta) = \theta + (K / 2 \pi) \sin(2 \pi \theta). Sometimes it will also be convenient to represent the circle map in terms of a mapping f(\theta): : \theta_ = f(\theta_i) = \theta_i + \Omega + \frac \sin(2 \pi \theta_i). We now proceed to listing some interesting properties of these circle endomorphisms. P1. f is monotonically increasing for K < 1, so for these values of K the iterates \theta_i only move forward in the circle, never backwards. To see this, note that the derivative of f is: : f'(\theta) = 1 + K \cos(2 \pi \theta) which is positive as long as K < 1. P2. When expanding the recurrence relation, one obtains a formula for \theta_n: : \theta_n = \theta_0 + n \Omega + \frac \sum_^n \sin(2 \pi \theta_i). P3. Suppose that \theta_n = \theta_0 \bmod 1, so they are periodic fixed points of period n. Since the sine oscillates at frequency 1 Hz, the number of oscillations of the sine per cycle of \theta_i will be M = (\theta_n - \theta_0) \cdot 1, thus characterizing a phase-locking of n : M. P4. For any p \in \mathbb, it is true that f(\theta + p) = f(\theta) + p, which in turn means that f(\theta + p) = f(\theta) \bmod. Because of this, for many purposes it does not matter if the iterates \theta_i are taken modulus 1 or not. P5 (translational symmetry). Suppose that for a given \Omega there is a n : M phase-locking in the system. Then, for \Omega' = \Omega + p with integer p, there would be a n : (M + np) phase-locking. This also means that if \theta_0, \dots, \theta_n is a periodic orbit for parameter \Omega, then it is also a periodic orbit for any \Omega' = \Omega + p, p \in \mathbb. P6. For K = 0 there will be phase-locking whenever \Omega is a rational. Moreover, let \Omega = p/q \in \mathbb, then the phase-locking is q : p.


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 ''θn'' 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 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 ...
. :\omega=\lim_\frac. which is also sometimes referred to as the map
winding number In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of turn ...
. The phase-locked regions, or Arnold tongues, are illustrated in yellow in the figure to the right. Each such V-shaped region touches down to a rational value Ω =  in the limit of ''K'' → 0. The values of (''K'',Ω) in one of these regions will all result in a motion such that the rotation number ''ω'' = . 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'' = . One reason the term "locking" is used is that the individual values ''θn'' 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 ''θn''. This ability to "lock on" in the presence of noise is central to the utility of the 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 In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null s ...
at ''K'' = 0, to a set of non-zero measure for ''K'' ≠ 0. The largest tongues, ordered by size, occur at the
Farey fraction In mathematics, the Farey sequence of order ''n'' is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, which when in lowest terms have denominators less than or equal to ''n'', arranged in or ...
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 In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. Th ...
. One can show that for ''K<1'', the circle map is a diffeomorphism, there exist only one stable solution. However as ''K>1'' this holds no longer, and one can find regions of two overlapping locking regions. For the circle map it can be shown that in this region, no more than two stable mode locking regions can overlap, but if there is any limit to the number of overlapping Arnold tongues for general synchronised systems is not known. The circle map also exhibits subharmonic routes to chaos, that is, period doubling of the form 3, 6, 12, 24,....


Chirikov standard map

The Chirikov standard map is related to the circle map, having similar recurrence relations, which may be written as :\begin \theta_ &= \theta_n + p_n + \sin(2\pi\theta_n)\\ p_ &= \theta_ - \theta_n \end with both iterates taken modulo 1. In essence, the standard map introduces a momentum ''pn'' which is allowed to dynamically vary, rather than being forced fixed, as it is in the circle map. The standard map is studied in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
by means of the
kicked rotor The kicked rotator, also spelled as kicked rotor, is a paradigmatic model for both Hamiltonian chaos (the study of chaos in Hamiltonian systems) and quantum chaos. It describes a free rotating stick (with moment of inertia I) in an inhomogeneou ...
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
.


Applications

Arnold tongues have been applied to the study of * Cardiac rhythms - see and * Synchronisation of a resonant tunneling diode oscillators


Gallery


See also

* Sturmian word


Notes


References

* * * - ''Provides a brief review of basic facts in section 2.12''. * - ''Performs a detailed analysis of
heart The heart is a muscular organ in most animals. This organ pumps blood through the blood vessels of the circulatory system. The pumped blood carries oxygen and nutrients to the body, while carrying metabolic waste such as carbon dioxide t ...
cardiac rhythms in the context of the circle map.'' *


External links


Circle map
with interactive Java applet {{DEFAULTSORT:Arnold Tongue Chaotic maps