In mathematics, particularly in dynamical systems, an Arnold tongue is a phenomenon observed in bifurcation diagrams when the value of a certain parameter constrains the orbit of the dynamical system to a smaller region; in particular, Arnold tongues occur when these regions enlarge monotonically as the parameter increases, causing regions that are somewhat similar to tongues in the bifurcation diagram. 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 based on some quantity, and it is often of interest to study this relation. For instance, the outset of a tumor triggers in the area a series of substance 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 and phase-locked loops and other electronic oscillators, as well as in cardiac rhythms and heart arrhythmias. 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 were first investigated for a family of dynamical systems on the circle first defined by Andrey Kolmogorov. Kolmogorov proposed this family as a simplified model for driven mechanical rotors, and specifically, for a free-spinning wheel weakly coupled by a spring to a motor. The circle map also provides a simple model of the phase-locked loop in electronics, of mechanically or acoustically coupled musical instruments, and of cardiac tissue. The map exhibits certain regions of its parameters where it is locked to the driving frequency, commonly referred to as phase-locking or mode-locking. The circle map is given by iterating the map where is to be interpreted as polar angle such that its value lies between 0 and 1. 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.
Mode locking
For small to intermediate values of K, 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. which is also sometimes referred to as the map winding number. The phase-locked regions, or Arnold tongues, are illustrated in yellow in the figure above. Each such V-shaped region touches down to a rational value Ω = in the limit of K → 0. The values of in one of these regions will all result in a motion such that the rotation number ω = . For example, all values of 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, 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 at K = 0, to a set of non-zero measure for K ≠ 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. 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,....
The Chirikov standard map is related to the circle map, having similar recurrence relations, which may be written as 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 by means of the kicked rotorHamiltonian.
