In mathematics, particularly in dynamical systems, Arnold tongues (named after Vladimir Arnold) 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 shapes that resemble tongues, in which case they are called Arnold tongues.

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.

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.

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.

Overview

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 the 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.

Example applications

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 outset of a tumor 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 and phase-locked loops and other electronic oscillators, as well as in cardiac rhythms, heart arrhythmias and cell cycle.

For applications to cardiac rhythms, see and . For applications to synchronisation of a resonant tunneling diode oscillators, see .

The circle map family

The circle map family is a family of dynamical systems in which Arnold tongues appear. They are much simpler than real biological dynamical systems, as well as many others, but despite their simplicity, they exhibit many of the same phenomena that appears in the more complicated dynamical systems. Consequently, they are usually used in studies of the Arnold tongue.

.

modulo means the system is in a state in which the driven oscillator is advanced in phase by 0.1 cycles, while the driving oscillator is advanced in phase by 0.5 cycles. Equivalently, the driven oscillator is retarded in phase by 0.9 cycles, while the driving oscillator is retarded in phase by 0.5 cycles. This is why we state that we are taking modulo 1.

because it is notationally convenient to operate with real numbers.

Now, if we take a periodic strobing snapshot, because the physical system does not change its driving effect strength over time.

is the circle map family.

Arnold's circle map

The particular circle map originally studied by Arnold, and which continues to prove useful even nowadays, is:

is called coupling strength.

, then the map has two fixed points, one stable and one unstable. The attractor basin of the stable fixed point is the whole circle, minus the unstable fixed point.

Chirikov standard map

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 rotor Hamiltonian.

Mode locking

The mode locking diagram

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.

Conversely, in the complement of the phase-locked region, the θn series never settles into a rational orbit, thus we call it the ergodic set.

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, or the asymptotic phase velocity.

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.

Properties of the mode locking diagram

. Consequently, it is standard to only plot the vertical half-strip bounded by the 3 lines.

There is a mode-locked region for every rational number .

For every reduced rational number. For example:

• The shape of the mode-locked region above 0/1 is a triangle with straight edges at the lowest order.

• The shape of the mode-locked region above 1/2 is a triangle with two curved edges, each being a parabola at the lowest order.

• .

In other words, the largest tongues, ordered by size, occur at the Farey fractions.

.

regions.

, we obtain the same asymptotic phase velocity. In particular, this means that the dynamics can mode-lock to at most one mode. Graphically, this means the Arnold tongues do not overlap in this region.

. Graphically, this means the Arnold tongues may overlap in this region. For the circle map, in this region, no more than two stable mode locking regions can overlap. That is, at most two Arnold tongues can overlap at any point in this region.

region, the circle map maps the rationals, a set of measure zero at K=0, to a set of non-zero measure for K≠0.

, and taking a cross-section through the diagram, we obtain a one-dimensional fractal with the same scaling behavior as the fat Cantor set. Consequently, its fractal dimension is 1.

Fixing K and taking a cross-section through the diagram, so that ω is plotted as a function of Ω. This gives a "Devil's staircase", a shape that is the same as the Cantor function in terms of its scaling behavior.

It is unknown if there is an upper bound to how many Arnold tongues can overlap for a generic map.

Routes to chaos

point produces a phase transition.

, the ergodic set becomes a fractal of dimension ~0.87.

, the bifurcation diagram around this paragraph is obtained, where we can observe periodic orbits, period-doubling bifurcations as well as possible chaotic behavior.

The circle map also exhibits subharmonic routes to chaos, that is, period doubling of the form 3, 6, 12, 24,....

Other properties of the Arnold's circle map

Consider the general family of circle endomorphisms:

:

We now proceed to listing some interesting properties of these circle endomorphisms.

P1. is:

.

P2.:

P3.. Since the sine oscillates at frequency 1, thus characterizing a phase-locking.

P4. or not.

P5 (translational symmetry)..

P6..

Deriving the circle map

at which y(t) reaches zero.

is zero, thus yielding:

we obtain the circle map discussed previously:

above representing the concentration of a certain substance.

is reset exactly.

See also

• Phase-locked loop

• Kuramoto model

• Sturmian word

Notes

References

• - Provides a brief review of basic facts in section 2.12.

• - Performs a detailed analysis of heart cardiac rhythms in the context of the circle map.

External links

• Circle map with interactive Java applet