Arnold tongue

In mathematics, particularly in dynamical systems, an Arnold tongue (named after Vladimir Arnold[1][2]) is a phenomenon observed in bifurcation diagrams when the value of a certain parameter ${\displaystyle \alpha }$ constrains the orbit of the dynamical system to a smaller region; in particular, Arnold tongues occur when these regions enlarge (or shrink) monotonically as the parameter increases, causing regions that are somewhat similar to tongues in the bifurcation diagram.[3]

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[4] 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 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.[3]

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

Bifurcation diagram for ${\displaystyle \Omega }$ held fixed at ${\displaystyle 1/3}$. ${\displaystyle K}$ goes from ${\displaystyle 0}$ at bottom to ${\displaystyle 4\pi }$ at top, and the orbits are shown in the interval ${\displaystyle [-0.5,0.5]}$ instead of ${\displaystyle [0,1]}$. Black regions correspond to Arnold tongues.

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.[5]

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 ${\displaystyle x}$ in the real line that should be interpreted modulo ${\displaystyle 2\pi }$ (or any other constant), with the result (a number in ${\displaystyle [0,2\pi )}$) representing the angle at which the point is located in the circle. With this in mind, the family of circle maps is given by:[6]

${\displaystyle \theta _{i+1}=g(\theta _{i})+\Omega }$

where ${\displaystyle \Omega }$ is the oscillator's "natural" frequency and ${\displaystyle g(\theta _{i})}$ is a periodic function that yields the influence caused by the external oscillator. Note that if ${\displaystyle g(\theta _{i})=\theta _{i}}$ the particle simply walks around the circle at ${\displaystyle \Omega }$ units at a time.

The particular circle map originally studied by Arnold,[1] and which continues to prove useful even nowadays, is:

${\displaystyle \theta _{i+1}=\theta _{i}+\Omega +{\frac {K}{2\pi }}\sin(2\pi \theta _{i})}$

where ${\displaystyle K}$ is called coupling strength, and ${\displaystyle \theta _{i}}$ should be interpreted modulo ${\displaystyle 1}$. For ${\displaystyle K=0}$ and ${\displaystyle \Omega }$ irrational, the map reduces to an irrational rotation. This map displays very diverse behavior depending on the parameters ${\displaystyle K}$ and ${\displaystyle \Omega }$; if we fix ${\displaystyle \Omega =1/3}$ and vary ${\displaystyle K}$, the bifurcation diagram around this paragraph is obtained, where we can observe periodic orbits, period-doubling bifurcations as well as possible chaotic behavior.

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 θ_n 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.

${\displaystyle \omega =\lim _{n\to \infty }{\frac {\theta _{n}}{n}}.}$[7]

which is also sometimes referred to as the map winding number.

Some of the Arnold tongues for the standard circle map, ε = K/2π

Rotation number as a function of Ω with K held constant at K = 1

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 = p/q 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 = p/q. 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 = 1/2. One reason the term "locking" is used is that the individual values θ_n 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 θ_n. 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 p/q. 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,....

Chirikov standard map

The Chirikov standard map is related to the circle map, having similar recurrence relations, which may be written as

${\displaystyle {\begin{aligned}\theta _{n+1}&=\theta _{n}+p_{n}+{\frac {K}{2\pi }}\sin(2\pi \theta _{n})\\p_{n+1}&=\theta _{n+1}-\theta _{n}\end{aligned}}}$

with both iterates taken modulo 1. In essence, the standard map introduces a momentum p_n 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.

Applications

Arnold tongues have been applied to the study of

• Cardiac rhythms - see Glass, L. et al. (1983) and McGuinness, M. et al. (2004)
• Synchronisation of a resonant tunneling diode oscillators[8]

Gallery

Circle map showing mode-locked regions or Arnold tongues in black. Ω varies from 0 to 1 along the x-axis, and K varies from 0 at the bottom to 4π at the top. The redder the color, the longer the recurrence time.

Rotation number, with black corresponding to 0, green to 1/2 and red to 1. Ω varies from 0 to 1 along the x-axis, and K varies from 0 at the bottom to 2π at the top.

See also

• Sturmian word

Notes

1. Arnol'd, V.I. (1961). "Small denominators. I. Mapping the circle onto itself". Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 25 (1): 21–86. Section 12 in page 78 has a figure showing Arnold tongues.
2. Translation to english of Arnold's paper: S. Adjan; V. I. Arnol'd; S. P. Demuškin; Ju. S. Gurevič; S. S. Kemhadze; N. I. Klimov; Ju. V. Linnik; A. V. Malyšev; P. S. Novikov; D. A. Suprunenko; V. A. Tartakovskiĭ; V. Tašbaev. Eleven Papers on Number Theory, Algebra and Functions of a Complex Variable. 46. American Mathematical Society Translations Series 2.
3. Jensen, M.H.; Krishna, S. (2012). "Inducing phase-locking and chaos in cellular oscillators by modulating the driving stimuli". FEBS Letters. 586 (11): 1664–1668. arXiv:1112.6093. doi:10.1016/j.febslet.2012.04.044. PMID 22673576. S2CID 2959093.
4. Gérard, C.; Goldbeter, A. (2012). "The cell cycle is a limit cycle". Mathematical Modelling of Natural Phenomena. 7 (6): 126–166. doi:10.1051/mmnp/20127607.
5. Glass, L. (2001). "Synchronization and rhythmic processes in physiology". Nature. 410 (6825): 277–284.
6. Glass, L.; Perez, R. (1982). "Fine structure of phase locking". Physical Review Letters. 48 (26): 1772.
7. Weisstein, Eric. "Map Winding Number". MathWorld. Retrieved 20 June 2016.
8. Romeira, B.; Figueiredo, J.M.; Ironside, C.N.; Slight, T. (2009). "Chaotic dynamics in resonant tunneling optoelectronic voltage controlled oscillators". IEEE Photonics Technology Letters. 21 (24): 1819–1821. Bibcode:2009IPTL...21.1819R. doi:10.1109/LPT.2009.2034129. S2CID 41327316.

References

• Weisstein, Eric W. "Circle Map". MathWorld.
• Boyland, P.L. (1986). "Bifurcations of circle maps: Arnol'd tongues, bistability and rotation intervals". Communications in Mathematical Physics. 106 (3): 353–381. Bibcode:1986CMaPh.106..353B. doi:10.1007/BF01207252. S2CID 121088353.
• Gilmore, R.; Lefranc, M. (2002). The Topology of Chaos: Alice in Stretch and Squeezeland. John Wiley & Sons. ISBN 0-471-40816--6. - Provides a brief review of basic facts in section 2.12.
• Glass, L.; Guevara, M.R.; Shrier, A.; Perez, R. (1983). "Bifurcation and chaos in a periodically stimulated cardiac oscillator". Physica D: Nonlinear Phenomena. 7 (1–3): 89–101. Bibcode:1983PhyD....7...89G. doi:10.1016/0167-2789(83)90119-7. - Performs a detailed analysis of heart cardiac rhythms in the context of the circle map.
• McGuinness, M.; Hong, Y.; Galletly, D.; Larsen, P. (2004). "Arnold tongues in human cardiorespiratory systems". Chaos. 14 (1): 1–6. Bibcode:2004Chaos..14....1M. doi:10.1063/1.1620990. PMID 15003038.

External links

• Circle map with interactive Java applet