Arnold tongue

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 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. 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.[6]

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 ${\displaystyle 2\pi }$, representing the angle at which the point is located in the circle. When the modulo is taken with a value other than ${\displaystyle 2\pi }$, the result still represents an angle, but must be normalized so that the whole range ${\displaystyle [0,2\pi ]}$ can be represented. With this in mind, the family of circle maps is given by:[7]

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

where ${\displaystyle \Omega }$ is the oscillator's "natural" frequency and ${\displaystyle g}$ is a periodic function that yields the influence caused by the external oscillator. Note that if ${\displaystyle g(\theta )=\theta }$ for all ${\displaystyle \theta }$ the particle simply walks around the circle at ${\displaystyle \Omega }$ units at a time; in particular, if ${\displaystyle \Omega }$ is irrational the map reduces to an irrational rotation.

The particular circle map originally studied by Arnold,[8] 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 ${\displaystyle 1}$. 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.

Depiction of the simple model where the circle map arises 'naturally'. The red line is ${\displaystyle y(t)}$ and is reset every time it reaches the sinusoidal black line.

Another way to view the circle map is as follows. Consider a function ${\displaystyle y(t)}$ that decreases linearly with slope ${\displaystyle a}$. Once it reaches zero, its value is reset to a certain oscillating value, described by a function ${\displaystyle z(t)=c+b\sin(2\pi t)}$. We are now interested in the sequence of times ${\displaystyle \{t_{n}\}}$ at which y(t) reaches zero.

This model tells us that at time ${\displaystyle t_{n-1}}$ it is valid that ${\displaystyle y(t_{n-1})=c+b\sin(2\pi t_{n-1})}$. From this point, ${\displaystyle y}$ will then decrease linearly until ${\displaystyle t_{n}}$, where the function ${\displaystyle y}$ is zero, thus yielding:

${\displaystyle {\begin{aligned}0&=y(t_{n-1})-a\cdot (t_{n}-t_{n-1})\\[0.5em]0&=\left[c+b\sin(2\pi t_{n-1})\right]-at_{n}+at_{n-1}\\[0.5em]t_{n}&={\frac {1}{a}}\left[c+b\sin(2\pi t_{n-1})\right]+t_{n-1}\\[0.5em]t_{n}&=t_{n-1}+{\frac {c}{a}}+{\frac {b}{a}}\sin(2\pi t_{n-1})\end{aligned}}}$

and by choosing ${\displaystyle \Omega =c/a}$ and ${\displaystyle K=2\pi b/a}$ we obtain the circle map discussed previously:

${\displaystyle t_{n}=t_{n-1}+\Omega +{\frac {K}{2\pi }}\sin(2\pi t_{n-1}).}$

Glass, L. (2001) argues that this simple model is applicable to some biological systems, such as regulation of substance concentration in cells or blood, with ${\displaystyle y(t)}$ above representing the concentration of a certain substance.

In this model, a phase-locking of ${\displaystyle N:M}$ would mean that ${\displaystyle y(t)}$ is reset exactly ${\displaystyle N}$ times every ${\displaystyle M}$ periods of the sinusoidal ${\displaystyle z(t)}$. The rotation number, in turn, would be the quotient ${\displaystyle N/M}$.[7]

Consider the general family of circle endomorphisms:

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

where, for the standard circle map, we have that ${\displaystyle g(\theta )=\theta +(K/2\pi )\sin(2\pi \theta )}$. Sometimes it will also be convenient to represent the circle map in terms of a mapping ${\displaystyle f(\theta )}$:

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

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

P1. ${\displaystyle f}$ is monotonically increasing for ${\displaystyle K<1}$, so for these values of ${\displaystyle K}$ the iterates ${\displaystyle \theta _{i}}$ only move forward in the circle, never backwards. To see this, note that the derivative of ${\displaystyle f}$ is:

${\displaystyle f'(\theta )=1+K\cos(2\pi \theta )}$

which is positive as long as ${\displaystyle K<1}$.

P2. When expanding the recurrence relation, one obtains a formula for ${\displaystyle \theta _{n}}$:

${\displaystyle \theta _{n}=\theta _{0}+n\Omega +{\frac {K}{2\pi }}\sum _{i=0}^{n-1}\sin(2\pi \theta _{i}).}$

P3. Suppose that ${\displaystyle \theta _{n}=\theta _{0}{\bmod {1}}}$, so they are periodic fixed points of period ${\displaystyle n}$. Since the sine oscillates at frequency 1 Hz, the number of oscillations of the sine per cycle of ${\displaystyle \theta _{i}}$ will be ${\displaystyle M=(\theta _{n}-\theta _{0})\cdot 1}$, thus characterizing a phase-locking of ${\displaystyle n:M}$.[7]

P4. For any ${\displaystyle p\in \mathbb {N} }$, it is true that ${\displaystyle f(\theta +p)=f(\theta )+p}$, which in turn means that ${\displaystyle f(\theta +p)=f(\theta ){\bmod {1}}}$. Because of this, for many purposes it does not matter if the iterates ${\displaystyle \theta _{i}}$ are taken modulus ${\displaystyle 1}$ or not.

P5 (translational symmetry).[9][7] Suppose that for a given ${\displaystyle \Omega }$ there is a ${\displaystyle n:M}$ phase-locking in the system. Then, for ${\displaystyle \Omega '=\Omega +p}$ with integer ${\displaystyle p}$, there would be a ${\displaystyle n:(M+np)}$ phase-locking. This also means that if ${\displaystyle \theta _{0},\dots ,\theta _{n}}$ is a periodic orbit for parameter ${\displaystyle \Omega }$, then it is also a periodic orbit for any ${\displaystyle \Omega '=\Omega +p,p\in \mathbb {N} }$.

To see this, note that the recurrence relation in property 2 would become:

${\displaystyle {\begin{aligned}\theta _{n}'&=\theta _{0}+n\Omega '+{\frac {K}{2\pi }}\sum _{i=0}^{n}\sin(2\pi \theta _{i})\\&=\theta _{0}+n(\Omega +p)+{\frac {K}{2\pi }}\sum _{i=0}^{n}\sin(2\pi \theta _{i})\\&=\theta _{n}+np,\end{aligned}}}$

so since ${\displaystyle \theta _{n}-\theta _{0}=M}$ due to the original phase-locking, now we would have ${\displaystyle \theta _{n}'-\theta _{0}=\theta _{n}+np-\theta _{0}=M+np}$.

P6. For ${\displaystyle K=0}$ there will be phase-locking whenever ${\displaystyle \Omega }$ is a rational. Moreover, let ${\displaystyle \Omega =p/q\in \mathbb {Q} }$, then the phase-locking is ${\displaystyle q:p}$.

Considering the recurrence relation in property 2, a rational ${\displaystyle \Omega =p/q}$ implies:

${\displaystyle \theta _{n}=\theta _{0}+n{\frac {p}{q}}}$

and equality modulus ${\displaystyle 1}$ will hold only when ${\displaystyle n(p/q)}$ is an integer, and the first ${\displaystyle n}$ that satisfies this is ${\displaystyle n=q}$. Consequently:

${\displaystyle \theta _{q}=\theta _{0}+p}$

${\displaystyle (\theta _{q}-\theta _{0})=p}$

meaning a ${\displaystyle q:p}$ phase-locking.

For irrational ${\displaystyle \Omega }$ (which leads to an irrational rotation), it would be necessary to have ${\displaystyle n\Omega =k}$ for integers ${\displaystyle n}$ and ${\displaystyle k}$, but then ${\displaystyle \Omega =k/n}$ and ${\displaystyle \Omega }$ is rational, which contradicts the initial hypothesis.

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

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.

${\displaystyle \omega =\lim _{n\to \infty }{\frac {\theta _{n}}{n}}.}$[10]

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 to the right. 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,....

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.

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[11]

• Sturmian word

• 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.

• Circle map with interactive Java applet