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Arkady Pikovsky and Michael Rosenblum (2007), Scholarpedia, 2(12):1459. doi:10.4249/scholarpedia.1459 revision #137076 [link to/cite this article]
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Curator: Michael Rosenblum

in-phase synchronization anti-phase synchronization
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synchronization with an arbitrary phase shift no synchrony
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In a classical context, synchronization (from Greek \(\mathbf \sigma \acute{\upsilon}\nu\ :\) syn = the same, common and \(\mathbf \chi\rho \acute{o} \nu o \varsigma\ :\) chronos = time) means adjustment of rhythms of self-sustained periodic oscillators due to their weak interaction; this adjustment can be described in terms of phase locking and frequency entrainment. The modern concept also covers such objects as rotators and chaotic systems; in the latter case one distinguishes between different forms of synchronization: complete/identical, phase, generalized, etc. Synchronization phenomena in large ensembles of coupled systems often manifest themselves as collective coherent regimes appearing via non-equilibrium phase transitions.



Figure 1: An original drawing of Huygens illustrating his experiments with pendulum clocks

The history of synchronization goes back to the 17th century when the famous Dutch scientist Christiaan Huygens reported on his observation of synchronization of two pendulum clocks which he had invented shortly before: "... It is quite worth noting that when we suspended two clocks so constructed from two hooks imbedded in the same wooden beam, the motions of each pendulum in opposite swings were so much in agreement that they never receded the least bit from each other and the sound of each was always heard simultaneously. Further, if this agreement was disturbed by some interference, it reestablished itself in a short time. For a long time I was amazed at this unexpected result, but after a careful examination finally found that the cause of this is due to the motion of the beam, even though this is hardly perceptible."

Another important observation of synchrony of organ pipes was described by Lord Rayleigh in his "Theory of Sound". Being, probably, the oldest scientifically studied nonlinear effect, synchronization was understood only in 1920-ies when E. V. Appleton and B. Van der Pol systematically -- theoretically and experimentally -- studied synchronization of triode electronic generators.

Phase of oscillations and its dynamics

Synchronization properties of periodic self-sustained oscillators are based on the existence of a special variable, phase \(\phi\) (see Phase Models). Mathematically, self-sustained oscillations correspond to a stable limit cycle in the state space of an autonomous continuous-time dynamical system. The phase \(\phi\) can be introduced as the variable parametrizing the motion along this cycle. One can always choose phase in a way that it grows uniformly in time, \(\tag{1} {d\phi\over dt}=\omega_0 \,, \)

where \(\omega_0\) is the natural frequency of oscillations. The phase is neutrally stable: it's perturbations neither grow nor decay, this corresponds to the invariance of solutions of autonomous dynamical systems with respect to time shifts. Contrary to this, the amplitude of oscillations has a definite stable value (for systems which can be described in terms of energy, this value is determined by a balance between energy influx and dissipation).

Due to the neutral stability of the phase, already a small perturbation (e.g. external periodic forcing or coupling to another system) can cause large deviations of the phase -- contrary to the amplitude, which is only slightly perturbed due to the transversal stability of the cycle. Thus, with a relatively small forcing one can adjust the phase and the frequency of oscillations without influencing the amplitude, this adjustment is the essence of the synchronization phenomenon.

Synchronization by external forcing

The simplest setup for observation of synchronization is when a periodic force is applied to an autonomous self-sustained oscillator. Examples of such a situation include radio-controlled clocks (relatively non-precise clocks are made perfect being adjusted by a periodic radio signal), cardiac pacemakers (heart beats are paced by a sequence of pulses from an electronic generator) and circadian rhythms (internal clocks of an organism are locked by the 24 h day-night cycle).

Phase approximation for weak forcing

Stability of amplitudes and neutral stability of phases suggests to describe the effect of a small forcing in the framework of the so-called phase approximation, where only the dynamics of the phase is followed (see Kuramoto (1984) for details). Considering the simplest case of a limit cycle oscillator, driven by a periodic force with frequency \( \omega\) and amplitude \( \varepsilon\ ,\) one can write the equation for the perturbed phase dynamics in the form \(\tag{2} {d\phi\over dt}=\omega_0 +\varepsilon Q(\phi,\omega t)\,, \)

where the \(2\pi\)-periodic in its both arguments coupling function \(Q\) depends on the form of the limit cycle and of the forcing. Expanded into a Fourier series, function \(Q\) contains fast oscillating and slowly varying, resonant terms. The latter can be gathered as \(q(\phi-\omega t)\ .\) Thus, performing averaging over fast oscillations, one obtains the following basic equation for the dynamics of the phase difference \(\tag{3} {d\Delta\phi\over dt}=-(\omega-\omega_0) +\varepsilon q(\Delta\phi)\,, \)

Figure 2: Left: Schematic view of the Arnold tongue - the region on the plane of parameters of the external force where synchronization occurs. Right: Dependence of the observed frequency on the external one (at a constant amplitude of forcing) exhibits a synchronization plateau.

where \(\Delta\phi=\phi-\omega t\) is the difference between the phases of the oscillations and of the forcing. Function \(q\) is \(2\pi\)-periodic, and in the simplest case \(q(\cdot)=\sin(\cdot)\) Eq. (3) is called the Adler equation. One can easily see that on the plane of parameters of the external forcing "frequency mismatch \((\omega-\omega_0)\) -- amplitude \(\varepsilon\)" there is a region \(\varepsilon q_{min}<\omega-\omega_0<\varepsilon q_{max}\) where Eq. (3) has a stable stationary solution that exactly corresponds to phase locking (the phase \(\phi\) just follows the phase of the forcing, i.e. \(\phi=\omega t +\mbox{constant}\)) and frequency entrainment (the observed frequency of the oscillator \(\Omega=\langle \dot{\phi}\rangle \) exactly coincides with the forcing frequency \(\omega\)). This region is called synchronization region, or Arnold tongue.

High order locking

Figure 3: (a) A sketch of Arnold tongues. (b) Devil's staircase for a fixed amplitude of the forcing (dashed line in (a))

If the frequencies of the oscillator and of the force fulfill \(n\omega\approx m\omega_0\ ,\) then the dynamics of the generalized phase difference \(\Delta\phi=m\phi-n\omega t\) is described by the equation similar to Eq.~(3), namely by \({d(\Delta\phi)/dt}=-(n\omega-m\omega_0) +\varepsilon \tilde q(\Delta\phi)\ .\) Synchronous regime then means perfect entrainment of the oscillator frequency at the rational multiple of the forcing frequency, \(\Omega={n\over m}\omega\ ,\) as well as phase locking \(m\phi=n\omega t +\mbox{constant}\ .\) The overall picture for a broad range of forcing frequencies is presented by a family of triangular-shaped synchronization regions touching the \(\omega\)-axis at the rationals of the natural frequency \({m\over n}\omega_0\) ( Figure 3(a)).

Moderate and strong forcing

This picture is preserved for moderate forcing, although now the shape of the tongues generally differs from being exactly triangular. The phase of oscillations does not exactly follow that of the forcing, instead the condition \(|m\phi-n\omega t|<\mbox{const}\) holds that means nevertheless the full entrainment of frequencies \(\Omega=\frac{n}{m}\omega\ .\) For a fixed amplitude of the forcing \(\varepsilon\) and varied driving frequency \(\omega \) one observes different phase locking intervals where the motion is periodic, whereas in between them it is quasiperiodic. The curve \(\Omega\) vs. \(\omega\) thus consists of horizontal plateaus at all possible rational frequency ratios; this fractal curve is called devil's staircase ( Figure 3(b)). A famous example of such a curve is the voltage--current plot for a Josephson junction in an ac electromagnetic field; here synchronization plateaus are called Shapiro steps. Note that a junction can be considered as a rotator (rotations are maintained by a dc current); this example demonstrates that synchronization properties of rotators are very close to those of oscillators.

For very strong forcing the dynamics is no more close to that in the phase approximation and many extra features, e.g. a transition to chaos, may occur.

Two coupled oscillators

Synchronization of two coupled self-sustained oscillators can be described in a way similar to the case of periodic forcing. A weak interaction affects only the phases of two oscillators \(\phi_1\) and \(\phi_2\ ,\) and Eq. (1) generalizes to \(\tag{4} {d\phi_1\over dt}=\omega_1+\varepsilon Q_1(\phi_1,\phi_2) \,,\qquad {d\phi_2\over dt}=\omega_2+\varepsilon Q_2(\phi_2,\phi_1) \,. \)

For the phase difference \(\Delta\phi=\phi_2-\phi_1\) one obtains after averaging an equation of the type of (3). This is performed by expanding \(Q_{1,2}\) in a double Fourier series, neglecting all fast oscillating terms and keeping only slow terms depending on the phase difference. (More on the relationship between the coupling function and the resulting phase model can be read in Phase Models and Phase Response Curve).

Synchronization now means that two nonidentical oscillators start to oscillate with the same frequency (or, more generally, with rationally related frequencies). This common frequency usually lies between \(\omega_1\) and \(\omega_2\ .\) It is worth mentioning that locking of the phases and frequencies implies no restrictions on the amplitudes, in fact the synchronized oscillators may have very different amplitudes and waveforms (e.g., oscillations may be relaxation (integrate-and-fire) or quasiharmonic).

Figure 4: Synchronization states for two coupled oscillators with nearly in-phase (left) and nearly anti-phase lockings (\(\alpha\) is some oscillatory observable, e.g. a deviation of a pendulum for coupled clocks in the Huygens experiment)

For coupled oscillators of similar waveforms one often observes (like Huygens observed deviations of pendula in his experiments) the phase shift between the synchronous motions. One distinguishes different forms of synchronous dynamics: in-phase, when the phase shift is nearly zero, anti-phase when it is close to \(\pi\ ,\) and out-of-phase for the other values of the difference.

An interesting phenomenon that may be observed for strong diffusive coupling is the oscillation death, when oscillations extinct in both systems. This happens because the coupling tends to equalize the states of the system bringing extra dissipation in them.

Ensembles of coupled oscillators

Mutual synchronization can be observed also in large populations of oscillating systems, consisting of many hundreds or thousands of units. It appears then as a collective coherent mode, examples are a simultaneous flashing of fireflies sitting on trees and a rhythmic applause in a large audience. In some cases such a coherent mode is not desirable, an example is the lateral swaying of the Millennium Bridge in London that occurred on the opening day, which appeared due to emerging synchrony of pedestrians' steps.

Figure 5: Left: Below critical coupling the oscillators are not synchronized and their phases are uniformly distributed over interval \((0,2\pi)\ .\) Above critical coupling (right panel) at least a part of oscillators is entrained by a periodic mean field with amplitude \( M \) oscillating with some mean frequency.

The mutual synchronization in a large populations of oscillators can be treated as a nonequilibrium phase transition, the amplitude of the mean field (\( M \) in Figure 5) serving as an order parameter. The simplest model in the phase approximation is the Kuramoto model which is a direct generalization of Eqs. (4) to the case when each unit is coupled with all \(N\) oscillators in the population\[ \dot\phi_k=\omega_k+\frac{\varepsilon}{N}\sum_{j=1}^N\sin(\phi_j-\phi_k)\;,\qquad k=1,2,\dots N\;, \]

The Kuramoto model describes a situation where uncoupled limit cycle oscillators are almost identical with a narrow distribution of frequencies, and the coupling is weak (see Phase Models).

If the distribution of natural frequencies \(\omega_k\) is broad, small coupling cannot synchronize oscillators and they remain independent. At some critical coupling \(\varepsilon_c\ ,\) however, a transition to collective synchrony occurs (the transition is sharp in the thermodynamic limit of very large \(N\)), where a periodic global mode appears. With further increase of coupling, this mode entrains more and more oscillators. See Kuramoto (1984) for details and Acebron et al. (2005) for further references.

Effects of noise

Synchronization effects are quite robust against noise. The influence of a small noise on a synchronous state reduces typically to appearance of rather rare phase slips. In the case of periodically forced oscillations this means that synchrony is observed for a long time interval, which is interrupted by a slip event, where the forced system performs one cycle more (or one cycle less) compared to the forcing. This means that the natural noise-induced phase diffusion is highly suppressed. This is of course of advantage if one uses synchronization to increase stability of oscillator's frequency, e.g., to improve the quality of clocks.

Quite counterintuitive, common noise acting on noninteracting oscillating systems can synchronize them (see Coherence Resonance). This happens because the noise-induced dynamics of the phase is stable, and the phase follows, although implicitly, the pattern of the noisy forcing. If one and the same noise acts on two (or several) nearly identical systems, both phases follow the same pattern of noise and therefore are close to each other, although they remain irregular functions of time. In neurosciences one observes this synchronization by common noise as a reliable response of a neural oscillator (e.g., of a single neuron) on a repeatedly applied external pre-recorded noisy forcing.

Chaotic systems

In dissipative chaotic systems case one distinguishes between different forms of synchronization: complete, phase, master-slave, etc. (see Synchronization of Chaotic Oscillators)

Weak coupling: phase synchronization

Figure 6: Illustration of phase synchronization in two coupled Roessler systems. Both oscillators remain chaotic what can be seen from their phase portraits. The phases of oscillators are shifted but remain adjusted (middle panel) while the amplitudes are nearly uncorrelated.

Quite often one can find a projection of the strange attractor that looks like a smeared limit cycle; phase is then introduced as a variable that gains \(2\pi\) with each rotation. These rotations are nonuniform due to chaos, what can be modelled by an effective noise in phase dynamics. If this noise is small (i.e. the rotations are rather uniform), the mean frequency of the system can be entrained by a periodic forcing while the chaos is preserved. If two or more chaotic oscillators with different natural frequencies interact, their mean frequencies can be adjusted while the amplitudes remain chaotic and only weakly correlated, see Pikovsky et al. (2001) for details.

Strong coupling: complete synchronization

Another type of chaotic synchronization --- complete synchronization --- can be observed for identical chaotic systems of any type (maps, autonomous or driven time-continuous systems). In the simplest case of two diffusively coupled in all variables systems the dynamics is described by \(\tag{5} {d\vec{x}\over dt}=\vec{F}(\vec{x})+\varepsilon (\vec{y}-\vec{x}) \,,\qquad {d\vec{y}\over dt}=\vec{F}(\vec{y})+\varepsilon (\vec{x}-\vec{y}) \,, \)

where \(\varepsilon\) is the coupling parameter. The regime when \(\vec{x}(t)=\vec{y}(t)\) for all \(t\) is called complete synchronization; because in this state the diffusive coupling vanishes, the dynamics is the same as if the systems were uncoupled. Although such symmetric solution exists for all \(\varepsilon\ ,\) it is stable only if the coupling is sufficiently strong.

To find the critical value of the coupling one linearizes Eqs. (5) near the synchronized state and obtains for the mismatch \(\vec{v}(t)=\vec{y}(t)-\vec{x}(t)\) the linearized system \(\tag{6} {d\vec{v}\over dt}=J(t)\vec{v}-2\varepsilon \vec{v} \,, \)

where \(J(t)=\partial F / \partial x\) is the Jacobian at the chaotic solution \(\vec{x}(t)\ .\)

Figure 7: Above the synchronization threshold the variables of two coupled systems are identical and chaotic in time (left panel). Below threshold the identity is broken while some degree of correlation between \(x\) and \(y\) remains (right panel).

With the ansatz \(\vec{v}=e^{-2\varepsilon t}\vec{u}\) one can get rid of the last term on the r.h.s. of (6); the resulting equation coincides with the linearized equation for small perturbations of the solutions of an individual chaotic oscillator. Thus, \(\vec{u}\) grows proportionally to the maximal Lyapunov exponent \(\lambda\) of a single system, and the critical coupling is \(\varepsilon_c={\lambda\over 2}\ .\) Complete synchronization occurs if \(\varepsilon>\varepsilon_c\ ,\) i.e. when the divergence of trajectories of interacting systems due to chaos is suppressed by the diffusive coupling. For weak coupling \(\varepsilon<\varepsilon_c\ ,\) the states of two systems are different, \(\vec{x}(t)\neq \vec{y}(t)\ .\) Some other forms of synchronization in chaotic systems (e.g., generalized, master-slave) are similar to the complete one; in all these cases synchronization appears if the coupling is strong enough.

Recommended Reading

Huygens (Hugenii), Ch. (1673) Horologium Oscillatorium, Parisiis, France: Apud F. Muguet; English translation: (1986) The Pendulum Clock, Ames: Iowa State University Press

Kuramoto, Y. (1984) Chemical Oscillations, Waves and Turbulence, Berlin: Springer

Blekhman, I. I. (1988) Synchronization in Science and Technology, NY: ASME Press

Glass, L. (2001) Synchronization and rhythmic processes in physiology. Nature, 410:277--284

Pikovsky, A., Rosenblum, M., and Kurths, J. (2001) Synchronization. A Universal Concept in Nonlinear Sciences, Cambridge: Cambridge University Press

Strogatz, S. (2003) SYNC. The emerging science of spontaneous order. New York: Hyperion

Acebron, J. A. et al. (2005) The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77:137

Internal references

See Also

Arnold Tongues, Burst Synchronization, Coherence Resonance, Chain of Oscillators, Desynchronization, Fast Threshold Modulation, Gait Control, Generalized Synchronization, Isochron, Kuramoto Model, Malkin Theorem, Mean-Field Synchronization, Neuronal Synchronization, Periodic Orbit, Phase Model, Resonance, Phase Response Curve, Pulse Coupled Oscillators, Slowly Coupled Oscillators, Synchronization of Chaotic Oscillators, Synchrony Measures, Weakly Coupled Oscillators

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