# Talk:Synchronization

Comments of the reviewer

The authors have revised the article according to my previous comments. Two issues, however, remain to be addressed.

First, following Eq. 4, I think it is helpful of the authors show how the functions Q_1 and Q_2 (other authors call them "Gamma" or "H") are derived from the interaction function and the phase-response curve of the isolated oscillators. If the authors choose not to do it, at least they should give a reference to the article "Phase model" (or "Weakly coupled oscillators").

Second, I asked the authors to note that a "true" asynchronous state (zero synchrony measure) is obtained only in the limit N -> infinity, whereas for a finite N there is always some residual synchrony (the synchrony measure scales like 1/sqrt(N), but it is not zero). This result stems from the law of large numbers, and is valid for every model with noise, including the Kuramoto model (without noise, there may be "splay state" for which the synchrony measure decays faster with N, but this is an outlier). If the authors claim that "this system appears to be not correct", they should at least bring a reference (or present their own calculation) showing that the synchrony measure can be zero for finite N with even a small amount of noise. I think this cannot be done.

Response of the authors to the second comment:
We agree that a complex quantity introduced by Kuramoto as the order parameter in his model obeys finite-size fluctuations for finite number of oscillators N. However, we disagree that this quantity is "THE synchrony measure", moreover we neither introduce nor use this quantity in our article. For finite systems certainly other "synchrony measures" are more appropriate. For example, for N=2 one can measure the difference of the frequencies of two oscillators, at a certain coupling strength this diference vanish what means transition to synchrony. Below the transition the frequencies of the oscillators are different, what means that there is no synchrony (a quasiperiodic motion). This understanding of synchrony can be applied to any finite N. In a similar way one describes phase transitions in equilibrium statistical physics. One introduces an order parameter that in the thermodynamic limit vanish for temperatures beyond the critical one. For finite systems there are of course fluctuations of the order parameter (e.g. fluctuations of magnitization) due to finite-size effects. However we have never seen that such a state is called "with residual ferromagnetism". [For an example of a system where the Kuramoto order parameter vanishes for finite N, take the Kuramoto-Sakaguchi model with identical frequency and repulsive coupling].

Second Review:

Review Report for the Scholarpedia contribution on "Synchronization"

This is a very nice article and I have only some general remarks (that the authors may take into account or not) and some technical points (that should be corrected):

General remarks

-- In the beginning four cases of oscillating pendulum clocks are shown (in motion).

My understanding is that both cases in the upper row are examples for 'locking'. Presented in this way the reader might think that 'locking' (shown in the second row) is yet another case (different from the cases shown in the first row). I am afraid that this may lead to some confusion.

-- When listing different types of synchronization the authors mention: complete synchronization,

generalized synchronization, master-slave synchronization, phase synchronization, etc...

Here I want two make the following remarks: -- notions like 'master-slave' (addressing the way of coupling) and 'phase synch' (characterizing features of the resulting dynamics) should in my opinion not appear in the same list because they have different (logical) relations.

-- I don't like the name 'master-slave' (because I am against 'slavery' )

-- And in some sense I also consider the notion of 'complete' synchronization to be misleading. Of course, 'complete' makes sense in contrast to phase synchronization because all variables are synchronized (and not only the phases). But what about generalized synchronization? There also all variables are synchronized (just in a more sophisticated way) and in this sense generalized synchronization would be a special case of complete synchronization. If this is what the authors mean then it is o.k. for me. But probably they mean by 'complete synchronization' what other authors call 'identical synchronization' (and the latter is in my opinion more precise)

Technical points:

-- 'sysnchrony of organ pipes' should be 'synchrony of ...'

-- 'The phase is neutrally stable: it's perturbations neither grow no decay,'

I think it should be ' neither grow nor decay'

-- 'function Q contains fast oscillating and slow varying, resonant terms. '

I think it should be ' ...and slowly varying, resonant terms'

-- ' Devil's staircase '

I think it should be 'Devil's Staircase' or 'devil's staircase' (ot is Devil here the name of the devil ? ;-)

-- 'self-sustained oscillators can be described in a similar way to the case of periodic forcing'

I think it should be '.. in a way similar to the case ..' (but I am also not a native speaker)