# Talk:Synchronization of extended chaotic systems

## Contents |

## Reviewer A (Dr. Haye Hinrichsen)

In my opinion the article addresses an interesting topic which establishes a connection between two different fields of Statistical Physics, namely, the dynamics of chaotic maps and the field of non-equilibrium phase transitions. The article is well-written, clearly structured and readable even for people who are not so familiar with the topic. I think that it is a very nice contribution which is fully suitable for publication in Scholarpedia.

I have a few minor remarks:

1) Instead of "adsorbing states" most authors speak of "absorbing states", but I think that it is not necessary to change it because there is no risk of a misunderstanding.

2) Eq. (2) is true only if the system starts with a "homogeneously active" state, which here corresponds to a globally desynchronized situation. I inserted the initial condition in the text.

3) Remark: The unified framework proposed by Munoz and Pastor-Satorras is a highly plausible conjecture but to my knowledge a rigorous proof is still missing.

4) Remark: In Eq. (3) the power law is truncated (the sum runs only up to L). According to my own experience with anomalous DP the truncation leads to scaling corrections which make the numerical analysis more difficult. We found that cleaner results are obtained if the sum runs (in principle) to infinity, winding many times around the periodic chain.

5) In Fig. 5 I would have drawn the red lines only in the range where they are actually valid, i.d. the left one (mean field limit) from zero to 0.5 and the right one (short-range diffusive limit) from sigma_c to the right boundary, but this is certainly a matter of taste.

## Response to Reviewer A

We thank the Referee for his careful reading of our manuscript.

Here are the answers to his main questions/remarks:

1) Instead of "adsorbing states" most authors speak of "absorbing states", but I think that it is not necessary to change it because there is no risk of a misunderstanding.

Answer: We agree

2) Eq. (2) is true only if the system starts with a "homogeneously active" state, which here corresponds to a globally desynchronized situation. I inserted the initial condition in the text.

Answer: Thank to the Referee for his update

3) Remark: The unified framework proposed by Munoz and Pastor-Satorras is a highly plausible conjecture but to my knowledge a rigorous proof is still missing.

Answer: As suggested by the Referee we have modified the text as follows

"Remarkably, Muñoz and Pastor-Satorras (2003) proposed a Langevin equation mimicking a Kardar-Parisi-Zhang equation (KPZ) with an attractive wall, able to encompass both the linear and nonlinear phenomenologies. This represents a highly plausible conjecture for an unified theoretical framework for the analysis of the ST in diffusively coupled chaotic extended systems."

4) Remark: In Eq. (3) the power law is truncated (the sum runs only up to L). According to my own experience with anomalous DP the truncation leads to scaling corrections which make the numerical analysis more difficult. We found that cleaner results are obtained if the sum runs (in principle) to infinity, winding many times around the periodic chain.

Answer: Indeed we used periodic boundary conditions allowing for winding many times around the periodic chain. We have clarified this issue by writing just after Eq.(3)

"where $\sigma$ controls the range of the interactions, $\eta$ is a suitable normalization factor. Periodic boundary conditions are assumed and the sum in (3) has to be understood as extended to infinity by winding around the periodic chain."

5) In Fig. 5 I would have drawn the red lines only in the range where they are actually valid, i.d. the left one (mean field limit) from zero to 0.5 and the right one (short-range diffusive limit) from sigma_c to the right boundary, but this is certainly a matter of taste.

Answer:

We decided to leave the Figure as it is for major clarity.

## Reviewer B (Dr. Miguel A. Muñoz)

This is a nice review article on the relation between the non-linear problem of synchronization of spatially extended systems and the field of non-equilibrium phase transitions. It is clear, well written, and the list of references is fair. It certainly deserves to be published in Scholarpedia.

Minor comments:

1- Maybe the concept of "coupled map lattices" should be more explicitly introduced (apart from the link to the Wikipedia) to make the paper more readable (and self-contained) to a broader audience.

2- The discussion about long-range effects appears twice (one short in the Introduction, one extended) and is, thus, a bit repetitive. I have the impression that the article puts relatively a bit too much emphasis on this specific aspect.

3- I think I understand what the authors mean by:
"induces a flow of information in bit space"
but I do not think that this explanation/sentence is sufficiently clear.

4- Together with Eq. 1, it might be nice to include in the discussion
the possibility of two identical replicas synchronized by the presence
of a common external noise.

5- It is always nice to have figures, but I am not sure the employed scaling plots are really useful to the reader.

## Response to Reviewer B

We thank the Referee for his positive Report and we have addressed below all the questions he raised.

1- We agree with the referee. As suggested we have slightly expanded the description of CMLs in the introduction which now reads:

*The critical properties of STs have been mainly studied for diffusively coupled map lattices (CML) (Kaneko, 1993), which are prototype models for systems exhibiting spatio-temporal chaos. Analogously to discrete-time maps (such as the logistic or the Henon map), which somehow reproduce continuous flows, CMLs can be considered as time and space discrete versions of partial differential equations.*

2- We have now reduced by half the part of the introduction related to long-range effects so to avoid putting too much emphasis on this aspect.

3- We agree with the referee. We have now modified the sentence as follows:

*
In deterministic chaos, sensitive dependence on initial conditions leads to an exponential amplification of infinitesimally small errors, thus inducing a flow of information in bit space from "insignificant" towards "significant" digits.*

Mentioning explicitly the exponential amplification of error should avoid confusion in the interpretation of the "bit space".

4- We have added also the model corresponding to two replicas driven by the same spatio-temporal noise.
The text has been modified as follows:

*
As a matter of fact, the first studies of chaotic STs have been performed by considering two replicas of one-dimensional CMLs driven by the same realization of spatio-temporal noise, namely:
\begin{eqnarray}
x_i(t+1) &=& F(\tilde{x_i}(t))+\sigma \xi_i(t)\quad, \nonumber\\
y_i(t+1) &=& F(\tilde{y_i}(t))+\sigma \xi_i(t) \quad ,
\tag{1}
\end{eqnarray}
where $\xi_i(t)$ is a stochastic variable zero average and unitary distributed in the interval $[-1;1]$, for $\sigma > \sigma_c$ the two replicas eventually synchronizes. The universality classes describing the observed STs are the same identified for model (1) (Baroni et al., 2001). Therefore in the following we will limit to the latter model.
*

5- We agree that it is not always easy to find proper and easy to read visualisations especially for more technical aspects in articles written for a broad audience. Frankly speaking we would like to keep also these more technical illustrations hoping that they will stimulate deeper investigations in the readers of this article.