Exploring Transition from Stability to Chaos through Random Matrices

Round 1

Reviewer 1 Report

Dear authors!

The paper is dedicated to establishing a new criterion of chaotic behavior based on Wishart-like matrices. The paper presents good theoretical background, but some remarks should be made.

1. First of all, and critical, I think a lack of comparison with some well-studied chaos criteria like those based on Hurst and Lyapunov exponent. A good work 

DOI:10.1016/j.physa.2017.08.159

can clarify how this comparison could be made.

2. There are some typos, e.g. "he" instead of "The" after equation (1). Please read out the text.

3. Please explain h in formula (4) and how they are computed before (or right after) this formula

4. Coefficients of the Wishart-like matrix for the Chirikov map should be explicitly shown. Otherwise, a reader is confused how to calculate them. Thus, the results become non-repeatable. Formulas given for the Ising model do not help to understand a transition to Chirikov map.

5. What is MP-law? Please give an explanation in the text.

6. Literature references are insufficient, 1/4 of all references are self-references, which is higher than recommended. Please add a brief review on recent results in this field and add some more studies on properties of Chirikov map and other simple maps. I recommend the following works: 

https://doi.org/10.1016/j.physa.2018.06.100

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

In this paper, the authors investigate the relationship between time series generated by simple map iterations that exhibit chaotic behavior and the spectral properties of Wishart-like matrices constructed from these series. In essence, the authors wants to find out whether chaotic behavior can be detected in these spectra, thus providing an alternative way to study chaotic phenomena

Therefore, I recommend publication of the manuscript in Dynamics, Chaotic Dynamics in Discrete Time Systems  as a research article. However, I would like the authors to address some points that might improve the readability of the article:

Major points

11       The study carried out by the authors should be generalized to other dynamical systems, such as the logistic map, the Henon map and also to continuous dynamical systems, such as the nonlinear Lorenz system.

22      The stability analysis performed by the authors must be supported by the analysis of the Furier spectrum or the Lyapunov exponent.

33       Another tool that could contribute to the understanding of the dynamics is the bifurcation diagram as a function of the K parameter,

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

The paper treats a niche topic, yet is it exceptionally well written and pleasant to read. I spottet a number of small mistakes that need to be corrected before getting into production:

- After equation (1): "he" -> "The"

- After equation (3): "must works" -> "must work"; "In section" -> delete

- Equation (5): "Pr(...)" I guess it should rather be "f(...)"

- Equation (6): missing d\lambda_2 d\lambda_3 ... d\lambda_N under the integration sign

- Caption of Figure 3: Fix "Poincaré" and a missing reference.

Comments for author File: Comments.pdf

Quality of English language is mostly fine.

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Dear authors! Thank you much for carefully correcting the manuscript following my humble suggestions. Now, I suppose, the paper is ready for publication.

Author Response

We are thrilled to have incorporated the changes in accordance with your invaluable suggestions.

With warm regards,

Roberto (on behalf of all authors)