Stability and Bifurcations in Banks and Small Enterprises—A Three-Dimensional Continuous-Time Dynamical System

Round 1

Reviewer 1 Report

Report

      

This is a review for the Manuscript Number: jrfm-2142218-" Stability and Bifurcations in a Bank and Small Enterprise Three-Dimensional Continuous-Time Dynamical System"

 

     In this paper the authors studied a three-dimensional continuous-time Lotka-Volterra dynamical system, that describes the role of government in interactions with banks and small enterprises. In Italy, the main objective of government economic intervention during the COVID-19 emergency was to maintain the proper operation of the bank-enterprise system. the authors reviewed the effectiveness of measures introduced in response to lockdowns for the COVID-19 pandemic in order to avoid a further credit crunch. By applying bifurcation theory to the system the authors produced evidence of the existence of periodic solutions Hopf and zero-Hopf bifurcating from a saddle-focus in a special region of the parameter space. A numerical analysis is performed.

Comments and Suggestions for Authors:

1.      The topic of this paper is interesting. Therefore, it can be published subject to the following Major modifications:

2.      The original papers should be much better presented.

3.      The English writing of the paper is required to be improved. Please check the manuscript carefully for typos and grammatical errors. The reviewer found some typos and grammatical errors  including punctuation, semicolon, and other structures, must be carefully reviewed.

4.      The authors have to add some real applications to compare with other published articles.

5.      The authors have to compare their results with the other literature and present the advantages.

6.      What is the novelty of the proposed work? Please list the motivation and   contribution in the introduction section.

7.      We suggest expanding the Introduction part in proportion to the importance of the topic.

8.      As this paper  “Stability and Bifurcations in a Bank and Small Enterprise Three-Dimensional Continuous-Time Dynamical System’’ can be interesting for the researchers in many field of applied mathematics. So, the authors can improve the introduction by the recent developments in this field.  Also, the reference section can strengthen by including some of the following papers:

I.          New Fractal Soliton Solutions and Sensitivity Visualization for Double-Chain DNA Model.

II.          Lyapunov stability and wave analysis of Covid-19 omicron variant of real data with fractional operator.

III.          Stability Of Nonlinear Fractional Delay Differential Equations,

Recommendation: After my careful reading, the topic discussed is novel, interesting, significant and important for the related areas. Scientific sound and the problems in the paper is solved in a successful way. Therefore, I recommend the publication of this paper in '' JRFM" after the modifications.

With many thanks and best regards.

Comments for author File: Comments.pdf

Author Response

Dear reviewer,

In the meantime, we would like to thank you for your invaluable suggestions and guidance. We followed your instructions. Now, the paper seems to be much improved, and certainly so from a formal standpoint.

Specifically, regarding your first three points: the paper’s linguistic, morphological and syntactic aspects were thoroughly reviewed and revised. The grammar and punctuation were corrected and the text now appears to be more fluent, clear and elegant.

Concerning your points 4 to 7: the introduction section was supplemented, pointing out and better specifying our model’s real applications, the results it offers, its innovative approach and its advantages over what has to date been already offered in the scientific literature on the subject.

Then, in response to your point 8, we strengthened the reference section, especially regarding the mathematical literature, with some additional citations, including all those that you so kindly suggested.

With much gratitude, we send you our kindest regards,

The authors: Marco Desogus and Beatrice Venturi

Reviewer 2 Report

In the present article, the authors discuss a three-dimensional continuous-time Lotka-Volterra dynamical system that describes the role of government in interactions with banks and small enterprises. The authors review the effectiveness of measures introduced in response to lockdowns for the COVID-19 pandemic in order to avoid a further credit crunch in Italy. By applying bifurcation theory to the system we produce evidence of the existence of periodic solutions Hopf and zero-Hopf bifurcating from a saddle-focus in a special region of the parameter space. The authors also provide numerical simulation.

The problem of study is good. Terminologies used in the paper are standard. But the authors have written paper in very rough style; specially, mathematical part is not well written. This paper needs major improvements. The following are some suggestions and comments here and rest of them inside manuscript given by yellow color (see attachment):

w  Writing style of paper must be improve; specially, mathematical part.

w  Align all equations numbering in the right side.

w  Everywhere write S as “(S)”.

w  Conclusion: It should be very short. It seems both Conclusion and Discussion. Separate Discussion write before Conclusion as new section.

w  References: Improve writing style of references.

w  Appendix: Improve writing style it.

Rest of comments and suggestions are mentioned in inside the manuscript by yellow colour.

Note: The highlight all changes by color in revision.

Comments for author File: Comments.pdf

Author Response

Dear reviewer,

First of all, we would like to thank you for your invaluable suggestions and guidance. We followed your instructions. Now, the paper seems to be much improved, and certainly so from a formal standpoint.

Specifically, regarding your first points: the paper’s linguistic, morphological and syntactic aspects were thoroughly reviewed and revised. The grammar and punctuation were corrected and the text now appears to be more fluent, clear and elegant. Special attention was also given to the style of the mathematical parts. Furthermore, the writing and presentation of the appendix and reference section have also been improved.

Additionally, we made all the textual corrections, according to the items you so kindly pointed out to us, for which we would like to thank you once again, for your patience, consideration and generous assistance. 

The numbers of the equations have all been aligned toward the right margin for clarity. In all cases where the system S was mentioned, we have corrected with the label (S).

Our conclusion was divided into two paragraphs, namely into a discussion section followed by the actual conclusion, per your indispensable advice.

With much gratitude, we send you our kindest regards,

The authors: Marco Desogus and Beatrice Venturi

Round 2

Reviewer 1 Report

Recommendation: After my careful reading, the topic discussed is novel, interesting, significant and important for the related areas. Scientific sound and the problems in the paper is solved in a successful way. Therefore, I recommend the publication of this paper in '' JRFM"

Reviewer 2 Report

The authors have implemented all suggestions and comments properly. We recommend this paper for possible publication in JRFM but needs little punctuation corrections in the paper as fullstop, comma, etc in mathematical expressions.