Global and Local Behavior of the System of Piecewise Linear Difference Equations xn+1 = |xn| − yn − b and yn+1 = xn − |yn| + 1 Where b ≥ 4
Round 1
Reviewer 1 Report
First of all, I want to say that the work which has been performed by the authors is interesting and could be shared with other specialists via the Journal. The main result of the paper is an investigation of the global and local behavior of the special system of piecewise linear difference equations.
The manuscript contains the statement of the problem and the methods for its solution, which are presented clearly. The motivation of the work and contribution of the authors to the topic is indicated. Limitations of the method applicability are marked.
The paper can be accepted for publication in the present form.
Author Response
Thank you very much for the comments. However, due to the other reviewers that need us to improve the presentation of the paper, we then modified our writing by specific each case to the specific region so that the presentation does not involve several subcases and subsubcases. Please see file attached.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The paper is aimed at considering a very specific case of the system of piecewise linear difference equations which will be a significant contribution in this field. All results are new and develop the theory of difference equations. In the introduction authors thoroughly reviewed some specific cases studied before and similar to the current system. Mostly they mentioned cases studied by W. Tikjha in numerous articles. In Section 2 they considered the global behavior for $b=4$. Local behavior for $b>5$ is studied in Section 3.
The paper comprises a list of Theorems Lemmas: section 1: Lemma 1, Section 2: Lemma 2 - Lemma 9 and Theorem 1, Section 3: Lemma 10 - Lemma 19. It is rather unclear whether Theorem 1 is the main result of the paper or what is the motivation for specifying that statement as a theorem. The proof of theorem 1 is based on Lemmas 2-9, which are split into a big range of special cases. Undoubtedly, the authors have done a meticulous job by considering and reviewing all of them. However, this job didn't need specific mathematical tools, and all proofs lack originality, their proofs are very similar and based on elementary methods. As a result, the paper has been blown up to 41 pages. Admittedly, each case might be tremendously interesting in a frame of the general theory of such equations but my suggestion is that for this journal the article should be revised significantly. For example, cases, subcases, subsubcases, etc. need to be organized in a more friendly way. Additionally, the majority of cases are proved similarly, thus I strongly recommend developing general rules and methods which will be suitable and applicable to them. The readers are supposed to know the contribution to the theory of difference equations although it is a hard job to balance between details and clearness.
However, the advantages of this article are that all lemmas and cases have clear, distinct, and complete proofs, and the style is clear and understandable easily. It is also appreciated that the authors left some open questions for further research which is very fruitful for other researchers.
Author Response
Thank you very much for the valuable comments. Please see our responses in the file attached.
Author Response File: Author Response.pdf
Reviewer 3 Report
Minor corrections
Authors presents a mathematical model for dynamic systems
Comments
- The work has a new results and high mathematical tools
- I suggest a systematization of the cases as well as possible.
- Also I suggest that introduction in the annexes of a part of the material.
- Please improve the Conclusions section to emphasize more clearly the originality and novelty of the paper. It is not very clear from the contents of the paper
Comments for author File: Comments.pdf
Author Response
Thank you very much for the valuable comments. Please see our responses in the file attached.
Author Response File: Author Response.pdf
Reviewer 4 Report
The aim of this article is to study the system of piecewise linear difference equations x_{n+1}=|x_n|-y_n-b and y_{n+1}=x_n-|y_n|+1 where n \geq 0. A global behavior for b=4 shows that all solutions become the equilibrium point. For a large value of |x_0| and |y_0|, the authors prove that (i) If b=5, then it the solution becomes the equilibrium point and (ii) if b \geq 6, then the solution becomes the periodic solution of prime period 5.
It seems that the authors attempted a lot to make this paper and its content, but unfortunately the presentation of the paper is to much poor. From the title, to abstract and the the body of the paper. I suggest them to consult with an expert to rewrite it before submitting somewhere else.
Author Response
Thank you very much for the valuable comments. Please see our responses in the file attached.
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
The authors have made significant work by improving the paper but numerous subcases are still presented in some parts of the draft. Frankly speaking, the paper doesn't contain serious mathematical progress in the field of the theory of difference equations and based upon the high-school level calculations and might be fulfilled easily. Although it has an exhaustive list of cases, it is merely a matter of meticulous calculation. I suggest that the paper still lacks originality and uniqueness. The purposes and applications of the research still remain unclear.
Author Response
(Please see also the attached file)
Thank you for the valuable comments and thank you for giving us a chance to do a second revision.
Comments: The authors have made significant work by improving the paper but numerous subcases are still presented in some parts of the draft.
Response: At first, we would like to give as much details as possible so that the material is easy to understand. However, for the second revision, we collapse several cases again using the results from the previous lemmas that we have been proved. Please see Lemma 8 on pages 6-11 and Lemma 9 on pages 11-22.
Comments: Frankly speaking, the paper doesn't contain serious mathematical progress in the field of the theory of difference equations and based upon the high-school level calculations and might be fulfilled easily. Although it has an exhaustive list of cases, it is merely a matter of meticulous calculation. I suggest that the paper still lacks originality and uniqueness.
Response: We did try to emphasis again that the absolute value function is not differentiable. Then, to analyze this system of difference equations, one need to find an alternative way rather than the method based on differentiability. We also try to give that even though we use only a fundamental method, we can obtain some insight in the analysis of this type of system of difference equations in general and also obtain a contrast behavior rather than other $b \neq 4, 5$. See the Conclusion and discussion section on page 27.
Comments: The purposes and applications of the research still remain unclear.
Response: - The purposes of this paper is given on the Introduction section. See lines 64-70 on pages 2-3.
- Some applications as an insight of our fundamental method are also given on the Conclusion and discussion section. See lines 370-374 on page 27.
Author Response File: Author Response.pdf
Round 3
Reviewer 2 Report
The authors have made progress in improving the article. I recommend them to rewrite the whole paper and think how the length of the paper might be reduced significantly.
