Time-Delay Synchronization and Anti-Synchronization of Variable-Order Fractional Discrete-Time Chen–Rossler Chaotic Systems Using Variable-Order Fractional Discrete-Time PID Control

Round 1

Reviewer 1 Report

A PID control issue for variable-order fractional discrete-time of Chen-Rossler chaotic systems was discussed in this paper such that the estimation error is stabilized. Through the synchronization and anti-synchronization between the Chen and Rossler chaotic systems, the control problem is investigated via the Lyapunov function to ensure the guarantee of converged energy. Although it is an interesting topic, the bad typesetting and the disordered symbols cause huge difficulty for understanding the contribution of this paper. Moreover, the correction is not easily judged. Thus, this paper cannot be recommended for acceptance. Some comments are provided as follows:

1. Some typos and missing symbols are furtherly checked and modified for increasing the readability.
2. Is any assumption to guarantee the existence of the quality (4)?
3. Some qualities are repeated that may be eliminated after the first time for defining the qualities.
4. The reason of choosing Rossler chaotic system as the slave system should be discussed.
5. Referring to [8], some differences such as the related signals, appears in designing the PID controller in this paper.
6. The quality of all figures should be improved such that the readers easily understand the responses.

Author Response

The organization and presentation of the article has been substantially improved, as well as the analytical results obtained, are justified and are summarized in a theorem, duly justified in the same article and the analytical results are illustrated by examples, based on the analytical results obtained and on the simulations that we carry out, our results on synchronization and anti-synchronization for discrete dynamical systems of variable fractional order with time delay are encouraging.

 

We have given answers to each of the questions from (1) to (6), the corrections have already been made, the article was organized as mentioned above, from what has just been written in the lines above, in particular, in In question (5), relative to reference [8], we use a PID, similar to the one used in that reference, this control is widely used in control theory and extended to discrete fractional order control systems.

Reviewer 2 Report

In this manuscript, the authors presented the synchronization and anti-synchronization between the fractional variable order in discrete time of Chen and Rossler for chaotic systems with time delay, to guarantee, that the error between systems approaches zero, the authors also use, the well-known fractional and discrete order PID control theory, as can be seen in the illustrations, the results are satisfactory and the authors guarantee synchronization and anti-synchronization, via, Lyapunov stability analysis for discrete fractional variable-order systems. Moreover, the authors use the variable-order fractional-discrete Lyapunov stability theory and the required condition is done to ensure the stability of error dynamics. Implementation of the proposed controller including this PID Control, in the synchronization and the anti-synchronization of chaotic systems are demonstrated. The authors illustrate the simulation with examples to demonstrate that the proposed scheme, with the variable-order discrete-time controller, is satisfactory and efficient using this method. As far as I known, both the proposed scheme and the simulation are correct. However, the overall quality of the English language needs to be carefully edited by someone with expertise in technical English editing paying particular attention to English grammar and sentence structure. Once the above concerns are fully addressed, the paper could be accepted for publication in Mathematics.

Author Response

The organization and presentation of the article has been substantially improved, as well as the analytical results obtained, are justified and are summarized in a theorem, duly justified in the same article and the analytical results are illustrated by examples, based on the analytical results obtained and on the simulations that we carry out, our results on synchronization and antisynchronization for discrete dynamical systems of variable fractional order with time delay are encouraging.

 

Reviewer 3 Report

The article Time-Delay Synchronization and anti-Synchronization of Variable-Order Fractional Discrete-Time of Chen-Rossler Chaotic Systems Using Variable-Order Fractional-Discrete Time PID Control is an original. As the Authors wrote – we investigate the synchronization and anti-synchronization between the fractional variable order in discrete time of Chen and Rossler for chaotic systems with time delay, to guarantee, that the error between systems approaches zero, we also use, the well-known fractional and discrete order PID control theory, as can be seen in the illustrations, the results are satisfactory and we guarantee synchronization and anti-synchronization, via, Lyapunov stability analysis for discrete fractional variable order systems. The Authors present a solution to this problem and illustrate the simulation.

In my review, I refer to the formal page of the article and mainly editorial side. I do not engage in polemics and discussions on the mathematical considerations because it determines the originality of the issue in the article. I accept the above with appreciation.

The following observation require to draw attention and response from the Authors:

1. According to the reviewer, the title Time-Delay Synchronization and anti-Synchronization of Variable-Order Fractional Discrete-Time of Chen-Rossler Chaotics Systems Using Variable-Order Fractional-Discrete Time PID Control should be corrected with the word Chaotics and changed to the word Chaotic (Title).
2. I would like to draw your attention to the lack of the cited content, which the Authors refer to with reference to publications [1], [2] and [3]. It seems necessary to present the essence of the citations of these items. Referring only to the position itself leads to the notion of its lack of content (Introduction Section).
3. The introduction section does not include the composition and content of individual sections of the article, which is usually the case in the last sentences of this section. This description of the content of the article is used in MDPI publications. (Introduction Section).
4. Sections 6 and 7 are too short in their content. According to the reviewer, the statements contained in these sections should be placed directly after the solutions that relate to them (Sections 6 and 7).
5. Conclusions are a description of the results of the conducted research and should be their precise description. Referring to the formulas contained in the text of the article makes it difficult to understand the content of the conclusions. The reviewer proposes to minimize these references to the formulas (Conclusions).

In my review, I do not include more comments other than those presented in the review form. The formal part of the article after a small correction will be fully compliant with the requirements of the MDPI. In the article, the Authors have used 9 items of references, in large numbers from recent years. The text of the article is clear. The overall impression of the article is good. I accept the above with appreciation.

Author Response

The organization and presentation of the article has been substantially improved, as well as the analytical results obtained, are justified and are summarized in a theorem, duly justified in the same article and the analytical results are illustrated by examples, based on the analytical results obtained and on the simulations that we carry out, our results on synchronization and antisynchronization for discrete dynamical systems of variable fractional order with time delay are encouraging.

 

Round 2

Reviewer 1 Report

None further comment for this manuscript.

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.