Multi-State Synchronization of Chaotic Systems with Distributed Fractional Order Derivatives and Its Application in Secure Communications

Round 1

Reviewer 1 Report

In this work, the authors achieved multi-state synchronization of a fractional-order system with distributed order. Some good results were obtained but I have the following comments:

1)      The presentation style in this work should be edited including fixing the language and word duplications, and the literature review.

2)      Where are the chaotic states in this work? Figure 3 does not seem to show chaotic states, otherwise, the authors should calculate the corresponding Lyapunov exponents.

3)      I advise the authors to give a statement on the existence and uniqueness of the proposed system in the revised version.

4)      The references are incomplete; The authors should add some related new references on the following topics:

i)                    Chaotic attractors that exist only in the fractional-order case;

ii)                   Advanced Applications of Fractional Differential Operators to Science and Technology;

iii)                 The novel fractional-order systems and their recent potential applications; For example, applications to linear control;

iv)                 The stability problem of novel fractional-order system such as the fractional Routh-Hurwitz criteria;

v)                   The applications of Routh-Hurwitz to some fractional epidemic models like COVID-19;

vi)            The multi-stability in fractional order systems like Self-Excited and Hidden Chaotic Attractors;

vii)               Control and synchronization of fractional-order chaotic satellite systems;

viii)         Complex dynamics and control of novel physical models;

                  Viiii) The discretization process of fractional-order systems such as discretized fractional  model of quasi-periodic plasma perturbations.

Author Response

Reviewer 1 comments:

In this work, the authors achieved multi-state synchronization of a fractional-order system with distributed order. Some good results were obtained but I have the following comments:

1)      The presentation style in this work should be edited including fixing the language and word duplications, and the literature review.

Ans: The requested modifications were applied to the text.

2)      Where are the chaotic states in this work? Figure 3 does not seem to show chaotic states, otherwise, the authors should calculate the corresponding Lyapunov exponents.

Ans: The system is chaotic and the phase curve and Lyapunov exponents show this. They were added to Figure 3 in the paper.

3)      I advise the authors to give a statement on the existence and uniqueness of the proposed system in the revised version.

 Ans: In this article, the main goal is to develop a multi-mode synchronization method for systems with distributed fractional order derivatives. Duffing's chaotic system may have been used in various cases of synchronization, but it has not been used in the case where the derivative is of distributed fractional-order despite the unknown delay. Please refer to Discussion Section.

4)      The references are incomplete; The authors should add some related new references on the following topics:

1. i)    Chaotic attractors that exist only in the fractional-order case;

1. ii)   Advanced Applications of Fractional Differential Operators to Science and Technology;

iii) The novel fractional-order systems and their recent potential applications; For example, applications to linear control;

1. iv) The stability problem of novel fractional-order system such as the fractional Routh-Hurwitz criteria;

1. v)  The applications of Routh-Hurwitz to some fractional epidemic models like COVID-19;
2. vi) The multi-stability in fractional order systems like Self-Excited and Hidden Chaotic Attractors;

vii) Control and synchronization of fractional-order chaotic satellite systems;

viii) Complex dynamics and control of novel physical models;

      Viiii) The discretization process of fractional-order systems such as discretized fractional model of quasi-periodic plasma perturbations.

Ans: The important aim of this article is to investigate synchronization of chaotic systems with distributed fractional order derivatives. There may be various chaotic systems, which is not the main purpose of this article. The Routh-Hurwitz stability analysis method is used for linear systems (fractional order and integer-order). The dynamics discussed in this article is nonlinear. Also in this article, the Lyapunov method which can be used for both linear and non-linear systems (fractional and integer-order) was used for stability analysis.

Overall, according to the opinion of the reviewer, the following paragraph and references were added. Please refer to Introduction Section.

In [25], chaotic absorbers that exist only in the fractional-order state have been investigated. Self-excited chaotic attractors were also considered. The results were supported by the calculation of the set of attractors, bifurcation, and Lyapunov exponent spectrum. Advanced applications of fractional differential operators in science and technology were studied in [26]. A new fractional-order chaotic system was introduced in [27]. Lyapunov exponent, Lyapunov spectrum, and bifurcation diagrams were calculated and chaotic attractors were investigated. Then, its application in linear control was explained. A non-standard finite difference scheme for unknown modelling and synchronization of a new fractional-order chaotic system including quadratic features was studied in [28]. In [29], the stability of Routh-Hurwitz and periodic Gaussian attractors in a fractional order model was investigated along with the application of N in the Covid-19 epidemic. In [30], dynamic analysis and adaptive synchronization were performed using a new fuzzy adaptive sliding mode control method. In [31], the control and synchronization problem of fractional-order chaotic satellite systems was studied using adaptive control and feedback techniques. The new Routh-Hurwitz stability conditions were applied for arbitrary orders. Also, the local stability of arbitrary orders was checked. The conditions for approximating the periodic solution in this model were discussed with Hopf's bifurcation theory [32]. The characteristics of nonlinear reverberation such as the presence of chaos were proved with the help of bifurcation diagrams, Lyapunov expressions, and Marotto's theorem [33].

[25] Matouk, A. E. (2022). Chaotic attractors that exist only in fractional-order case.Journal of Advanced Research.

[26] Matouk, A. E. (Ed.). (2020). Advanced applications of fractional differential operators to science and technology.IGI Global.

[27] Matouk, A. E. (2021). A Novel Fractional-Order System: Chaos, Hyperchaos and‎ Applications to Linear Control. Journal of Applied and Computational Mechanics, 7(2), 701-714.

[28] Baleanu, D., Zibaei, S., Namjoo, M., &Jajarmi, A. (2021). A nonstandard finite difference scheme for the modeling and nonidentical synchronization of a novel fractional chaotic system.Advances in Difference Equations, 2021(1), 1-19.

[29] Hassan, T. S., Elabbasy, E. M., Matouk, A. E., Ramadan, R. A., Abdulrahman, A. T., &Odinaev, I. (2022). Routh–Hurwitz Stability and Quasiperiodic Attractors in a Fractional-Order Model for Awareness Programs: Applications to COVID-19 Pandemic. Discrete Dynamics in Nature and Society, 2022.

[30] Jahanshahi, H., Yousefpour, A., Munoz-Pacheco, J. M., Moroz, I., Wei, Z., & Castillo, O. (2020). A new multi-stable fractional-order four-dimensional system with self-excited and hidden chaotic attractors: Dynamic analysis and adaptive synchronization using a novel fuzzy adaptive sliding mode control method. Applied Soft Computing, 87, 105943.

[31] Kumar, S., Matouk, A. E., Chaudhary, H., & Kant, S. (2021). Control and synchronization of fractional‐order chaotic satellite systems using feedback and adaptive control techniques. International Journal of Adaptive Control and Signal Processing, 35(4), 484-497.

[32] Matouk, A. E., & Khan, I. (2020). Complex dynamics and control of a novel physical model using nonlocal fractional differential operator with singular kernel. Journal of Advanced Research, 24, 463-474.

[33] Djenina, N., Ouannas, A., Batiha, I. M., Grassi, G., Oussaeif, T. E., &Momani, S. (2022). A Novel Fractional-Order Discrete SIR Model for Predicting COVID-19 Behavior.Mathematics, 10(13), 2224.

Author Response File: Author Response.docx

Reviewer 2 Report

In this paper, the multi-state synchronization of distributed fractional chaotic systems in the presence of uncertainty, disturbance, time delays and unknown parameters is investigated. Parameter setting rules were obtained to converge the synchronization errors into zero with the presence of uncertainty.

Comments:

1-Add a comparison of the proposed method with other related algorithms.

2-Add more statistical criteria including PSNR, UACI, NPCR, histogram, and correlation analysis.

Author Response

Reviewer 2 comments:

In this paper, the multi-state synchronization of distributed fractional chaotic systems in the presence of uncertainty, disturbance, time delays and unknown parameters is investigated. Parameter setting rules were obtained to converge the synchronization errors into zero with the presence of uncertainty.

Comments:

1-Add a comparison of the proposed method with other related algorithms.

Ans: The comparison criteria for evaluating the synchronization method are generally as follows:

Existence of disturbance, uncertainty, unknown delay, unknown parameters and type of order in master and slave systems.

Table 1 compares our work with other studies on the subject of synchronization. Please refer to Discussion Section.

Table 1. Comparison of the proposed method with other related works

2-Add more statistical criteria including PSNR, UACI, NPCR, histogram, and correlation analysis.

Ans: PSNR, UACI, NPCR, histogram, and correlation analysis parameters are used to evaluate the quality of images as we did in reference [49]. In this article, secure communication is used for the signal. Usually, the performance evaluation criterion is signal of error tracking that used in reference [19]. So, we used this criterion in our research.

Author Response File: Author Response.docx

Round 2

Reviewer 2 Report

The figures are with very low quality. Make them in vector format.

Author Response

Dear Reviewer,

We improved the quality of the figures and attached a new version of the paper.

Thanks a lot.

Best regards

Dr.Alizadehsani

Author Response File: Author Response.docx