A Distributed-Order Fractional Hyperchaotic Detuned Laser Model: Dynamics, Multistability, and Dual Combination Synchronization

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In this article, the authors studied a distributed-order hyperchaotic detuned laser model using a modified Predictor-Corrector approach to obtain numerical solutions.

Please check the typos and language corrections throughout the manuscript. For example in Lemma 3.1 and 3.2 all fonts should be italics. “We noe” should be “We now” in line 301.

However, the motivation for employing a distributed-order formulation, as opposed to standard ordinary differentiation, is not clearly justified. Relevant references on the viscoelastic distributed-order (DO) operator and its application to laser models should be cited to strengthen the rationale.

The authors provided phase portraits to support their claims of multistability and coexistence of attractors for selected parameter sets and initial conditions. To further substantiate these findings, it is recommended to include basin of attraction plots.

A bifurcation diagram along with the corresponding Lyapunov exponent should be plotted to illustrate the dynamic behavior of the model more comprehensively.

The results should also be presented in a way that clearly distinguishes the behaviors under the order and distributed-order formulations in time series, particularly as σ approaches 1.

The analytical foundation for dual combination synchronization, especially concerning the coupling between four-order and two distributed-order response models, should be discussed in more detail.

Finally, the limitations of the current study and its implications for potential future experimental applications should be addressed.

Author Response

Please see our attached "Responses_fractalfract-3794874"

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

In this manuscript the authors propose the distributed-order hyper chaotic detuned laser model and investigate several dynamical features of it. Analytical results are supported by the numerical experiments. The results coming from the use of distributed-order fractional calculus are found similar to those of Caputo fractional derivative. 

The manuscript is in general clearly written. After an appropriate introduction, firstly analytical results are given, and the numerical treatment follows. A conclusion part wraps up the findings.  

In the light of the above considerations, in my opinion, the present manuscript has enough novelty to be published in Fractal and Fractional.

 

Author Response

Please see our reply to all 3 reviewers attached hereby.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Please find attached a PDF file with the review report.

Comments for author File: Comments.pdf

Author Response

Please see the attached Response to all 3 reviewers.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have addressed many of my earlier queries. However, resolving two important comments is still necessary before making a final decision.

Firstly, it seems that the authors are misinterpreting the concept of multistability. For example, any multistable system must exhibit different attractors (such as the coexistence of different periodic and chaotic attractors) depending on the initial conditions. The existence of multistability is confirmed through the basin of attraction. However, the material presented in the revised version does not demonstrate multistability. At least, the authors should provide phase portraits showing the coexistence of different periodic and chaotic attractors. Simply demonstrating the coexistence of distinct chaotic attractors is not related to multistability. Does the considered system exhibit any hidden attractors?

Second, plotting a bifurcation diagram is straightforward and essential for understanding the system's dynamics, yet the authors have designated it as future work.

Without addressing these two important points, this manuscript is not suitable for publication.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 3

Reviewer 1 Report

Comments and Suggestions for Authors

I leave the final decision regarding the manuscript to the editor.