Chaotic van der Pol Oscillator Control Algorithm Comparison

Round 1

Reviewer 1 Report

The paper discusses the control of dynamics of the damped van der Pol (VDPL) oscillator using a combined approach.  The paper fits well into the scope of the journal. However, I have several questions and remarks which can be found below.

1. The Authors should be careful while calling a general case of van der Pol oscillator chaotic. The conditions where van der Pol is truly chaotic should be clearly stated. It should be indicated that the Authors mean damped oscillator.

2. I recommend considering an output of another VDPL oscillator as an input signal for the system.

3. I recommend considering another chaotic noise case, e.g., noise from the Rossler\Lorenz oscillator.

4. van der Pol oscillator is a stiff system. Will the proposed methods work for larger values of M?

5. Explicit Runge-Kutta Methods are badly suited for solving stiff and chaotic systems. How does the choice of an ODE solver affect the simulation? Was this effect investigated during the experimental studies? How will various finite-difference models behave in this case?

6. To claim the system to be chaotic, the simple divergence of trajectories\sensitivity to initial conditions is not enough. There are at least three necessary conditions for chaos: sensitivity to the IC\parameters, trajectories density and topological mixing. Usually, to prove the presence of chaos the largest Lyapunov exponent is calculated. I highly recommend adding this type of analysis to prove that all considered cases of the van der Pol oscillator are truly chaotic and that they can stay in this mode during control procedures.

7. Please, clarify the role and purpose of Fig.1 in this manuscript.

I also recommend proofreading the manuscript to avoid at least such typos as "can der Pol oscillator" (Line 24)
Nevertheless, I like the study and believe it can be accepted after relatively moderate revisions.

Author Response

We thank the three peer reviewers for diligent efforts towards improving the manuscript’s description of the research. In accordance with Editorial guidance, in instance where it is impossible to address certain comments in the review reports, explanations are included in the in your appeal.

 

REVIEWER 1

1. The Authors should be careful while calling a general case of van der Pol oscillator chaotic. The conditions where van der Pol is truly chaotic should be clearly stated. It should be indicated that the Authors mean damped oscillator.

• Thank you for the wonderful suggestion. The chaotic nature due to initial conditions is emphasized in the introduction. Technically delving into the other necessary conditions for chaos was deemed unnecessary, following the cited examples of previous scholarship that this paper builds upon (such papers, such as Cooper & Heidlauf and particularly Zhai). To aid the readership, a revision was made to indicate that the Authors refer to a damped oscillator.

2. I recommend considering an output of another VDPL oscillator as an input signal for the system.

• This remains beyond the scope of this manuscript but makes a very interesting potential for sequels. Our group will add such to the list of future treatments.

3. I recommend considering another chaotic noise case, e.g., noise from the Rossler\Lorenz oscillator.

• The Lorenz system was investigated by colleagues in our group, and the manuscript is in peer review presently, while this companion research elaborated the same approaches applied to van der Pol. Taken together, we are elaborating the approach applied to all the chaotic systems we can consider (thus we can treat VDPL in the sequels as well.

4. van der Pol oscillator is a stiff system. Will the proposed methods work for larger values of M?

• System stiffness is an aspect the team is already investigating, and this work establishes a benchmark for comparative sequels specifically on the effects of system stiffness.

5. Explicit Runge-Kutta Methods are badly suited for solving stiff and chaotic systems. How does the choice of an ODE solver affect the simulation? Was this effect investigated during the experimental studies? How will various finite-difference models behave in this case?

• This prequel evaluation using Runge-Kutta will subsequently be compared to (at least) numerical differentiation, modified Rosenbrock integration, trapezoidal rule, and trapezoid rule with back differentiation. The comparison should discern the “stiffness” in application.

6. To claim the system to be chaotic, the simple divergence of trajectories\sensitivity to initial conditions is not enough. There are at least three necessary conditions for chaos: sensitivity to the IC\parameters, trajectories density and topological mixing. Usually, to prove the presence of chaos the largest Lyapunov exponent is calculated. I highly recommend adding this type of analysis to prove that all considered cases of the van der Pol oscillator are truly chaotic and that they can stay in this mode during control procedures.

• Great point. Illustration of chaotic behavior is included in Zhai’s prequel due to the importance of illustrating this key facet, where this manuscript is a first-order sequel.

7. Please, clarify the role and purpose of Fig.1 in this manuscript.

• Indeed, thanks. The figure is not effective illustrating the desired “initial impact” intended for potential readers scanning the cover page. The figure has been replaced with a hopefully more effective graphic.

8. I also recommend proofreading the manuscript to avoid at least such typos as "can der Pol oscillator" (Line 24)

• The keyword terms section was unfortunately skipped during the proofread, leading to the error not being found and corrected. It has since been addressed. We appreciate the reviewer’s attention to detail, since it has certainly helped the authors enhance the manuscript.

9. Nevertheless, I like the study and believe it can be accepted after relatively moderate revisions.

• Please consider using Susy’s review volunteer function to review the upcoming sequels.  We really appreciate the Reviewer’s diligence.

Reviewer 2 Report

Please check attachment.

Comments for author File: Comments.pdf

Author Response

We thank the three peer reviewers for diligent efforts towards improving the manuscript’s description of the research. In accordance with Editorial guidance, in instance where it is impossible to address certain comments in the review reports, explanations are included in the in your appeal.

REVIEWER 2

 

1. The research results can be detailed in the abstract with more sentences to help readers understanding main contributions quickly.

• The abstract was revised to better emphasize the main contributions.

2. The Figure 2 and 3 are not stated in main text. Please state the Figure 2 and 3 in main text.

• We accept the point and have implemented such with thanks to the Reviewer for the recommendation.

3. The θ did not define in equation (1).

• We accept the point and have implemented such with thanks to the Reviewer for the recommendation.

4. Please state each Figure for more details in main text. This is very important to help readers understanding your contribution quickly.

• We accept the point and have implemented such with thanks to the Reviewer for the recommendation.

5. Please be suggested to provide a conclusion section and state more explanations of the conclusion from corresponding studies.

• A conclusion section was portioned off as a sub header of the discussion section.
• More explanations from corresponding studies were not added, given that other studies on this subject have not focused on analyzing the effectiveness of specific online parameter estimation algorithms, instead focusing on showing that online parameter estimation works in a deterministic AI framework.

6. Please state how to eliminate modeling uncertainties and environmental disturbances in your research.

• We concede the suggestion is worth of investigation and following the publication of this benchmark work, sequels will be assigned to team members to address model the reviewer’s suggestion.

7. It should be “van der Pol oscillator” in line 24. Please revise “can” to “van”

• We accept the point and have implemented such with thanks to the Reviewer for the recommendation.

Reviewer 3 Report

This paper deals with a chaotic van der Pol oscillator control algorithm comparison. I think this paper is reviewe paper. right?

I would like to point out following .

1. Many reserchers have interesting for VDP oscillator. Thus, I hope authors explain  physical meaning of VDP and it's application field in introduction. I think it is necessary to beignner resercher.

2. Please add recently published paper or document in the referene.

3. What is main objective of this paper, it is not clear in introduction even though authors mentioned through a historical research in introduction.

4. What is the main differencing or novelty for the previous work and proposed method? I cannot find novelty compare to previous work.

5.  In the experimental result, authors should had better show the control result between input versus control by time series to easily understand by reader.

Author Response

We thank the three peer reviewers for diligent efforts towards improving the manuscript’s description of the research. In accordance with Editorial guidance, in instance where it is impossible to address certain comments in the review reports, explanations are included in the in your appeal.

REVIEWER 3

 

1. Many researchers have interesting for VDP oscillator. Thus, I hope authors explain physical meaning of VDP and its application field in introduction. I think it is necessary to beignner resercher.

• The recommendation is accepted completely. More details were added to the introduction to show greater context for van der Pol oscillators (i.e. discussing how they were developed by van der Pol for vacuum tubes).

2. Please add recently published paper or document in the referene.

• The Authors did not understand this recommendation.

3. What is main objective of this paper, it is not clear in introduction even though authors mentioned through a historical research in introduction.

• The recommendation is accepted completely, since overtness of such is paramount. The novelties sub section was simplified to better emphasize that the objective of the paper is investigating the effect of different online parameter estimation algorithms on adaptive control for the van der Pol system.

4. What is the main difference or novelty for the previous work and proposed method? I cannot find novelty compared to previous work.

• The recommendation is accepted completely. A new section was added at the end of the introduction to emphasize the novelties.

5. In the experimental result, authors should had better show the control result between input versus control by time series to easily understand by reader.

• The Authors did not understand this recommendation but remain open-minded if the recommendation could be re-phrased. We remain gratefully compliant to peer review suggestions, since such generally leads to improved manuscript quality.

Round 2

Reviewer 1 Report

Thank you for providing a revised version of your manuscript. I am satisfied with both revisions and a point-by-point reply letter, and can now recommend your paper for publication. I will certainly keep a close eye on your research updates and wish you every success in your work.

 

Reviewer 2 Report

Thanks for your responses.

Reviewer 3 Report

This paper is well revised according to reviewer's point out. Thus, I would like to decide as an "accept"