Limit Cycles of Polynomially Integrable Piecewise Differential Systems

Round 1

Reviewer 1 Report

Authors study limit cycles of polynomially integrable piecewise differential

systems. 

 

A list of issues and questions follow:

- Reference [3] has a typo in Champnys: please, fix it.

- What happens if one regularizes the system? Does any regularization preserve limit cycles, as proven in

  L. Dieci, C. Elia, D. Pi. Limit cycles for regularized discontinuous dynamical systems 

  with a hyperplane of discontinuity. Discrete and Continuous Dynamical Systems - B, 2017, 22(8): 3091-3112. doi: 10.3934/dcdsb.2017165?

  A mention on this is necessary, also in light of R. Huzak, K. Uldall Kristiansen,

  The number of limit cycles for regularized piecewise polynomial systems is unbounded,

  Journal of Differential Equations, 2023.

- The paper Tao Li, Jaume Llibre, Limit cycles in piecewise polynomial Hamiltonian systems allowing nonlinear switching boundaries,

  Journal of Differential Equations, 2023, is a strong generalization of the present work (also, by one of the author). Please, add it

  and mention how the present work gives some originality with respect to it: in fact, a slight perturbation of discontinuity manifold 

  would make the present problem nonlinear.

- What happens if one has a co-dimension 2 discontinuity manifold? Some recent work (e.g., F.V. Difonzo, A note on attractivity for 

  the intersection of two discontinuity manifolds, Opuscula Mathematica 40, no. 6 (2020), 685-702, https://doi.org/10.7494/OpMath.2020.40.6.685) 

  provides characterization of attractivity, which can be by sliding or by spiraling. In this last case,

  how would authors deal with such a higher co-dimensional phenomenon? Please, add a comment on this, since limit cycles can arise in particular

  spiraling dynamics. 

- Systems (26) and (27) do not seem to coincide for x=0. Please, remove x=0 from either definition.

- In (27), what is j? I think there is nor definition neither specification for it. Please, fix it.

- Would using number of zeros the first order Melnikov function provide some further insight? See, e.g., Xiaoyan Chen  and Maoan Han, 

  Number of Limit Cycles from a Class of Perturbed Piecewise Polynomial Systems, International Journal of Bifurcation and Chaos, 2021.

  

I think the paper presents nice computations and interesting results, but needs to be clearly enriched with more recent literature, more remarks

on possible extensions (co-dimension 2, other techniques for determining the number of limit cycles, nonlinear perturbation).

Once all these issues have been addressed it will be robust enough for Axioms.

 

Author Response

See the attached pdf file

Author Response File: Author Response.pdf

Reviewer 2 Report

Report on the paper "Limit cycles of polynomially integrable piecewise differential systems"

This paper deals with the number of algebraic limit cycles of a discontinuous piecewise linear differential systems separated by a straight line, with polynomial first integrals on both sides.

Two main results are presented. The first regards a classification of the planar linear differential systems with a polynomial first integral. The second establishes the number of limit cycles of the above-mentioned piecewise system.

The paper is well written. I strongly recommend it for publication.

 

Author Response

Please see the attached pdf file.

Author Response File: Author Response.pdf

Reviewer 3 Report

Axioms-2268643

Title: Limit cycles of polynomially integrable piecewise differential systems

Authors: Belén García, Jaume Llibre, Jesús S. Pérez del Río , Set Pérez-González

The authors study how many algebraic limit cycles have the discontinuous piecewise linear differential systems separated by a straight line, with polynomial first integrals on both sides, when at least one of the systems is Hamiltonian.

The paper is interesting, and well organized. However, it has about 30% of self-references, what is inadequate. The authors should add more references to have a lower percentage of self-references.

 In page 17, line 362: Figure 6, instead of Figure 4.

 

 

Author Response

Please see the attached pdf file.

Author Response File: Author Response.pdf

Reviewer 4 Report

Title: Limit cycles of polynomially integrable piecewise differential systems

 I have read this paper on the study of algebraic limit cycles in discontinuous piecewise linear differential systems separated by a straight line, with polynomial first integrals on both sides. The paper appears to be well-written, but I have some concerns.

1. Firstly, I noticed that one of the authors (the second author) has already published a paper with a similar title, "Limit Cycles for Planar Piecewise Linear Differential Systems via First Integrals". Could you please explain the difference between the two papers and what advantages the current work offers?"

2. Reference should be recent years.

Author Response

Please see the attached pdf file.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The authors have addressed some of the issues raised, but not even mentioning what happens in other regularizing cases and co-dim 2 case. Such remarks should be added to the paper to make it ready for publication.

Author Response

Thank you for your comments.