Talk:Perturbation theory (dynamical systems)
The authors have written a good and technically detailed overview of KAM theory, but this is a relatively narrow problem area and does not appropriately represent the scope of coverage that should be spanned in an article on the broad topic of "Perturbation Theory". Good detail is given for KAM theory, normal forms and chaotic dynamics, but the authors tend to gravitate to just their own sphere of expertise, and their own works as the primary references.
In an review article on perturbation theory, I'd expect to see many topics discussed, including:
Forms of perturbation expansions
Regular vs Singular perturbation problems
Bifurcation theory (a la Golubitsky, Schaeffer, Stewart)
Algebraic problems and perturbations of eigenvalues
dissipative vs conservative dynamics
stability analysis (linearized and weakly nonlinear methods)
I hope that the authors will broaden their overview and expand their references and links to include a broad class of problems and a better sampling of the historical development of the area and its applications in various areas of applied science
Basically, I agree with the comments of the other reviewer. This is a well written and technically detailed overview of certain areas of perturbation theory. However, it is primarily concerned with the authors’ own area of expertise and does not cover other important areas of perturbation theory; singular perturbation theory is one example. The article is written for a rather sophisticated audience and the article should be useful for those interested in KAM theory and normal forms. As the other reviewer comments, I too would encourage the authors to broaden their overview and provide a better sampling of the historical developments of the area and its applications in various areas of applied science. Another possibility would be to change the title of the article and then expand on those topics covered in the article. In this case, it would be useful to provide more background material and more examples of applications.
Today we have been editting the Perturbation Theory article of Scholarpedia. We have incooperated most of the remarks of the referees B and C by adding references (including scholarpedia links), a paragraph on asympotic expansions and another one on bifurcations. If the referees mean us to write a quite different article, we don't agree to this. We could imagine another scholarpedia entry, for instance titled Classical Perturbation Theory, maybe with a historic introduction, in which all their points of interest are addressed. In fact, some of these points deserve an entry of their own, e.g., Singular Perturbation Theory and Algebraic Problems and Perturbations of Eigenvalues. We are willing to write a first draft of such a paper, but could also imagine that the referees B and C are suitable authors. We feel that this gives an appropriate reply to the comments of referee B, whereas referee C is more or less saying the same things. To be more detailed:
Forms of perturbation expansions (addressed in the new paragraph, could be further expanded in the proposed new article)
Asymptotic analysis (addressed in one of the new paragraphs, could be further expanded in the proposed new article)
Regular vs Singular perturbation <http://www.scholarpedia.org/article/Singular_Perturbation_Theory> problems (should be a separate article)
Bifurcation theory <http://www.scholarpedia.org/article/Bifurcations> (a la Golubitsky, Schaeffer, Stewart) (in one of the new paragraphs and also addressed further in the text (in the section on Parametrized KAM Theory) and by quite a few references)
Algebraic problems and perturbations of eigenvalues <http://www.scholarpedia.org/article/Eigenvalues_And_Eigenvectors> (does not belong to our additions, but could be linked to in the proposed new article)
dissipative vs conservative dynamics (this is already addressed throughout our article)
stability <http://www.scholarpedia.org/article/Stability> analysis (linearized and weakly nonlinear methods) (does not belong to our additions, but could be linked to in the proposed new article)
Please let us know what you think of this!
Heinz Hanßmann and Henk Broer
Prof.dr H.W. Broer
University of Groningen
Department of Mathematics
Reviewer: Review of perturbation theory article (revised)
I respect the authors' careful and technically sophisticated presentation in the article they have written. I also respect the authors' freedom to write about whatever focus areas they prefer. However I will not give my approval to have this article represent the area of "Perturbation Theory". It is far too narrowly focused to represent this field. Rather than having other articles with different qualified titles "Classical Perturbation Theory, Perturbation theory of eigenvalues", it should be the current text that should be shifted to a different entry, for example "KAM theory and perturbation theory". Sections 1-1.3 might be appropriate for an article on general perturbation theory (and I appreciate some revisions that the authors have made here in response to the previous review), most of the rest is directed in far too specialized a way for an "encyclopedia entry". For comparison, see the WIKIPEDIA article on perturbation theory: http://en.wikipedia.org/wiki/Perturbation_theory which is far more comprehensive in its treatment of different areas, applications and history of the field.
The Wikipedia article has set a standard by which this article (and Scholarpedia) will be judged. The authors and the editors should consider this fact and evaluate how best to proceed in such a way to make best use of the authors' sophisticated technical contribution.
I agree that "KAM theory and perturbation theory" is a much more appropriate title. The current title is too general for this article for reasons described in the earlier reviews.
We have noticed that you have changed the title of our article to 'Perturbation theory (dynamical systems)' and announced two further papers on perturbation theory in quantum mechanics and statistical mechanics. Let us say that we like this solution, in fact this fits with the intention we had when writing the article! We expect that in this way the problems of the referees have been resolved.
We think that it is now time to accept our article.
Heinz Hanßmann and Henk Broer