Talk:Period doubling

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    Reviewer #1

    The proposed article is very informative and certainly useful. However, to facilitate the readability, I would suggest to split it into entries: (1) a simple article on the period-doubling bifurcation of maps in the spirit of the existing articles "Saddle-node bifurcation for maps" and "Neimark-Sacker bifurcation"; (2) a more advanced article on the universality of cascades of period-doublings, which can be based on the proposed text. In such articles, one also should not try to define all basic notions (like dynamical system, center manifold, etc.) from scratch but rater give links to the existing entries in Scholarpedia.

    Author's Response

    Thank you very much for your review on the period doubling article. James Meiss has been our contact throughout the long and laborious process of writing this text, which is in fact the first of three planned articles.

    This means in particular that we have already extracted the short story out of a longer one (and the resulting size was approved by the editor). Besides, period doubling is quite different from other local bifurcations for a number of important reasons:

    (1) Many systems are prone to not only exhibit one period doubling bifurcation, but a cascade of such bifurcations. Hence there is a global bifurcation pattern attached to the local one in a way that is quite different from other bifurcation phenomena.

    (2) Period doubling has brought to dynamics renormalization techniques. The impact of these techniques on dynamical systems theory culminates in the analysis of universality of certain bifurcation diagrams and of certain invariant Cantor sets (or other objects, most often measures). However, even topological universality and topological renormalization have become standard tools in the hands of many researchers, not only in one-dimensional dynamics but also in the study of billiards and in the investigation of deep questions about surfaces.

    (3) A proper understanding of period doubling requires the broader viewpoint of regarding as dynamical systems the actions of groups, semi-groups, or even pseudo groups. This is in a spirit that goes back to the origins of modern dynamical systems and that is reflected in the seminal works of several contemporary mathematicians -- e.g., Dennis Sullivan with his famous dictionary between rational maps and Kleinian groups (which lead to the groundbreaking works of Sullivan, McMullen and Lyubich on geometric rigidity and universality).

    For the reasons above, it seemed to us -- and this was validated by the editor -- that we should use more pages for this specially important bifurcation (and for a part of what comes with it). We are here in a domain of mathematics where there are many more unknown answers (and perhaps not yet even the right questions). The paper as it stands represents about one quarter of the original paper we wrote, a paper that went much further into the mathematics of renormalization in dimension one and above (mostly dimension two, in the context of the Hénon model). Again, we wish to stress that this is the sort of length upon which we agreed with the editor before plunging into the hard work not only of writing a first draft but also of editing, polishing, choosing very carefully what should be in the first part versus what should go to the other parts (or be ignored).

    Finally, let us turn to another important suggestion that you make. If we can get help for cross-referencing and hyperlinks, or if someone in the editorial office could do that (with as much of our help as needed) we would be happy to have such easy links in our text. However, we stress that, due to the broad viewpoint we take, as described in (3) above, we would need to continue defining the terms we use at our level of exposition and originality. This means that your suggested complement to our text could be made (almost or completely) automatically, by simply creating hyperlinks for all relevant words for which there is an article in the Scholarpedia.

    Sincerely yours,

    Pierre Coullet, Edson de Faria, and Charles Tresser

    Editor's Response

    The suggestion that more links be included in the article is being worked on!

    Jim Meiss

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