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A cobweb diagram is a visualization tool that allows one to qualitatively study the iterates of a self-map of the real line based on the graph of the function; here is an example:

enter image description here

(Here the map is the Boole's transformation $x\mapsto x-1/x$. The link to the interactive Desmos graph is https://www.desmos.com/calculator/20yftwjmys.)

Other names for this construct include: zig-zag, staircase, graphical analysis, Verhulst diagram,..., the latter being from https://en.wikipedia.org/wiki/Cobweb_plot (the other terms are from e.g. Rudin, Devaney, Hasselblatt-Katok, Strogatz).

Question: Who first used cobweb diagrams in a publication? In particular, I would be interested in the verification or falsification of the attribution to Verhulst.


Added: According to the paper Prof. Friedland referenced, Verhulst never used cobweb diagrams, at least in his work on the logistic map; which seems reasonably attributable to him. Since the logistic map is a popular choice to introduce cobweb diagram, it seems there has been a transference of attribution.


Added: To continue, here is a relevant page from Legendre's 1816 monograph Essai sur la théorie des nombres (2e) (https://gallica.bnf.fr/ark:/12148/bpt6k62826k/f609.item.zoom#):

enter image description here

The context is approximation of roots; and two arbitrary curves are allowed (see also https://math.stackexchange.com/q/4435177/169085). Rosa, in the paper referenced by Prof. Friedland also mentions Bidone's work which precedes that of Legendre, but also that it's a possibility that already this was folklore. Rosa claims (p.35) that Fourier and Galois, together with Legendre, are regarded as the pioneers of the method, due to their interest and contributions.

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    $\begingroup$ So you ask about this type of diagram, regardless of whether it is called "cobweb"? Or you ask only about the name "cobweb"? $\endgroup$ Feb 6, 2023 at 18:26
  • $\begingroup$ @GeraldEdgar I am asking about the type of diagram independently of what it is called. $\endgroup$
    – Alp Uzman
    Feb 7, 2023 at 1:05

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According to Alessandro Rosa, ``An episodic history of the staircased iteration diagram", ANTIQUITATES MATHEMATICAE Vol. 15(1) 2021, p. 3–90, doi: 10.14708/am.v15i1.7056, a version of the diagram can be traced back to Adrien-Marie Legendre, in 1822.

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