Editing Logistic map (section)

===After May's research===
May also drew attention to the paper by using the term "chaos", which was used by Tien-Yen Li and James York in their paper "Period three implies chaos".<!--[361]--> Although some disagree, Li and York's paper is considered the first to use the word "chaos" as a mathematical term, and is credited with coining the term "chaos" to refer to deterministic, chaotic behavior.<!--[362]--> Li and York completed the paper in 1973, but when they submitted it to ''The American Mathematical Monthly'', they were told that it was too technical and that it should be significantly rewritten to make it easier to understand, and it was rejected. <!--[ 363 ]--><!--[ 361 ]--> The paper was then left unrevised. <!--[ 361 ]--> However, the following year, in 1974, May came to give a special guest lecture at the University of Maryland where Lee and York were working, and talked about the logistic map. <!--[ 361 ]--> At the time, May did not yet understand what was happening in the chaotic domain of the logistic map, but Lee and York were also unaware of the period-doubling cascade of the logistic map. <!--[ 354 ]--> Excited by May's talk, Lee and York caught up with May after the lecture and told him about their results, which surprised May. <!--[ 364 ]--> Lee and York quickly rewrote the rejected paper, and the resubmitted paper was published in 1975. <!--[ 365 ]-->

[[File:Mitchell J Feigenbaum - Niels Bohr Institute 2006.jpg|thumb|Mitchell Feigenbaum (photographed in 2006)]]

Also, around 1975, mathematical physicist Mitchell Feigenbaum noticed a scaling law in which the branching values converged in a geometric progression when he looked at the period-doubling cascade of the logistic map, and discovered the existence of a constant, now called the Feigenbaum constant, through numerical experiments. <!--[ 366 ]--> May and George Oster had also noticed the scaling law, but they were unable to follow it in depth. <!--[ 139 ]--> Feigenbaum discovered that the same constant also appeared in the sine map shown in equation (4–1), and realized that this scaling law had a universality that went beyond the logistic map. <!--[ 367 ]--> In 1980, a rigorous proof of this result was given by Pierre Collé, Jean-Pierre Eckman, Oscar Rumford, and others. <!--[ 368 ]--> At about the same time as Feigenbaum, or later, physicists discovered the same period doubling cascade and the Feigenbaum constant in real life, and chaos, which had previously been seen as a strictly mathematical phenomenon, had a major impact on the field of physics as well. <!--[ 369 ]-->

However, there is criticism of the tendency to downplay research results from before the popularity of chaos, and to attribute many of those results to rediscoverers who used the logistic map, etc. <!--[ 352 ]--> May himself respects the existence of previous research, but positions his own achievement as not being "the first to independently discover the strange mathematical behavior of quadratic maps", but as being one of the "last researchers to emphasize their broad implications in science". <!--[ 353 ]--> Mathematician Robert Devaney states the following before explaining the logistic map in his book:<!--[ 54 ]-->

{{quote|This means that by simply iterating the quadratic function <math>f_{\lambda}(x) = \lambda x(1 - x )</math> (also known as the logistic map), we can predict the fate of the initial population <math>x_0</math>. This sounds simple, but I dare to point out that it was only in the late 1990s, after the efforts of hundreds of mathematicians, that the iteration of this simple quadratic function was fully understood.}}