Editing Logistic map (section)

== Research history ==
===Before Chaos was named===
Before the iteration of maps became relevant to dynamical systems, mathematicians Gaston Julia and Pierre Fatou studied the iteration of complex functions. <!--[ 345 ]--> Julia and Fatou's work was broad, focusing on analytic functions. <!--[ 346 ]--> In particular, they studied the behavior of the following complex quadratic function, also shown in equation (6–3), in the 1920s. <!--[ 336 ]-->

{{NumBlk|:|<math>{\displaystyle z_{n+1}=z_{n}^{2}+c}</math>|{{EquationRef|7-1}}}}

Julia and Fatu also recognized chaotic behavior in Julia sets, but because there was no computer graphics at the time, no one followed suit and their research stalled. <!--[ 347 ]--> Research on complex dynamical systems then declined until the late 1970s, and it was not until the appearance of Benoit Mandelbrot and others that the rich dynamical behavior exhibited by maps on the complex plane was noticed. <!--[ 348 ]-->

[[File:Stanislaw Ulam ID badge.png|thumb|Stanislaw Ulam]]
[[File:John von Neumann ID badge.png|thumb|John von Neumann]]
In 1947, mathematicians Stanislaw Ulam and John von Neumann wrote a short paper entitled "On combination of stochastic and deterministic processes" in which they{{huh?|reason=There is obviously text missing here|date=May 2025}}

{{NumBlk|:|<math>{\displaystyle f(x)=4x(1-x)}</math>|{{EquationRef|7-2}}}}

They pointed out that pseudorandom numbers can be generated by the repeated composition of quadratic functions such as.{{huh?|reason=There is obviously text missing here|date=May 2025}} <!--[ 349 ]--> In modern terms, this equation corresponds to the logistic map with {{mvar|r}} = 4. <!--[ 350 ]--> At that time, the word "chaos" had not yet been used, but Ulam and von Neumann were already paying attention to the generation of complex sequences using nonlinear functions. <!--[ 330 ]--> In their report, Ulam and von Neumann also clarified that the map (7–2) and the tent map are topologically conjugate, and that the invariant measure of the sequence of this map is given by equation (3–17).<!--[ 267 ]-->

There have since been some detailed investigations of quadratic maps of the form with arbitrary parameter {{mvar|a}}. <!--[ 351 ]--> Between 1958 and 1963, Finnish mathematician Pekka Mylberg developed the{{huh?|reason=There is obviously text missing here|date=May 2025}}

{{NumBlk|:|<math>{\displaystyle f(x)=x^{2}-\lambda }</math>|{{EquationRef|7-3}}}}

This line of research is essential for dynamical systems, and Mühlberg has also investigated the period-doubling branching cascades of this map, showing the existence of an accumulation point λ = 1.401155189....<!--[352 ]--> Others, such as the work of the Soviet Oleksandr Sharkovsky in 1964, the French Igor Gumowski and Christian Mila in 1969, and Nicholas Metropolis in 1973, have revealed anomalous behavior of simple one-variable difference equations such as the logistic map. <!--[ 353 ]-->

===Robert May's research===
[[File:BobMayHarvard.jpg|thumb|Robert May (photographed in 2009)]]
Later, in the early 1970s, mathematical biologist Robert May encountered the model of equation (1–2) while working on an ecological problem. <!--[ 354 ]--> May introduced equation (1–2), i.e., the logistic map, by discretizing the logistic equation in time. <!--[ 355 ]--> He mathematically analyzed the behavior of the logistic map, and published his results in 1973 and 1974. <!--[ 356 ]--> Numerical experiments were performed on the logistic map to investigate the change in its behavior depending on the parameter {{mvar|r}}. <!--[ 357 ]-->

In 1976, he published a paper in ''Nature'' entitled "Simple mathematical models with very complicated dynamics". <!--[ 26 ]--> This paper was a review paper that focused on the logistic map and emphasized and drew attention to the fact that even simple nonlinear functions can produce extremely complex behaviors such as period-doubling bifurcation cascades and chaos. <!--[ 358 ]--> This paper in particular caused a great stir and was accepted by the scientific community due to May's status as a mathematical biologist, the clarity of his research results, and above all, the shocking content that a simple parabolic equation can produce surprisingly complex behavior. <!--[ 359 ]--> Through May's research, the logistic map attracted many researchers to chaos research and became such a famous mathematical model that it is said to have restarted the flow of chaos research. <!--[ 360 ]-->

===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.}}