Editing Logistic map (section)

===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 ]-->