This project can be used to approximate Fiegenbaum's constant. Use the Minimum R and Maximum R inputs to explore small sections of the map to find the values of R where bifurcation (splitting) occurs. The ratio of successive R values at points of bifurcation approximates Feignbaum's constant. I have used this program to compute the Feigenbaum constant as 4.66. To see the complete map, Set Minimum R at 2.95 and Maximum R at 3.95. Just as Pi is the quotient of ratios (circumference to diameter) where both numerator and denominator represent lengths, Feigenbaum's contant is also a ratio of lengths. To use this project to approximate Feigenbaum's constant see Notes and Credits below.
See the book 'Chaos - Making a New Science' by James Gleick for the story of how Mitchell Feignebaum, while a physicist at Los Alamos National Laboratory, discovered the constant that bears his name and is one of the cornerstones of chaos theory. The first bifurcation, R1, occurs at R = 3.00. Set min R to 3.4, max R to 3.5, and observe that the second bifurcation,2.95 R2, occurs at R =3.45. Set min R to 3.5, max R to 3.6, and observe that the third bifurcation, R3, occurs at 3.546. Compute (R2 – R1)/(R3 – R2) to get an approximation to Feigenbaum's constant. See my other Logistic Equation projects: An Abstract Fish Population Model https://scratch.mit.edu/projects/195253979/ The Logistic Equation – Cobweb Form https://scratch.mit.edu/projects/194143589/ y = Rx(1 – x) As a Time Series https://scratch.mit.edu/projects/195475210/
