# Logistic map: Feigenbaum diagram

March 17, 2012
By

(This article was first published on mages' blog, and kindly contributed to R-bloggers)

The other day I found some old basic code I had written about 15 years ago on a Mac Classic II to plot the Feigenbaum diagram for the logistic map. I remember, it took the little computer the whole night to produce the chart.

With today’s computers even a for-loop in a scripting language like R takes only a few seconds.
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``logistic.map <- function(r, x, N, M){  ## r: bifurcation parameter  ## x: initial value  ## N: Number of iteration  ## M: Number of iteration points to be returned  z <- 1:N  z[1] <- x  for(i in c(1:(N-1))){    z[i+1] <- r *z[i]  * (1 - z[i])  }  ## Return the last M iterations   z[c((N-M):N)]}## Set scanning range for bifurcation parameter rmy.r <- seq(2.5, 4, by=0.005)system.time(Orbit <- sapply(my.r, logistic.map,  x=0.001, N=1000, M=300))##   user  system elapsed (on a 2.4GHz Core2Duo)##  1.834   0.011   1.840 Orbit <- as.vector(Orbit)r <- sort(rep(my.r, (M+1)))plot(Orbit ~ r, pch=".")``
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Let’s not forget when Mitchell Feigenbaum started this work in 1975 he did this on his little calculator!

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