The Ikeda’s Galaxy

June 17, 2014
By

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

Chaos is the score upon which reality is written (Henry Miller)

Nonlinear dynamical systems are an enormous seam of amazing images. The Ikeda Map is an example of strange attractor which represents the movement of particles under the rules of certain differential equations.

I have drawn the trajectories followed by of 200 particles under the 2D-Ikeda Map with the same tecnique I used in this previous post, resulting this nice galaxy:

ikeda0.918

Wold you like to create your own galaxies? Here you have the code:

u=0.918 #Parameter between 0 and 1
n=200 #Number of particles
m=40 #Number of iterations
ikeda=data.frame(it=1,x1=runif(n, min = -40, max = 40), y1=runif(n, min = -40, max = 40))
ikeda$x2=1+u*(ikeda$x1*cos(0.4-6/(1+ikeda$x1^2+ikeda$y1^2))-ikeda$y1*sin(0.4-6/(1+ikeda$x1^2+ikeda$y1^2)))
ikeda$y2=  u*(ikeda$x1*sin(0.4-6/(1+ikeda$x1^2+ikeda$y1^2))+ikeda$y1*cos(0.4-6/(1+ikeda$x1^2+ikeda$y1^2)))
for (k in 1:m)
{
df=as.data.frame(cbind(rep(k+1,n),
ikeda[ikeda$it==k,]$x2,
ikeda[ikeda$it==k,]$y2,
1+u*(ikeda[ikeda$it==k,]$x2*cos(0.4-6/(1+ikeda[ikeda$it==k,]$x2^2+ikeda[ikeda$it==k,]$y2^2))-ikeda[ikeda$it==k,]$y2*sin(0.4-6/(1+ikeda[ikeda$it==k,]$x2^2+ikeda[ikeda$it==k,]$y2^2))),
u*(ikeda[ikeda$it==k,]$x2*sin(0.4-6/(1+ikeda[ikeda$it==k,]$x2^2+ikeda[ikeda$it==k,]$y2^2))+ikeda[ikeda$it==k,]$y2*cos(0.4-6/(1+ikeda[ikeda$it==k,]$x2^2+ikeda[ikeda$it==k,]$y2^2)))))
names(df)=names(ikeda)
ikeda=rbind(df, ikeda)
}
plot.new()
par(mai = rep(0, 4), bg = "gray12")
plot(c(0,0),type="n", xlim=c(-35, 35), ylim=c(-35,35))
apply(ikeda, 1, function(x) lines(x=c(x[2],x[4]), y=c(x[3],x[5]), col = paste("gray", as.character(min(round(jitter(x[1]*80/(m-1)+(20*m-100)/(m-1), amount=5)), 100)), sep = ""), lwd=0.1))

To leave a comment for the author, please follow the link and comment on their blog: Ripples.

R-bloggers.com offers daily e-mail updates about R news and tutorials on topics such as: Data science, Big Data, R jobs, visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, git, hadoop, Web Scraping) statistics (regression, PCA, time series, trading) and more...



If you got this far, why not subscribe for updates from the site? Choose your flavor: e-mail, twitter, RSS, or facebook...

Comments are closed.

Sponsors

Mango solutions



RStudio homepage



Zero Inflated Models and Generalized Linear Mixed Models with R

Quantide: statistical consulting and training



http://www.eoda.de









ODSC

CRC R books series











Contact us if you wish to help support R-bloggers, and place your banner here.

Never miss an update!
Subscribe to R-bloggers to receive
e-mails with the latest R posts.
(You will not see this message again.)

Click here to close (This popup will not appear again)