The Ikeda’s Galaxy

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))
Advertisements

3 thoughts on “The Ikeda’s Galaxy

  1. Maybe a little faster, and I think easier to read:
    newit<- rep(k+1,n)
    tn <- 0.4-6/(1+ikeda[ikeda$it==k,]$x2^2+ikeda[ikeda$it==k,]$y2^2)
    cutikeda<-ikeda[ikeda$it==k,]
    df <- as.data.frame(cbind(newit, cutikeda$x2, cutikeda$y2, 1+u* (cutikeda$x2*cos(tn) – cutikeda$y2 * sin(tn) ), u* (cutikeda$x2*sin(tn) + cutikeda$y2 * cos(tn) )

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s