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:

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))

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*Related*

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) )

Thanks!