# Chaotic Galaxies

Tell me, which side of the earth does this nose come from? Ha! (ALF)

Reading about strange attractors I came across with this book, where I discovered a way to generate two dimensional chaotic maps. The generic equation is pretty simple:

$x_{n+1}= a_{1}+a_{2}x_{n}+a_{3}x_{n}^{2}+a_{4}x_{n}y_{n}+a_{5}y_{n}+a_{6}y_{n}^{2}$
$y_{n+1}= a_{7}+a_{8}x_{n}+a_{9}x_{n}^{2}+a_{10}x_{n}y_{n}+a_{11}y_{n}+a_{12}y_{n}^{2}$

I used it to generate these chaotic galaxies:

Changing the vector of parameters you can obtain other galaxies. Do you want to try?

```library(ggplot2)
library(dplyr)
#Generic function
attractor = function(x, y, z)
{
c(z[1]+z[2]*x+z[3]*x^2+ z[4]*x*y+ z[5]*y+ z[6]*y^2,
z[7]+z[8]*x+z[9]*x^2+z[10]*x*y+z[11]*y+z[12]*y^2)
}
#Function to iterate the generic function over the initial point c(0,0)
galaxy= function(iter, z)
{
df=data.frame(x=0,y=0)
for (i in 2:iter) df[i,]=attractor(df[i-1, 1], df[i-1, 2], z)
df %>% rbind(data.frame(x=runif(iter/10, min(df\$x), max(df\$x)),
y=runif(iter/10, min(df\$y), max(df\$y))))-> df
return(df)
}
opt=theme(legend.position="none",
panel.background = element_rect(fill="#00000c"),
plot.background = element_rect(fill="#00000c"),
panel.grid=element_blank(),
axis.ticks=element_blank(),
axis.title=element_blank(),
axis.text=element_blank(),
plot.margin=unit(c(-0.1,-0.1,-0.1,-0.1), "cm"))
#First galaxy
z1=c(1.0, -0.1, -0.2,  1.0,  0.3,  0.6,  0.0,  0.2, -0.6, -0.4, -0.6,  0.6)
galaxy1=galaxy(iter=2400, z=z1) %>% ggplot(aes(x,y))+
geom_point(shape= 8, size=jitter(12, factor=4), color="#ffff99", alpha=jitter(.05, factor=2))+
geom_point(shape=16, size= jitter(4, factor=2), color="#ffff99", alpha=jitter(.05, factor=2))+
geom_point(shape=46, size= 0, color="#ffff00")+opt
#Second galaxy
z2=c(-1.1, -1.0,  0.4, -1.2, -0.7,  0.0, -0.7,  0.9,  0.3,  1.1, -0.2,  0.4)
galaxy2=galaxy(iter=2400, z=z2) %>% ggplot(aes(x,y))+
geom_point(shape= 8, size=jitter(12, factor=4), color="#ffff99", alpha=jitter(.05, factor=2))+
geom_point(shape=16, size= jitter(4, factor=2), color="#ffff99", alpha=jitter(.05, factor=2))+
geom_point(shape=46, size= 0, color="#ffff00")+opt
#Third galaxy
z3=c(-0.3,  0.7,  0.7,  0.6,  0.0, -1.1,  0.2, -0.6, -0.1, -0.1,  0.4, -0.7)
galaxy3=galaxy(iter=2400, z=z3) %>% ggplot(aes(x,y))+
geom_point(shape= 8, size=jitter(12, factor=4), color="#ffff99", alpha=jitter(.05, factor=2))+
geom_point(shape=16, size= jitter(4, factor=2), color="#ffff99", alpha=jitter(.05, factor=2))+
geom_point(shape=46, size= 0, color="#ffff00")+opt
#Fourth galaxy
z4=c(-1.2, -0.6, -0.5,  0.1, -0.7,  0.2, -0.9,  0.9,  0.1, -0.3, -0.9,  0.3)
galaxy4=galaxy(iter=2400, z=z4) %>% ggplot(aes(x,y))+
geom_point(shape= 8, size=jitter(12, factor=4), color="#ffff99", alpha=jitter(.05, factor=2))+
geom_point(shape=16, size= jitter(4, factor=2), color="#ffff99", alpha=jitter(.05, factor=2))+
geom_point(shape=46, size= 0, color="#ffff00")+opt
```

## 3 thoughts on “Chaotic Galaxies”

1. No, those curiosities from topology have constructor algorithms which can be found here and there (including wolfram), but these are simple 2-D chaotic “orbit” plots.