Animating the Mandelbrot Set - r

Question

I have always been interested in learning about how computers are able to animate the "Mandelbrot Set" (https://en.wikipedia.org/wiki/Mandelbrot_set).
I found this website (https://www.dandelbrot.com/post/the-mandelbrot-set-in-r/) that shows how to create Mandelbrot Set:
mandelbrot_generator <- function(
p = 2,
q = 1,
xmin = -2.1, # minimum x value
xmax = 0.8, # maximum x value
nx = 500,
ymin = -1.3, # minimum y value
ymax = 1.3, # maximum y value
ny = 500,
n = 100,
showplot = TRUE, # If TRUE then display image,
showvals = FALSE,
cols = colorRampPalette(c("black","cyan","cyan3","black"))(11))
{
# variables
x <- seq(xmin, xmax, length.out=nx)
y <- seq(ymin, ymax, length.out=ny)
c <- outer(x,y*1i,FUN="+")
z <- matrix(0.0, nrow=length(x), ncol=length(y))
k <- matrix(0.0, nrow=length(x), ncol=length(y))
for (rep in 1:n) {
index <- which(Mod(z) < 2)
z[index] <- z[index]^p + c[index]*q
k[index] <- k[index] + 1
}
if (showplot==TRUE) { image(x,y,k,col=cols, xlab="Re(c)", ylab="Im(c)")}
if (showvals==TRUE) {return(k)}
}
Here is the plot:
mandelbrot_generator(p=2, q=1)
Does anyone know how to make an "animation" using the above code, so that it looks like this?
(https://en.wikipedia.org/wiki/Mandelbrot_set#/media/File:Mandelbrot_sequence_new.gif)
I always wondered : how are these animations made? I understand that a single picture of the Mandelbrot Set can be made, but how do these "zooming" animations work? Is this simply done by changing the "axis" (i.e. scale) of the picture?
The above picture is made from x = (-2.1, 0.8) and y = (-1.3, 1.3) - my understanding is that if we wanted to make a "zooming animation", we would "shrink" these ranges at each frame?
For example:
Frame 1: x = (-2.1, 0.8) and y = (-1.3, 1.3)
Frame 2 : x = (-1.9, 0.6) and y = (-1.1, 1.1)
Frame 3 : x = (-1.4, 0.3) and y = (0.7, 0.7)
etc.
Is this correct? Could we use this logic to make a series of "Mandelbrot frames" - and then fade/transition between these frames, giving them the illusion of being animated?
Thank you!

Answer 1

First, you should change a little your code like
mandelbrot_generator <- function(n){
p=2
q=1
xmin = -2.1 # minimum x value
xmax = 0.8 # maximum x value
nx = 500
ymin = -1.3 # minimum y value
ymax = 1.3 # maximum y value
ny = 500
showplot = TRUE # If TRUE then display image,
showvals = FALSE
cols = colorRampPalette(c("black","cyan","cyan3","black"))(11)
# variables
x <- seq(xmin, xmax, length.out=nx)
y <- seq(ymin, ymax, length.out=ny)
c <- outer(x,y*1i,FUN="+")
z <- matrix(0.0, nrow=length(x), ncol=length(y))
k <- matrix(0.0, nrow=length(x), ncol=length(y))
for (rep in 1:n) {
index <- which(Mod(z) < 2)
z[index] <- z[index]^p + c[index]*q
k[index] <- k[index] + 1
}
if (showplot==TRUE) { image(x,y,k,col=cols, xlab="Re(c)", ylab="Im(c)")}
if (showvals==TRUE) {return(k)}}
Then you can use the images to create a gif file, following this this thread, for instance.
To some mathematical aspects of Mandelbrot set, please see this thread, and find more on SearchOnMath.