The logistic map in R

I am just trying to learn the logistic map, so here are some notes and the R code snippets.

Basically what the logistic map does, is a equation that returns the next value based on current value.

The formula is Xn+1 = R * Xn * (1-Xn).

  • Xn is the value of X in the nth iteration.
  • Xn+1 is the next value of Xn, which we calculate.
  • R is the parameter.

This equation has been used in several real life problems to model the situation. One example is population growth.

In its crude form, it looks like this when coding in R.

R=3

x0=0.5
x1=R*x0*(1-x0)
x2=R*x1*(1-x1)
x3=R*x2*(1-x2)
x4=R*x2*(1-x3)
x5=R*x2*(1-x4)
x6=R*x2*(1-x5)

We can try to plot it and iterate over many rounds through a loop.

R=3
a=0.5

points<-c(a)


for (i in 1:30){
  b<-R*a*(1-a)
  a=b
  points[i]<-a
}

plot(points)
 

Screenshot 2024-05-30 at 5.22.03 AM.png

We see a stabilisation with R being 3. This is also call a fixed point when there is a stabilisation to a value.

Let's try with R being 3.4.

R=3.4
a=0.5

points<-c(a)


for (i in 1:30){
  b<-R*a*(1-a)
  a=b
  points[i]<-a
}

plot(points)

Now we see a bistable state, i.e. X value fluctuating up and down.

Screenshot 2024-05-30 at 5.28.57 AM.png

Apparently, at R of approximately 3.5699456, not far from 3.4, something strange starts to happen.


R=3.5699456 
a=0.5

points<-c(a)


for (i in 1:30){
  b<-R*a*(1-a)
  a=b
  points[i]<-a
}

plot(points)

The bistable states starts to be changed.

Screenshot 2024-05-30 at 5.32.57 AM.png

And when R is 3.65, some kind of messy plot emerges.

Screenshot 2024-05-30 at 5.36.43 AM.png

At R at 3.999999999999...

Screenshot 2024-05-30 at 5.39.13 AM.png

Finally, at R = 4.0. We get back another fixed point.

Screenshot 2024-05-30 at 5.41.30 AM.png



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