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](https://plus.maths.org/content/maths-minute-logistic-map).


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](https://files.peakd.com/file/peakd-hive/snippets/23tHbKgySUppzvx8UUFyCGXkn6ej9cBMJWVf2edgyuyvFgaBZge9DJZmaGk2AhLAHsN6Y.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](https://files.peakd.com/file/peakd-hive/snippets/23t8CfKPvq6siMpQSRxVxWaYWmRND59F7WSTD7RDZV9WBox9f6zYN8KT82V9vcPX2ZVeK.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](https://files.peakd.com/file/peakd-hive/snippets/23t8CfKPvrR3VDy1crkoFEbTbrKwKQjLN98ddVYUJVRxDYhAVH4GpJMwwzPT3eggrXxQK.png)

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

![Screenshot 2024-05-30 at 5.36.43 AM.png](https://files.peakd.com/file/peakd-hive/snippets/23sxozwpQz1dWV2oBNvU29zDESeC8Bgcx4PDwKCSWKPrLeTxA9uzz8ttCXeKo6FFup5wH.png)
 
At R at 3.999999999999...

![Screenshot 2024-05-30 at 5.39.13 AM.png](https://files.peakd.com/file/peakd-hive/snippets/23tSyz4YxQSgdZeqZd7ZewMuSqoBtxswrVEk2XxKxyoXmNXoWC1pouCsj4GALsuJrnWsH.png)
  
Finally, at R = 4.0. We get back another fixed point.

![Screenshot 2024-05-30 at 5.41.30 AM.png](https://files.peakd.com/file/peakd-hive/snippets/23tSyz4YxQSghaZUQkwzGvF8j2BybJxzUNhsMgAfP6dBkioTnGyUhMorraZB3NZhFJP5r.png)