Contraction mapping of the logistic map

I wish to find the values for which the logistic map behaves as a contraction map

$$x_{n+1}=rx_n(1+x_n)\equiv F(x;r)$$

i.e, I wish to find for which $$r$$, the mapping above admits a unique fixed point $$F(x^*;r)=x^*$$ using the contraction map theorem.

It is easy to find the fixed points, and it exists in any text book, as well as Banach's fixed point theorem, however I have no idea on how to apply this theorem over the example of the logistic map.

The most helpful reference I've found is in here, but I still can't figure out what is required.

https://www.math.ucdavis.edu/~hunter/book/ch3.pdf

Can any one please suggest how to approach this problem?

• How about solving $x=rx(1+x)$ by hand? – John B May 9 at 21:13
• @JohnB, I would like to show it via contraction mapping – jarhead May 10 at 6:48

So, provided that the logistic function $$f \colon [0,1] \to [0,1]$$ given by $$f(x) = rx(1-x)$$ is a contraction map, you can show it has a unique fixed point.
If $$r < 1$$, then $$f$$ as above is a contraction map (show that $$f'(x) < 1$$ on $$[0,1]$$). Hence for $$r < 1$$, it has a unique fixed point (it'll be $$x = 0$$, when the population dies out).
If $$r \geq 1$$, it is no longer a contraction map on $$[0,1]$$. For example, if $$r = 2$$, then $$f(0) = 0$$ and $$f(1/4) = 3/8$$. So $$|f(0) - f(1/4)| = 3/8 > 1/4 = |0-1/4|$$. Hence it is not a contraction map, and Banach tells us nothing. This shouldn't be too surprising, since for $$r = 2$$ it has two fixed points ($$x = 0$$ and $$x = 1/2$$).
I leave it as an exercise to show in general that it isn't a contraction map for $$r \geq 1$$. So for $$r \geq 1$$, the fixed point theorem doesn't tell you anything; alternate means are required.
• Consider that $rx(1-x) = x \iff rx(1-x)-x = 0$ - now you’re tasked with finding the roots of a quadratic polynomial. How many roots can a quadratic polynomial have and how can you decide whether they are in the interval [0,1]? You can do this with some basic high school algebra. – Chris May 11 at 12:01