(This is part of exercise 10.7.8 in Strogatz's Nonlinear Dynamics and Chaos that I am struggling to solve)
QUESTION:
Consider the map $x_{n+1} = f(x_n,r)$ where $f(x_n,r)=-r+x-x^2$. We use the notation $f^2(x,r)=f(f(x,r))$. Plot $f(x,r)$ and $f^2(x,r)$ and observe that they look like scaled versions of each other for $x$ small and $r$ small. Let us show this more concretely.
Expand $f^2(x,r)$ around $x=0$ keeping terms through $\mathcal{O}(x^2)$. Now rescale $x$ and $r$ to put this new map into the normal form $F(X,R) \approx -R + X + X^2$.
MY ATTEMPTS:
Taking the Taylor expansion of $f^2(x,r)$, we get
$f^2(x,r) = -r(r+2) + (2r+1)x -2(r+1)x^2 + \mathcal{O}(x^3)$.
Let's cut off the higher-order terms and equal it to the normal form
$ T(x,r) := -r(r+2) + (2r+1)x -2(r+1)x^2 = -R + X - X^2. $
Now, as we are asked to scale $x$ and $r$, we assume $X = \alpha x$ and $R=\beta r$. But from here, I am not able to find $\alpha$ or $\beta$. I thought maybe I could ignore second-order terms of $r$ but this is also a dead-end.