What should $\mu$ be here in order for the logistic map to be stable?

Question

I think this might be an error in A Survey of Computational Physics: Introductory Computational Science by Landau. On page 292, he mentions that in order for the logistic map to be stable, we must have

$$\left | \frac{df}{dx} \right |_{x_*}<1$$

The logistic map is given by

$$x_{i+1}=\mu x_i(1-x_i)$$

For the fixed point $x_*=\frac{\mu-1}{\mu}$, he claims that the map is stable if $\mu<3$. But shouldn't he have written that the map is stable if $1<\mu < 3$? My reasoning was that

$$\left | 2-\mu \right | <1$$ $$\to 2-\mu <1 \quad ; \quad -2+\mu < 1$$ $$\to \mu>1 \quad ; \quad \mu <3$$

So why didn't he write $\mu >1$ as a second condition for the map's stability?

Answer 1

You're right, but.... It's usual to consider the logistic map as a map on $[0,1]$. I don't know whether the author makes this restriction explicit in the text. With that restriction, $x=(\mu-1)/\mu$ is not in the domain for $\mu<1$, so it doesn't count as a fixed point. So as far as fixed points given by that formula go, we're only considering $\mu\ge1$ anyway, and in that range we have stability for $\mu<3$.