Fixed points and dynamics of $x-x^2$, $x+x^3$, $x-x^3$ and $\tan x$ For each of the following functions, show that $0$ is a fixed point, and the derivative of the function is $1$ at $x=0$. Describe the dynamics of points near $0$. Is $0$ attracting, repelling, semi-attracting, or none of these? (a) $f(x)=x-x^2$ (b) $g(x)=x+x^3$ (c) $h(x)=x-x^3$ (d) $k(x)=\tan x$ I can easily show that $0$ is a fixed point as well as the derivative of the function is $1$ at $x=0$. This part of the problem is fairly trivial. But I'm not sure what the dynamics are or how to describe them or the last portion about the attracting, repelling, and so forth. Any help here?

To do this you want to look at what happens to points a little to the right, and a little to the left of $0$. So let $\epsilon >0$ be given and consider

(a) $f(\epsilon)=\epsilon-\epsilon^2<\epsilon$ so the points a little to the right are being drawn _towards_ $0$. Similarly a point $-\epsilon$ (slightly to the left of $0$) gets mapped to $f(-\epsilon)-\epsilon -\epsilon^2<-\epsilon$ are being forced _away_ from $0$, so it does half attracting, half repelling.

(b) We do the same procedure getting $f(\epsilon)=\epsilon+\epsilon^3>\epsilon$ and $f(-\epsilon) = -\epsilon -\epsilon^3<-\epsilon$ both push away from $0$ which makes it a repelling fixed point.

You should try (c) and (d) for yourself using an appropriate polynomial approximation for $\tan x$ in (d). $\tan x\approx x+{x^3\over 3}$ is good enough, for example.