Attractors and stability.

Question

Does anyone who studied chaos get this?

I get everything preceding the red box, but I don't understand how it makes x* an attractor.

The notes we have don't explain the theory very well, nor wikipedia.

Thank you.

For anyone who can't read the image, he's my latex attempt:

Let $f(x^*)=x^*$ and $f$ be smooth at $x^*$

Let $x_t=x^*+\delta_t: |\delta_t|<<1$

Expand $f$ in Taylor series at $x^*$

$x_{t+1}=f(x_t)=f(x^*+\delta_t)=f(x^*)+f'(x^*)\delta_t+\frac{1}{2}f''(x^*)\delta_t^2+O(\delta_t^3$

$\delta_{t+1}+x^*=x^*+f'(x^*)\delta_t+\frac{1}{2}f''(x^*)\delta_t^2+O(\delta_t^3)$

This gives

$\delta_{t+1}=f'(x^*)\delta_t+O(\delta_t^2)$

and so $x^*$ is an attractor if $|f'(x^*)|<1$

Answer 1

Nevermind, I get it now.

Taking any point within distance $\delta$ of $x^*$, it is shown that this distance $\delta$ grows smaller with time if $|f'(x^*)|<1$. This shows that $x^*$ attracts nearby points and is therefore an attractor.

Answer 2

I think the idea is that as long as $|f^\prime(x^*)|<1$, then $|\delta_{t+1}|$<$|\delta_t|$. This means that with each step, your perturbation is smaller than it was in the previous step. Therefore your solution is always approaching $x^*$, and so $x^*$ is an attractor. (Caveat: I'm really just an interested amateur in chaos theory.)