I am having difficulty understanding how epsilon is chosen to prove that a dynamical system is attractive and/or stable. I have taken several analysis modules and was okay at proof writing, well now a year later I am doing 2 more and I seem to have lost my ability a bit!
The fact there is delta and epsilon is perhaps what is confusing me, although I have watched some videos on this and have seen it drawn out - ie an epsilon ball with some smaller boundaries, delta, within it and we want to show that $SI_\delta\subset I_\delta$ where $I_\delta$ is a compact invariant set. Sorry if this description doesnt make sense, hopefully it does or is clear what I am trying to say and any better description would be welcomed.
So anyway, here is a question which I am going to use as an example. It is a continuous time dynamical system. I have the solutions so I don't want any hints, I would like a good well explained explanation of how epsilon is found and how we choose whether delta is equal to epsilon, or half epsilon etc. I want to really understand what I am doing, so that I can write great proofs and not lose any marks in my exam.
Consider $\dot x=0$ on $X=\mathbb R$ with $I=\{0\}$
The solution is given by $x(t)=1$
so $S_tx=x$
We have that $I= \{ 0\}$ is compact as it is bounded and closed.
It is easily seen that it is invariant since $S_tB=B$
i.e $S_t0=0$.
Now this is where I begin to get a little stuck.
I want to first show that $I$ is not attractive - it attracts no points in any neighbourhood of I
it is show by contradiction:
Let W be some neighbourhood of $I$ such that there exists $x\in W:x\ne 0$ and $\epsilon=|x|/2:\forall t_0 \geq 0$, there exists $t=t_0$ such that
$dist(S_tx,I)=|x|=2\epsilon$
So why is $\epsilon=|x|/2$? Have they really just "chosen" this or is there a reason it has been chosen to be this?
Also, why is the distance $|x|$?
Now to show stability;
For all $\epsilon>0$, if we let $\delta=\epsilon$,
we have
$|x|<\delta \implies |S_tx|=|x|<|\delta|=\epsilon \implies$ $I$ is stable.
Why have they chosen $\delta = \epsilon$? Sometimes it can be half epsilon or 2 epsilon. I am confused how this is chosen.
Thanks!
theorems I am using
It would be great if they could be used instead of any alternative ones please so that I do not get confused.
Stability
Let $I$ be a compact and invariant set. I is called stable is for all $\epsilon > 0$ there exists a $\delta>0$ such that $S_tI_\delta \subset I_\epsilon \forall t\geq 0$
Attractivity
Let $I, H \subset X$ and let $I$ be compact and invariant.
$I$ attracts points of H/compact sets $B \subset H$ /bounded sets $B\subset H$ if
$dist(S_tB,I)\rightarrow 0$ as $t \rightarrow \infty$
