Let $f$ be $C^1([0,1])$ function. I'm looking for a good reference for the asymptotic behavior of $$a_{n+1}=a_n + f(a_n)$$ and it's relation to dynamical systems.
Asked
Active
Viewed 94 times
3

3It doesn't seem apparent to me that you even can iterate this in general. When you add $a_n$ and $f(a_n)$ you may very well leave $[0,1]$. – Nate Sep 27 '19 at 16:36

Very similar question [here](https://math.stackexchange.com/questions/176858/asymptoticbehaviorofiterativesequences?rq=1). – BAYMAX Sep 29 '19 at 09:10
2 Answers
2
As a first cut, I would write $a_{n+1}a_n \approx a'(x)$ so my guess would be $a'(x) \approx f(a(x))$.
marty cohen
 101,285
 9
 66
 160
2
Clearly, if $a_n \to L$, then $f(L)=0$.
Convergence will be guaranteed if $g:x \mapsto x+f(x)$ is a contraction, as long as $g([0,1]) \subseteq [0,1]$:
$g'(x)<1$ iff $2 < f(x) < 0$.
$g([0,1]) \subseteq [0,1]$ iff $x \le f(x) \le 1x$.
lhf
 208,399
 15
 224
 525