Given a (semi)dynamical system from $[0,1]$ to itself that behaves very simply (it either oscillates between $x_0$ and $1-x_0$ or stays fixed at a half), $x_{n+1}=1-x_n$, and perturbing it with small perturbations (by small I mean $\sum_n |\varepsilon_{n+1}|<\infty$, obviously non-zero infinitely often) yielding $x_{n+1}=1-x_n+\varepsilon_{n+1}$ (assume a further constraint on the perturbations that keeps it from $[0,1]$ to itself), are there techniques to predict how the perturbed one will behave as $n$ grows, given our knowledge of how the limiting system behaves? My guess here is that the perturbed non-autonomous system can only oscillate as $n$ grows to infinity, no matter what $x_0$ is, having performed some simulations, but it could also be that the set of initial points for which it tends to a half is very small, e.g. a null measure set. I can prove that the omega limit set of the perturbed system is invariant w.r.t. the limiting map, but this is still very weak information about the behaviour of the non-autonomous orbits, and definitely does not explain why convergence to a half never shows up in simulations.
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Check out what happens if $\varepsilon_1 = 1,000,000$ and $\varepsilon_j=0$ otherwise. – amsmath Oct 22 '21 at 14:08
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It goes without saying that the perturbations are non-zero infinitely often. – xyz Oct 22 '21 at 16:03