regular sexp:different fixpoints

[sheldonison]

When I was reading Dmitrie's & Henryk's (newest(?)) paper on superfunctions I tried to get an own impression about the differences of tetration when regular iteration is applied with different fixpoints. (see picture 4 at page 22)
....
There are two aspects which make me headscratching.
(1) I could naively easier accept, if one of the functions proceeds faster and the other one slower; maybe with some modification, for instance a turning point at the center or something like that. But we have permanently changing signs - contradicting the assumtion of a somehow smoothely increasing function. But ok, the behave of the difference can be caused by one of the involved, say by the high (repelling)-fixpoint-version tet4.
(2) But this seems also not to hold. If I assume that at least the tet2-function is smoothely increasing, then a first guess may be, that all differences of all orders should have monotonuous behave. But that's also not true: looking at differences of high order (>24) we find sinusoidal behave in the magnitude of <1e-24. Consequence: very likely also the tet2-function, although based on the attracting fixpoint, has a sinusoidal component in that interval 2<x<4.

For me, the entire "bummer" post (same topic), and this paper is really interesting. My personal opinion is on the left side, the "smoothest" superexp is developed from the fixed point of four, and on the right side the "smoothest" superexp is developed from the fixed point of two. In between, they're both a little bit "less smooth".