@mehwish Yes, I figured out what you meant. My Answer below already covers the case in your example. I was simply thrown off by the word "iteration".

@janhardo You can safely just ignore what I said about complex cases. I just felt a need to say it for the sake of completeness. The complex numbers are always "lurking" in the background when you're dealing with power series and analytic functions.

@mehwish Okay, I'm understanding better now. By Q², for example, you mean Q(x)*Q(x) (2nd power), not Q(Q(x)) (2nd iteration), right? I was thrown off by your initial use of the word "iteration".

Are you the author of the PDF that you attached?

@Rouben Rostamian  While I agree that it's a good idea to declare the index variable local, it doesn't seem to help in this case, as my procedure example shows. You can't declare an indexed variable local; you can only declare its "parent" or "stem" name.

Nor does this help to explain the anomaly, because it's manifested exactly the same way whether the index is global or local.

I don't think that anything is possible along those lines if you don't supply an explicit formula for Q.

@Carl Love There's a good chance that the Answer above is wrong due to the discontinuity issue pointed out by Kitonum. I also noticed the discontinuities and the infinity returned from some numeric integrals, but I ignored these because supericial plotting indicated that the discontinuities were removable. Perhaps that was foolish.

@Mohamed Abohamer What is your Maple version, such as Maple 2019, Maple 18, etc.?

@Mohamed Abohamer Thanks for the reformatting. I am working on it currently.

@mmcdara I am suggesting that you use AddBaseUnit to create units with arbitrary names such as Time, Length, Mass, etc., with corresponding dimensions time, length, mass, etc. Again, I have no experience with this, either mathematically or in Maple. But if you try it and run into trouble, I'm willing to help, as always.

@janhardo In this case, the series doesn't converge for all x, so you need to include an assuming clause:

S:= convert(ln(1+x), FPS);
value(S) assuming -1 < x, x <= 1;
             ln(1+x)

The interval of convergence of any power series is easily determined from the series itself. It is often smaller than the set of values that work in the corresponding function. In the real-number case, it is always an interval (possibly degenerate), never a more complicated set. In the complex-number case, it is always a circular disk (possibly degenerate), possibly including some or all of the disk's boundary.

@Gerd But if what you really want is the orbit, then the Orbit procedure is more efficient. It stops as soon as it detects the orbit rather than at an arbitrary spot like your 10. Also, it avoids raising the matrix to powers.

The first step is to use * instead of . for multiplication.

If you make those changes, I'll work further on your problem.

@Gerd Of course, the newer procedure cannot work in your Maple 18.

@Gerd  Just to give you a taste of the syntax updates over the years, the Orbit procedure could be shortened in Maple 2019 to

Orbit:= proc(L::Vector, M::Matrix, m::posint)
uses LA= LinearAlgebra, LAM= LinearAlgebra:-Modular;
local Mm:= LAM:-Mod(m, M, integer), L0:= LAM:-Mod(m, L, integer), L1:= LAM:-Copy(m,L0);
    [do L1 until LA:-Equal((L1:= LAM:-Multiply(m, Mm, L1)), L0)]   
end proc
:

Though the Maple 2018 syntax that I posted previously also continues to work.