Thanks but something is wrong.   Line 7 reads [0, 0, 0, 0, 1, 0, 0, 0, 0] but it should be [0, 0, 0, 0, 1, 0, 0, 0, 1]. There is one product of two indeterminates in the fourth element of eq and no summation which means that only one line should be added to the matrix with two ones (since we have a product of two indeterminates).  I guess when there is just a term and the term is a product, the procedure fails.   That seems the case since the number of lines of resulting matrix is 15 instead of 14.  Lines 7 and 8 should be one instead. 

@Kitonum Many many thanks.  1) That seems to be the case and 2) It worked as a sequence that can be easily be converted to a list.  

@Kitonum Many thanks.   Why do I need to use zip in this case?   I have used a similar command in one of my functions, ff:=(l::list)-> eval(f,vars~=l)), that works in Maple 14 without a problem.  (f and vars are inputs to the function).

As for the second option, it does not work when I try f([x,y,z]) and then f(f([x,y,z])).  

@Carl Love Many thanks.  I retyped the equation twice, but I didn't pay attention to the other command line.  

There is a singularity at 26.3 (p[w] tends to zero)  Any suggestions to check if there is a solution beyond that point?   

@acer Many thanks for the explanation.  The main idea was to have a nicer set of plots without using the options under the right click, that is, not a transient change as you have mentioned. 

@Kitonum Many thanks.   Definitely much better, but somehow maple shows a tabulate?  that counts how many times I issue the command (See figure below).  To get rid of it I had to use a colon at the end of the issued command.

@tomleslie Many thanks for the explanation.   

@Carl Love Thank you ever so much. As usual, I will have to go through the whole script to understand what is going on (For instance, from Types to NLMZR). The figures show exactly what I need, that is, no matter what precision one uses the resulting trajectory is very likely to be chaotic except for the case of the exact arithmetic result. In this case, the concept of a transient is dubious. 

@Carl Love Many thanks again.  My bad! My copy and paste wrecked your beautiful code.  I do apologize for that.  I have installed Maple to a new computer and forget to change the settings (you warned me about it years ago).  I will try not to do that again in future posts.

Thanks for the piece of code.  The only part I am not sure if when you calculate aaa.  Why 49? 

When using hf the number of digits is set to 15 and I need to have it changed accordingly (more or fewer periods).  

Why did I choose the fifth fixed point?  The first 7 values can be expressed as radicals when using allvalues.  The last five fixed points out of 7 are the values of the period-5 cycle.  I chose one of them for no particular reason. (I couldn't get Mathematica to show those values).

My biggest concern is that I used evalf to convert the initial condition, that is fp[5], to a float number (but the result could be complex with 0*I for some reason).  My guess was that Maple would use float all the way to the end and, since there are only multiplications of numbers smaller than one, there is no reason to think that Maple would come up with infinity or a complex number.   (My plan is to have only one NestList to do both symbolical and numerical iterations).  

@nm Many thanks.   I have tried unapply but I was definitely missing something. Actually I was adding stuff that shouldn't be there.  

@Carl Love Many thanks again.   

@Carl Love Thank you ever so much.  I had no problem to run them all.  Many thanks again.  

I have encapsulated the second solution as

logistic := y-> 4*y*(1-y);
logistic1:=proc(f,x,n::nonnegint)
local res,p,x0;
x0=x;
res:=seq((x0:=f(x0)),p=1..n);
return [res];
end proc:

When I issued logistic1(logistic,1/4,2), it worked.  Many thanks.

@Carl Love I have no problem to use for. I was wondering if Maple had something similar to NestList (Mathematica).   

@Carl Love Many thanks.  Nice piece of code.  And if I am only interested in having the map symbolically iterated (I am avoiding using floats).  

@vv Many thanks but I am already using (f@@p) to find the fixed points.   What I need is to have the system iterated as in the initial post. More than that, I need to have it iterated backwards (two solutions each time).