@Andiguys No, the plot command should not be inside the loop.

In the original double loop, you had s[Cv, tau0]:= Maximize(...). You changed that to s:= Maximize(...). If you change it back to s[Cv, tau0], then you will get the plot.

@charlie_fcl Like this:

(e,f):= ifelse(c<d, [a, 2*b], [3*a, b])[]

You could replace ifelse with piecewise (leaving everything else exactly as is) and get the same result, but IMO ifelse is better.

@charlie_fcl In Maple it can be done as follows, so it's likely to also work in Maple Flow:

(a,b):= (min,max)(c,d)

@Carl Love Okay, I couldn't resist the temptation to make the applyrule work, and I found something that works, although I can't explain why you were getting an infinite loop before.

applyrule(
    x::(anything^And(fraction, 2 &under denom))*y::(Not(1) &under denom)
    = y %* x,
    eq
);

@Carl Love I just added a small change to the code above. I replaced denom(R)=1 with S=1 or denom(R)=1. In other words, if there are no coefficients with denominators OR there are no square roots, then leave the product unchanged.

@SARBAST There is no attached file. Please try uploading it again.

@SARBAST I assume that you want to vary the initial values (x(0), y(0))  over [-6,6] x [-3,6]. We still need initial values z(0) and w(0) in order to do any computation.

@C_R The Wikipedia article "Attractor" contains a definition of basin of attraction. It's the mathematical generalization of the geological / geographical / hydrological concept of drainage basin or watershed.

@C_R I haven't run this yet, but in order to get such a slow and smooth animation to post, the OP very likely included a huge number of extra frames (10 times, or more) in the animation. These extra frames are not needed when viewing in a worksheet because you can slow down the animation with the toolbar controls. Not computing the extra frames should speed up the computation significantly.

@dharr As far as I can tell, the worksheet poly.mw attached to your Answer is the same as when it was originally attached. If this is not true, please include a comment in the worksheet to indicate the new part.

@Andiguys This is to diagnose the problem, not to fix it: Please insert this command after the double loop:

lprint(indices(s, pairs, indexorder));

Then execute the entire worksheet. Then post the entire executed worksheet.

@Andiguys s[1] isn't an equation, so it doesn't have an rhs. It's just a number. So, remove the first rhs.

@dharr p = sinnx (identically, not approximately) only for x in -2..2, because of the real domain of arcsin(x/2). I don't know why the OP is trying to extend the domain to -2.1..2.1.

simplify ignores the domain restriction, just as it does with sin(2*arcsin(x)). Indeed, sin(arcsin(x)) automatically simplifies to x.

@Andiguys Here is one way to get that 3D plot. Do this after the double loop. I needed to remove the higher values of Cv because Maximize rejected them.

CV:= [50, 55, 60, 65, 70]:
T0:= [0.3, 0.4, 0.5, 0.55, 0.6, 0.65, 0.7]:
S:= Matrix((nops(CV),nops(T0)), (i,j)-> s[CV[i],T0[j]][1]):
A:= (Cv,t0)-> CurveFitting:-ArrayInterpolation(
    [CV, T0], S, [[Cv], [t0]], method= spline
)[1,1]:
plot3d(A, (min..max)(CV), (min..max)(T0), labels= ['Cv', '`&tau;0`', 'TRC']);

This isn't the most efficient way to use ArrayInterpolation, but the syntax is already daunting, and it's sufficiently efficient for this purpose.
 

@Andiguys I didn't mean to put do at the end of the whole loop; I meant to put it and the end of the the line with the for:

for i from 1 to 10 do 
    i^2
od

od can be replaced by end or end do (they mean exactly the same thing), but not by do. The do on the first line cannot be replaced by anything.