@Carl Love Thank you for introducing the new commands. I can see the deeper side of Maple with them. 

@acer : Would you tell me where to find the information regarding how Maple traverses a given expression?

@Kitonum Thank you for the succinct answer. 

@nm The answer worked! Thank you. 

@acer Thank you for showing the richness of the Maple expressions. 

@Joe Riel : I did not realized that the system is stiff. The answer made the plot smooth, more reasonable. 

@acer : Your attention to the question is highly appreciated. 

@Kitonum : Your answer looks clear. Thank you for your attention to the question!

@acer : Your continued attention is appreciated. 

@Preben Alsholm : Thank you for your comment. Perhaps these points need to come under consideration:

1. It is obvious that my question is beyond the what Maple would concern. I made it clear in my question. 

2. I'm not sure the stability checking by eigenanalysis is pertaining to the set of equations in my question, because the set might not be a dynamic system. Also, I made it clear in my question. 

3. Your example shows the concept of SEP (Stable Equilibrium Point) and UEP (Unstable Equilibrium Point). By changing the sign, simply speaking, the maximum becomes minimum and the minimum becomes maximum. Thus a SEP flips into UEP and vice versa. What I wanted to check is the validity of the answers coming from the 'solve()' or 'fsolve()' commands, by using the eigenanalysis. Usually, a SEP is a viable solution in a real physical system. 

@Kitonum : Thank you for the answer! It is appreciated. 

@Acer: then I know this question is well beyond the extent of what Maple is concerned, but a usual way to tell a solution is stable is by looking at the eigenvalues of the Jacobian. 'Stable' solution must produce all negative real parts in the resulting eigenvalues. However, if the eigenvalues are calculated from the solutions, the results indicate that none of the solutions are stable. Or perhaps I might be wrong in the point that the stability criterion is applied to a wrong place (it is usually applied for a dynamic system, meaning a set of differential equations). If you don't mind, would you teach me a way to tell whether a solution is stable or not? Here is the worksheet where the eigenvalues of Jacobians are calculated. 
 

Download Jacobian.mw

@acer : Thank you for your further delving into the questions. Regarding the 'f9' and 'f10', it was my mistake when I manually copied the equations lines from another Maple worksheet. Here is where those lines are from:
 

Download AC_20200812.mw

@acer : Thank you for your two answers. Then, would you allow me to ask these following-up questions?

1) The dimension of the system is 10 (if we consider the real part and imaginary part separately) or 5 (if we consider the variables as complex numbers). As a result, the total number of solutions would be 10 or 5, but the worksheet presented only one. Is there any way to solicit more solutions (e.g., by setting initial values)?

2) Your complex numbers were decomposed to real/imaginary parts (by using your commands, 'K:=map(var->var=cat(re,var)+I*cat(im,var),variables):evalc(eval(map(eq->[Re(eq),Im(eq)][],{f1n2, f3n4, f5n6, f7n8, f9n10}),K)):'), then the 'solve()' command was attempted again. Unfortunately, it failed in a way similar to the one in my first question. Is there any way to make it work? Here is the worksheet where you can see the failure:

Download Q_ac_20200811.mw

@Preben Alsholm : Thank you for your further steps. Here is the worksheet where the 'expand()' command enabled the 'solve()' command. 
 

Download conjugate_sol_PrebenAlsholm.mw