This site is about using Maple to solve problems so first question: what exactly is Maple doing incorrectly?
If I go through your previous response more-or-less line by line, then you have to ask yourself some pretty basic questions, which I have itemised in the following
The general solver -LinearSolve() confirms the existence of a solution.
Incorrect. statement. LinearAlgebra:-LinearSolve demonstrate that there are an infinite number of solutions
The -IterativeFormula() suggests that this solution can be reached using one of the standard iterative methods (such Gauss-Seidel, Jacobi, etc).
Incorrect statement. Since there are an inifinite number of solutions, all the IterativeFormula() command does is find one of them
I guess, the fact that these methods can deal with the case of spectral radius of M being 1 is not so surprizing, since the singular matrix (I-M) from equation (5) of my write-up can be inverted using the Moore-Penrose pseudoinverse and still generate an (up-to-scale) unique solution.
Overcomplicating the issue: For a numerical process to generate one (of an infinite number of) solution(s), all it has to do is to assign an arbitrary value to one of the variables, throw away one of the redundant equations, and solve the remaining three independent (consistent) linear equations in three unknowns. Not exactly difficult
Presumably, however, the -IterativeFormula() applies a different type of iteration then the one you've coded in Example1.mw and Example2.mw above (where we start with an initial, model-consistent guess for the unknowns (of 0's) on the right hand side (RHS), calculate the left-hand side (LHS), plug those values again on the RHS, etc.), for which we have a convergence in Ex1 and a divergence in Ex2. (Presumably, the -IterativeFormula() plugs in the LHS values with some sort of 'damping' factor on the RHS to ensure the convergence to the existent solution. However, the existence of a solution that can be reached iteratively, e.g., using -IterativeFormula(), does not yet imply the stability of the equilibrium, right?).
Complete misrepresentation: the problem you posted in your very first worksheet was for a system of non-linear equations - lots of products of y*y etc. I was doubtful about the success of an iterative process in solving this non-linear system, which is why my original answer was headlined "use with care". At some point in the evolution of this thread, you morphed from a system of non-linear equations to a system of linear equations - I have no idea how/why this occurred, but if your original post had specified a system of linear equations, then no way would I have proposed the iterative solution in my original response. For systems of linear equations, there are several "better" methods of constructing an iterative solution - see for example the IterativeFormula() command
Hence, I wonder whether the fact of a divergence in Example2.mw (via a "mechanical" updating of the RHS using LHS) is some sort of indication that the equilibrium in Example2 is unstable and, if so, how can I formally tackle this question?
More-or-less meaningless: As stated above, I have little confidence that the iterative approach in my original response would work for non-linear system. For linear systems, far better iteration processes are avaliable. (See the documentation for the IterativeFormula() command.) I also have no idea what you mean by terms such as "equilbrium" or "unstable" in this context. All you have is two linear systems of four non-independent equations, each of which has an infinite number of solutions