This seems to not converge, no output after a day. Is there another way to do it, that will give output? It’s just a system of recursion equations, with boundary conditions at 1 and N (here N=3).
[1]: Wolfram Cloud Document
I also tried Reduce, it still thrashes
I would start by giving numerical values to the coefficients and see what happens.
This may or may not help you. If you try
FullSimplify[Collect[Eliminate[{
x[1]==f11[x[2],y[2]],
y[1]==f12[x[2],y[2]],
x[2]==fx[x[1],y[1],x[3],y[3]],
y[2]==fy[x[1],y[1],x[3],y[3]],
x[3]==f31[x[2],y[2],0],
y[3]==f32[x[2],y[2],0]},
{x[2],y[2],x[3],y[3]}],
{x[1],y[1]}]]
then it will quickly show you
expr1*x[1]+expr2*y[1]==0 &&
expr3*x[1]+expr4*y[1]==0 &&
expr5*x[1]+expr6*y[1]==0
Each of those six new expressions are large, but this has removed x[2],y[2],x[3],y[3] and perhaps you can see something in those expressions that will tell you something about the solution you are looking for.
OK, great, thanks!. That means either x[1]=y[1]=0 if the expressions are not equal, or if they are, an infinite number of solutions. The real problem is with xE .ne. 0, and using your idea I get something useful for some relations between the coefficients that give some unique solutions. Will try for N>3.
Solutions are such that the coefficients are not independent of each other.
I am trying a similar system, but.a bit different. Why is NSolve not able to substitute numerical values into the functions that are being minimized? I also tried FindRoot, with same issue.