I am trying to solve a high-dimensional nonlinear system of equations.
I believe the following three dimensional system is representative of the whole system, so I will restrict this question to that system:
$$ (x,y,z)\mapsto H(x,y,z):=(a-x^2, -cy-2bxy-c, -cz-by^2), $$ where $a,b,c>0$. For the three dimensional system, we can easily solve $H=0$ using the nonlinear echelon form. However, I am interested whether there are any deeper structures that would make some sort of numerical iteration converge.
For example, I would like to know if and why a fixed point iteration converges. For this purpose I tried checking whether $H$ is monotone, $$ \langle \begin{pmatrix}-b(x^2-x'^2)\\ -c(y-y')-2b(xy-x'y')\\ -c(z-z')-b(y^2-y'^2)\end{pmatrix},X-X'\rangle \leq -c\|X-X'\|^2,\quad c>0 $$ but didn't succeed.