Global convergence with Newton method for nonlinear systems

Question

Let $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ a function for which we seek a fixed point, $$f:(x,y)\mapsto\begin{pmatrix} -5x+2\sin x+2\cos y\\ -5y+2\sin y+2\cos x \end{pmatrix}. $$ I made a python program to solve this with the Newton's method $$X_{k+1}=X_k-(J_{X_k}f)^{-1}f(X_k)$$ and it worked with a random $x_0$ (so there maybe is global convergence). How can I prove that in this case the Newton's method is globally convergent (or not) ?

Answer 1

So, thanks to Cesarea I was able to visualize the answer to my question (to calculate $||\phi ||$ was not really interesting by hand), and there is indeed a global convergence.